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Micron Document

<!DOCTYPE html>

<title>Formelsammlung Trigonometrie</title>

</head>

<h1 id="firstHeading" class="firstHeading mw-first-heading">Formelsammlung Trigonometrie</h1>

</header>

</div>

<div class="mw-heading mw-heading2"><h2 id="Dreieckberechnung">Dreieckberechnung</h2></div>

Die folgende Liste enthält die meisten bekannten Formeln aus der Trigonometrie in der Ebene. Die meisten dieser Beziehungen verwenden <a href="Trigonometrische_Funktion" title="Trigonometrische Funktion">trigonometrische Funktionen</a>.

Dabei werden die folgenden Bezeichnungen verwendet: Das <a href="Dreieck" title="Dreieck">Dreieck</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ABC}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle ABC}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5e55b44cfd965fbdc7a328d5db8a35a619db0971.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.273ex; height:2.176ex;" alt="{\displaystyle ABC}" loading="lazy"> habe die Seiten <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=BC}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a=BC}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/944ad306da7031c91d6293d105e1fcc6fdb87b47.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.859ex; height:2.176ex;" alt="{\displaystyle a=BC}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b=CA}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b=CA}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5914a6e34b4d6a947cf7bd1e1234018329899ceb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.605ex; height:2.176ex;" alt="{\displaystyle b=CA}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c=AB}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c=AB}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/aad098a7276298d55c8196858f052aa1a0fa5da7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.612ex; height:2.176ex;" alt="{\displaystyle c=AB}" loading="lazy">, die <a href="Winkel" title="Winkel">Winkel</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b79333175c8b3f0840bfb4ec41b8072c83ea88d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \beta }</annotation>

</semantics>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \gamma }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a223c880b0ce3da8f64ee33c4f0010beee400b1a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }" loading="lazy"> bei den <a href="Ecke" title="Ecke">Ecken</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/47136aad860d145f75f3eed3022df827cee94d7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4fc55753007cd3c18576f7933f6f089196732029.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" loading="lazy">. Ferner seien <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0d1ecb613aa2984f0576f70f86650b7c2a132538.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}" loading="lazy"> der <a href="Umkreis" title="Umkreis">Umkreisradius</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \rho }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1f7d439671d1289b6a816e6af7a304be40608d64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }" loading="lazy"> der <a href="Inkreis" title="Inkreis">Inkreisradius</a> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho _{a}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \rho _{a}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cf5bcd5f18e417072a2f15baf57774af83e76672.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.304ex; height:2.176ex;" alt="{\displaystyle \rho _{a}}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho _{b}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \rho _{b}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e71b4c1f19dae23cb553215f66d17cdbe4853dca.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.14ex; height:2.176ex;" alt="{\displaystyle \rho _{b}}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho _{c}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \rho _{c}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/523e8c709cfeaef7ed497ce5be80144fdf2e9b70.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.146ex; height:2.176ex;" alt="{\displaystyle \rho _{c}}" loading="lazy"> die <a href="Ankreis" title="Ankreis">Ankreisradien</a> (und zwar die Radien der Ankreise, die den Ecken <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

</semantics>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/47136aad860d145f75f3eed3022df827cee94d7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" loading="lazy"> bzw. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4fc55753007cd3c18576f7933f6f089196732029.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" loading="lazy"> gegenüberliegen) des Dreiecks <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ABC}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle ABC}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5e55b44cfd965fbdc7a328d5db8a35a619db0971.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.273ex; height:2.176ex;" alt="{\displaystyle ABC}" loading="lazy">. Die Variable <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/01d131dfd7673938b947072a13a9744fe997e632.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}" loading="lazy"> steht für den halben Umfang des Dreiecks <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ABC}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle ABC}</annotation>

</semantics>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s={\frac {a+b+c}{2}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/787a98dac5681f383514fc1bd5b4d8e561a3fd21.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.94ex; height:5.343ex;" alt="{\displaystyle s={\frac {a+b+c}{2}}}" loading="lazy">.</dd></dl>

Schließlich wird die Fläche des Dreiecks <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ABC}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle ABC}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5e55b44cfd965fbdc7a328d5db8a35a619db0971.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.273ex; height:2.176ex;" alt="{\displaystyle ABC}" loading="lazy"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/545fd099af8541605f7ee55f08225526be88ce57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" loading="lazy"> bezeichnet. Alle anderen Bezeichnungen werden jeweils in den entsprechenden Abschnitten, in denen sie vorkommen, erläutert.

Es ist zu beachten, dass hier die Bezeichnungen für den Umkreisradius <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0d1ecb613aa2984f0576f70f86650b7c2a132538.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}" loading="lazy">, den Inkreisradius <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \rho }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1f7d439671d1289b6a816e6af7a304be40608d64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }" loading="lazy"> und die drei Ankreisradien <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho _{a}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \rho _{a}}</annotation>

</semantics>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \rho _{b}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e71b4c1f19dae23cb553215f66d17cdbe4853dca.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.14ex; height:2.176ex;" alt="{\displaystyle \rho _{b}}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho _{c}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \rho _{c}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/523e8c709cfeaef7ed497ce5be80144fdf2e9b70.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.146ex; height:2.176ex;" alt="{\displaystyle \rho _{c}}" loading="lazy"> benutzt werden. Oft werden davon abweichend für dieselben Größen auch die Bezeichnungen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4b0bfb3769bf24d80e15374dc37b0441e2616e33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0d1ecb613aa2984f0576f70f86650b7c2a132538.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{a}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r_{a}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/da5f0781a46ba20129a554237056b6ade78b956f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.151ex; height:2.009ex;" alt="{\displaystyle r_{a}}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{b}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r_{b}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8c22eac9d11ab84da51b891d29ee2e5de75eab1a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.986ex; height:2.009ex;" alt="{\displaystyle r_{b}}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{c}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r_{c}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5e41dc9dcda9fd0c224d94555361f02b73aacd2e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.993ex; height:2.009ex;" alt="{\displaystyle r_{c}}" loading="lazy"> verwendet.

<div class="mw-heading mw-heading3"><h3 id="Winkelsumme">Winkelsumme</h3></div>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha +\beta +\gamma =180^{\circ }}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/720f59b296cd42f5d2968b211dd56394a860c97a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.403ex; height:2.843ex;" alt="{\displaystyle \alpha +\beta +\gamma =180^{\circ }}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Sinussatz"><a href="Sinussatz" title="Sinussatz">Sinussatz</a></h3></div>

Formel 1:

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {a}{\sin \alpha }}={\frac {b}{\sin \beta }}={\frac {c}{\sin \gamma }}=2r={\frac {abc}{2F}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8b9f3e7bc934d0ede764bf31b2e1a0d4ec288e64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:34.994ex; height:5.843ex;" alt="{\displaystyle {\frac {a}{\sin \alpha }}={\frac {b}{\sin \beta }}={\frac {c}{\sin \gamma }}=2r={\frac {abc}{2F}}}" loading="lazy"></dd></dl>

Formel 2:

wenn <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha =90^{\circ }}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha =90^{\circ }}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/30f0fb8d6c2247b522d8bcb4a62b5c0a9f9ee7e7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.965ex; height:2.343ex;" alt="{\displaystyle \alpha =90^{\circ }}" loading="lazy">

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sin \beta ={\frac {b}{a}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/59d3b6233780d0dfb64cb62df635f935da9b7ebc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:9.739ex; height:5.343ex;" alt="{\displaystyle \sin \beta ={\frac {b}{a}}}" loading="lazy"></dd></dl>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sin \gamma ={\frac {c}{a}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8a0930aabd4b02f54f5a9d4bb1b7d72b077797a2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:9.669ex; height:4.676ex;" alt="{\displaystyle \sin \gamma ={\frac {c}{a}}}" loading="lazy"></dd></dl>

wenn <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta =90^{\circ }}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \beta =90^{\circ }}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a620541c07a5639eb98fb09c23d5de0dbb6c2209.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.81ex; height:2.676ex;" alt="{\displaystyle \beta =90^{\circ }}" loading="lazy">

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sin \alpha ={\frac {a}{b}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d1747314f67cce94eadaa090d0dc863b12522c21.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:9.895ex; height:4.843ex;" alt="{\displaystyle \sin \alpha ={\frac {a}{b}}}" loading="lazy"></dd></dl>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sin \gamma ={\frac {c}{b}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e1eb012ae1c2c877db8b2dfae670d32d924cf84b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:9.446ex; height:4.843ex;" alt="{\displaystyle \sin \gamma ={\frac {c}{b}}}" loading="lazy"></dd></dl>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \gamma =90^{\circ }}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/08ed45556d3a8174ad5d5a4b3f9e0451024aba9c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.74ex; height:2.843ex;" alt="{\displaystyle \gamma =90^{\circ }}" loading="lazy">

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sin \alpha ={\frac {a}{c}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/30a3bd270353e0f19f4f16054ac368730cbdc4d6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:9.895ex; height:4.676ex;" alt="{\displaystyle \sin \alpha ={\frac {a}{c}}}" loading="lazy"></dd></dl>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sin \beta ={\frac {b}{c}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/efbb10656691c5f28b422df92501cad81c2b45f1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:9.516ex; height:5.343ex;" alt="{\displaystyle \sin \beta ={\frac {b}{c}}}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Kosinussatz"><a href="Kosinussatz" title="Kosinussatz">Kosinussatz</a></h3></div>

Formel 1:

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a^{2}=b^{2}+c^{2}-2bc\ \cos \alpha }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e45e1928a8a7a05f206c821b17cdd92a37a253fb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:24.297ex; height:2.843ex;" alt="{\displaystyle a^{2}=b^{2}+c^{2}-2bc\ \cos \alpha }" loading="lazy"></dd></dl>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b^{2}=c^{2}+a^{2}-2ca\ \cos \beta }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1fb073ee8b8aa66a405deba589315d95f457d700.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:24.373ex; height:3.009ex;" alt="{\displaystyle b^{2}=c^{2}+a^{2}-2ca\ \cos \beta }" loading="lazy"></dd></dl>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c^{2}=a^{2}+b^{2}-2ab\ \cos \gamma }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/23935d3526342e0d4919166fbe9d703fb1251d0f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.294ex; height:3.176ex;" alt="{\displaystyle c^{2}=a^{2}+b^{2}-2ab\ \cos \gamma }" loading="lazy"></dd></dl>

Formel 2:

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha =90^{\circ }}</annotation>

</semantics>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \cos \beta ={\frac {c}{a}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/750e9a7e7b661c67cd5ea7bdbd96487bfa70748b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:9.995ex; height:4.676ex;" alt="{\displaystyle \cos \beta ={\frac {c}{a}}}" loading="lazy"></dd></dl>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \cos \gamma ={\frac {b}{a}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4c96eccc19f8aa6d899f0efdfd0fb0b8ad56d67f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:9.925ex; height:5.343ex;" alt="{\displaystyle \cos \gamma ={\frac {b}{a}}}" loading="lazy"></dd></dl>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \beta =90^{\circ }}</annotation>

</semantics>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \cos \alpha ={\frac {c}{b}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/01db57dd79cfdb8e2c6a389d171225ba0dc1d196.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:9.927ex; height:4.843ex;" alt="{\displaystyle \cos \alpha ={\frac {c}{b}}}" loading="lazy"></dd></dl>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \cos \gamma ={\frac {a}{b}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f1ae8c94d0b91a6bdf7ef0542828bcd91b5636df.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:9.925ex; height:4.843ex;" alt="{\displaystyle \cos \gamma ={\frac {a}{b}}}" loading="lazy"></dd></dl>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \gamma =90^{\circ }}</annotation>

</semantics>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a^{2}+b^{2}=c^{2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7ef0a5a4b8ab98870ae5d6d7c7b4dfe3fb6612e2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:12.336ex; height:2.843ex;" alt="{\displaystyle a^{2}+b^{2}=c^{2}}" loading="lazy"> (<a href="Satz_des_Pythagoras" title="Satz des Pythagoras">Satz des Pythagoras</a>)</dd></dl>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \cos \alpha ={\frac {b}{c}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/42ab6e3e280c970df88e4c7764510cda3fa4082e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:9.927ex; height:5.343ex;" alt="{\displaystyle \cos \alpha ={\frac {b}{c}}}" loading="lazy"></dd></dl>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \cos \beta ={\frac {a}{c}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5daa42754216029f8181ebee80e49c6bd8c30c4d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:9.995ex; height:4.676ex;" alt="{\displaystyle \cos \beta ={\frac {a}{c}}}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Projektionssatz">Projektionssatz</h3></div>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}a&=b\,\cos \gamma +c\,\cos \beta \\b&=c\,\cos \alpha +a\,\cos \gamma \\c&=a\,\cos \beta +b\,\cos \alpha \end{aligned}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f2e112c3038e976d2075857e9f62cf73e7b1527f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:21.512ex; height:8.843ex;" alt="{\displaystyle {\begin{aligned}a&=b\,\cos \gamma +c\,\cos \beta \\b&=c\,\cos \alpha +a\,\cos \gamma \\c&=a\,\cos \beta +b\,\cos \alpha \end{aligned}}}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Die_Mollweideschen_Formeln">Die <a href="Mollweidesche_Formeln" title="Mollweidesche Formeln">Mollweideschen Formeln</a></h3></div>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {b+c}{a}}={\frac {\cos {\frac {\beta -\gamma }{2}}}{\sin {\frac {\alpha }{2}}}},\quad {\frac {c+a}{b}}={\frac {\cos {\frac {\gamma -\alpha }{2}}}{\sin {\frac {\beta }{2}}}},\quad {\frac {a+b}{c}}={\frac {\cos {\frac {\alpha -\beta }{2}}}{\sin {\frac {\gamma }{2}}}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c12cfe54ce4f4742bfc388c37b8b5c4a3072633d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:58.626ex; height:8.843ex;" alt="{\displaystyle {\frac {b+c}{a}}={\frac {\cos {\frac {\beta -\gamma }{2}}}{\sin {\frac {\alpha }{2}}}},\quad {\frac {c+a}{b}}={\frac {\cos {\frac {\gamma -\alpha }{2}}}{\sin {\frac {\beta }{2}}}},\quad {\frac {a+b}{c}}={\frac {\cos {\frac {\alpha -\beta }{2}}}{\sin {\frac {\gamma }{2}}}}}" loading="lazy"></dd></dl>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {b-c}{a}}={\frac {\sin {\frac {\beta -\gamma }{2}}}{\cos {\frac {\alpha }{2}}}},\quad {\frac {c-a}{b}}={\frac {\sin {\frac {\gamma -\alpha }{2}}}{\cos {\frac {\beta }{2}}}},\quad {\frac {a-b}{c}}={\frac {\sin {\frac {\alpha -\beta }{2}}}{\cos {\frac {\gamma }{2}}}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ce914ac7c3ee93563308c729d1708c56c2feca4f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:57.86ex; height:8.843ex;" alt="{\displaystyle {\frac {b-c}{a}}={\frac {\sin {\frac {\beta -\gamma }{2}}}{\cos {\frac {\alpha }{2}}}},\quad {\frac {c-a}{b}}={\frac {\sin {\frac {\gamma -\alpha }{2}}}{\cos {\frac {\beta }{2}}}},\quad {\frac {a-b}{c}}={\frac {\sin {\frac {\alpha -\beta }{2}}}{\cos {\frac {\gamma }{2}}}}}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Tangenssatz"><a href="Tangenssatz" title="Tangenssatz">Tangenssatz</a></h3></div>

Formel 1:

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

<mfrac>

</mfrac>

</mrow>

<mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {b+c}{b-c}}={\frac {\tan {\frac {\beta +\gamma }{2}}}{\tan {\frac {\beta -\gamma }{2}}}}={\frac {\cot {\frac {\alpha }{2}}}{\tan {\frac {\beta -\gamma }{2}}}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bdd7ff080521d78d9889ea0822966194a1dda2d9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:28.942ex; height:8.843ex;" alt="{\displaystyle {\frac {b+c}{b-c}}={\frac {\tan {\frac {\beta +\gamma }{2}}}{\tan {\frac {\beta -\gamma }{2}}}}={\frac {\cot {\frac {\alpha }{2}}}{\tan {\frac {\beta -\gamma }{2}}}}}" loading="lazy"></dd></dl>

Analoge Formeln gelten für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {a+b}{a-b}}}">

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {a+b}{a-b}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/93415740a91d0b1807091264dae0b6d854ff4e65.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:5.904ex; height:5.676ex;" alt="{\displaystyle {\frac {a+b}{a-b}}}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {c+a}{c-a}}}">

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {c+a}{c-a}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/08af94c0c2fd7256196b86783eb70abc191b3284.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:5.913ex; height:5.176ex;" alt="{\displaystyle {\frac {c+a}{c-a}}}" loading="lazy">:

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

<mfrac>

</mfrac>

</mrow>

<mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {a+b}{a-b}}={\frac {\tan {\frac {\alpha +\beta }{2}}}{\tan {\frac {\alpha -\beta }{2}}}}={\frac {\cot {\frac {\gamma }{2}}}{\tan {\frac {\alpha -\beta }{2}}}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5ab81d0ef0db556f0300e99a4404d152994ff164.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:29.483ex; height:8.843ex;" alt="{\displaystyle {\frac {a+b}{a-b}}={\frac {\tan {\frac {\alpha +\beta }{2}}}{\tan {\frac {\alpha -\beta }{2}}}}={\frac {\cot {\frac {\gamma }{2}}}{\tan {\frac {\alpha -\beta }{2}}}}}" loading="lazy"></dd></dl>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

<mfrac>

</mfrac>

</mrow>

<mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {c+a}{c-a}}={\frac {\tan {\frac {\gamma +\alpha }{2}}}{\tan {\frac {\gamma -\alpha }{2}}}}={\frac {\cot {\frac {\beta }{2}}}{\tan {\frac {\gamma -\alpha }{2}}}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/61704ca0719701915d27e8c9762b91998e9ae889.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.671ex; width:29.394ex; height:8.676ex;" alt="{\displaystyle {\frac {c+a}{c-a}}={\frac {\tan {\frac {\gamma +\alpha }{2}}}{\tan {\frac {\gamma -\alpha }{2}}}}={\frac {\cot {\frac {\beta }{2}}}{\tan {\frac {\gamma -\alpha }{2}}}}}" loading="lazy"></dd></dl>

Wegen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tan(-x)=-\tan(x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \tan(-x)=-\tan(x)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/961705989441ed1a5aa05bf810320d4851b2db32.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.099ex; height:2.843ex;" alt="{\displaystyle \tan(-x)=-\tan(x)}" loading="lazy"> bleibt eine dieser Formeln gültig, wenn sowohl die Seiten als auch die zugehörigen Winkel vertauscht werden, also etwa:

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

<mfrac>

</mfrac>

</mrow>

<mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {a+c}{a-c}}={\frac {\tan {\frac {\alpha +\gamma }{2}}}{\tan {\frac {\alpha -\gamma }{2}}}}={\frac {\cot {\frac {\beta }{2}}}{\tan {\frac {\alpha -\gamma }{2}}}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5038d864c738e25fa32ff93a9c66bfce67c4dd21.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.671ex; width:29.394ex; height:8.676ex;" alt="{\displaystyle {\frac {a+c}{a-c}}={\frac {\tan {\frac {\alpha +\gamma }{2}}}{\tan {\frac {\alpha -\gamma }{2}}}}={\frac {\cot {\frac {\beta }{2}}}{\tan {\frac {\alpha -\gamma }{2}}}}}" loading="lazy"></dd></dl>

Formel 2:

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha =90^{\circ }}</annotation>

</semantics>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \tan \beta ={\frac {b}{c}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9260b12985d1251fb5ff22c8d45e50d5130760e2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:10.02ex; height:5.343ex;" alt="{\displaystyle \tan \beta ={\frac {b}{c}}}" loading="lazy"></dd></dl>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \tan \gamma ={\frac {c}{b}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c176a3c55e83a1c0d8e11d298700a7362a6a2c98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:9.95ex; height:4.843ex;" alt="{\displaystyle \tan \gamma ={\frac {c}{b}}}" loading="lazy"></dd></dl>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \beta =90^{\circ }}</annotation>

</semantics>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \tan \alpha ={\frac {a}{c}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f65fabc9497ebb14b6751877a91ea537c67f26b5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:10.399ex; height:4.676ex;" alt="{\displaystyle \tan \alpha ={\frac {a}{c}}}" loading="lazy"></dd></dl>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \tan \gamma ={\frac {c}{a}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/08ad06bf52c65829acac3804c59ba83762326ae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:10.173ex; height:4.676ex;" alt="{\displaystyle \tan \gamma ={\frac {c}{a}}}" loading="lazy"></dd></dl>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \gamma =90^{\circ }}</annotation>

</semantics>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \tan \alpha ={\frac {a}{b}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6190698fd29dcde8ede380d821a63b2d4bb16f45.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:10.399ex; height:4.843ex;" alt="{\displaystyle \tan \alpha ={\frac {a}{b}}}" loading="lazy"></dd></dl>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \tan \beta ={\frac {b}{a}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f1320d9c8aab9ff714e0215f9a930334662218b2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:10.243ex; height:5.343ex;" alt="{\displaystyle \tan \beta ={\frac {b}{a}}}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Formeln_mit_dem_halben_Umfang">Formeln mit dem halben Umfang</h3></div>

Im Folgenden bedeutet <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/01d131dfd7673938b947072a13a9744fe997e632.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}" loading="lazy"> immer die Hälfte des Umfangs des Dreiecks <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ABC}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle ABC}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5e55b44cfd965fbdc7a328d5db8a35a619db0971.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.273ex; height:2.176ex;" alt="{\displaystyle ABC}" loading="lazy">, also <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s={\frac {a+b+c}{2}}}">

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s={\frac {a+b+c}{2}}}</annotation>

</semantics>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s-a={\frac {b+c-a}{2}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/805906b78313058f280e780574d0f3bcbfb8b020.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:18.01ex; height:5.343ex;" alt="{\displaystyle s-a={\frac {b+c-a}{2}}}" loading="lazy"></dd></dl>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s-b={\frac {c+a-b}{2}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c6bd7bde06fbc044cb9730d7d947725402708a94.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:17.778ex; height:5.343ex;" alt="{\displaystyle s-b={\frac {c+a-b}{2}}}" loading="lazy"></dd></dl>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s-c={\frac {a+b-c}{2}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/140397740da1e53d188a32ebc8a4f399f2e74cfb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:17.787ex; height:5.343ex;" alt="{\displaystyle s-c={\frac {a+b-c}{2}}}" loading="lazy"></dd></dl>

<mrow>

<mrow>

</mrow>

</mrow>

<mrow>

<mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left(s-b\right)+\left(s-c\right)=a}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7b790e361a7dd45b6ae342b0c7ed87b7bf6238ef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.653ex; height:2.843ex;" alt="{\displaystyle \left(s-b\right)+\left(s-c\right)=a}" loading="lazy"></dd></dl>

<mrow>

<mrow>

</mrow>

</mrow>

<mrow>

<mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left(s-c\right)+\left(s-a\right)=b}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a4c9cc35cf083c5cecf5f4608d128e21a33ea4cb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.653ex; height:2.843ex;" alt="{\displaystyle \left(s-c\right)+\left(s-a\right)=b}" loading="lazy"></dd></dl>

<mrow>

<mrow>

</mrow>

</mrow>

<mrow>

<mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left(s-a\right)+\left(s-b\right)=c}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/81f17eccd84980303069f12ee45ab3c62310fc0a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.653ex; height:2.843ex;" alt="{\displaystyle \left(s-a\right)+\left(s-b\right)=c}" loading="lazy"></dd></dl>

<mrow>

<mrow>

</mrow>

</mrow>

<mrow>

<mrow>

</mrow>

</mrow>

<mrow>

<mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left(s-a\right)+\left(s-b\right)+\left(s-c\right)=s}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4377759001491bc6f483eb692990de8f36e311a9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.324ex; height:2.843ex;" alt="{\displaystyle \left(s-a\right)+\left(s-b\right)+\left(s-c\right)=s}" loading="lazy"></dd></dl>

<mfrac>

</mfrac>

</mrow>

<msqrt>

<mfrac>

<mrow>

<mrow>

</mrow>

</mrow>

<mrow>

<mrow>

</mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sin {\frac {\alpha }{2}}={\sqrt {\frac {\left(s-b\right)\left(s-c\right)}{bc}}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0a447976eb599b2d1fe0a5147926c9cd2c238687.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:25.697ex; height:7.509ex;" alt="{\displaystyle \sin {\frac {\alpha }{2}}={\sqrt {\frac {\left(s-b\right)\left(s-c\right)}{bc}}}}" loading="lazy"></dd></dl>

<mfrac>

</mfrac>

</mrow>

<msqrt>

<mfrac>

<mrow>

<mrow>

</mrow>

</mrow>

<mrow>

<mrow>

</mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sin {\frac {\beta }{2}}={\sqrt {\frac {\left(s-c\right)\left(s-a\right)}{ca}}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d2e189984b34d3d699ecc84f7cc9e2227305dba2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:25.773ex; height:7.676ex;" alt="{\displaystyle \sin {\frac {\beta }{2}}={\sqrt {\frac {\left(s-c\right)\left(s-a\right)}{ca}}}}" loading="lazy"></dd></dl>

<mfrac>

</mfrac>

</mrow>

<msqrt>

<mfrac>

<mrow>

<mrow>

</mrow>

</mrow>

<mrow>

<mrow>

</mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sin {\frac {\gamma }{2}}={\sqrt {\frac {\left(s-a\right)\left(s-b\right)}{ab}}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ac3176b303461e46a5f812dbaab293791127d501.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:25.694ex; height:7.509ex;" alt="{\displaystyle \sin {\frac {\gamma }{2}}={\sqrt {\frac {\left(s-a\right)\left(s-b\right)}{ab}}}}" loading="lazy"></dd></dl>

<mfrac>

</mfrac>

</mrow>

<msqrt>

<mfrac>

<mrow>

<mrow>

<mrow>

</mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \cos {\frac {\alpha }{2}}={\sqrt {\frac {s\left(s-a\right)}{bc}}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9077a7f58c3b1ef468cf4507f72f3bbc40d826f8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:20.528ex; height:7.509ex;" alt="{\displaystyle \cos {\frac {\alpha }{2}}={\sqrt {\frac {s\left(s-a\right)}{bc}}}}" loading="lazy"></dd></dl>

<mfrac>

</mfrac>

</mrow>

<msqrt>

<mfrac>

<mrow>

<mrow>

<mrow>

</mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \cos {\frac {\beta }{2}}={\sqrt {\frac {s\left(s-b\right)}{ca}}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1c6d5ca6641f397070a46ce77d90ecdd09afe1f7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:20.14ex; height:7.676ex;" alt="{\displaystyle \cos {\frac {\beta }{2}}={\sqrt {\frac {s\left(s-b\right)}{ca}}}}" loading="lazy"></dd></dl>

<mfrac>

</mfrac>

</mrow>

<msqrt>

<mfrac>

<mrow>

<mrow>

<mrow>

</mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \cos {\frac {\gamma }{2}}={\sqrt {\frac {s\left(s-c\right)}{ab}}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0879c030799bd86dd233826620d9c09f86b18c20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:20.08ex; height:7.509ex;" alt="{\displaystyle \cos {\frac {\gamma }{2}}={\sqrt {\frac {s\left(s-c\right)}{ab}}}}" loading="lazy"></dd></dl>

<mfrac>

</mfrac>

</mrow>

<msqrt>

<mfrac>

<mrow>

<mrow>

</mrow>

</mrow>

<mrow>

<mrow>

</mrow>

</mrow>

<mrow>

<mrow>

<mrow>

</mrow>

</mrow>

</mfrac>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \tan {\frac {\alpha }{2}}={\sqrt {\frac {\left(s-b\right)\left(s-c\right)}{s\left(s-a\right)}}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5ae1d36d5484cdd5cbd5f92ec2d4939e384c576b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:26.201ex; height:7.509ex;" alt="{\displaystyle \tan {\frac {\alpha }{2}}={\sqrt {\frac {\left(s-b\right)\left(s-c\right)}{s\left(s-a\right)}}}}" loading="lazy"></dd></dl>

<mfrac>

</mfrac>

</mrow>

<msqrt>

<mfrac>

<mrow>

<mrow>

</mrow>

</mrow>

<mrow>

<mrow>

</mrow>

</mrow>

<mrow>

<mrow>

<mrow>

</mrow>

</mrow>

</mfrac>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \tan {\frac {\beta }{2}}={\sqrt {\frac {\left(s-c\right)\left(s-a\right)}{s\left(s-b\right)}}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f2a38d91c436219140883746ca8fe45a9814442d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:26.277ex; height:7.509ex;" alt="{\displaystyle \tan {\frac {\beta }{2}}={\sqrt {\frac {\left(s-c\right)\left(s-a\right)}{s\left(s-b\right)}}}}" loading="lazy"></dd></dl>

<mfrac>

</mfrac>

</mrow>

<msqrt>

<mfrac>

<mrow>

<mrow>

</mrow>

</mrow>

<mrow>

<mrow>

</mrow>

</mrow>

<mrow>

<mrow>

<mrow>

</mrow>

</mrow>

</mfrac>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \tan {\frac {\gamma }{2}}={\sqrt {\frac {\left(s-a\right)\left(s-b\right)}{s\left(s-c\right)}}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/113f756b9fd39c502412ab4baa0839d1fbef440a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:26.198ex; height:7.509ex;" alt="{\displaystyle \tan {\frac {\gamma }{2}}={\sqrt {\frac {\left(s-a\right)\left(s-b\right)}{s\left(s-c\right)}}}}" loading="lazy"></dd></dl>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s=4r\cos {\frac {\alpha }{2}}\cos {\frac {\beta }{2}}\cos {\frac {\gamma }{2}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/587b54719bb9e19b0b0b5b426e7f78b85afa8e0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:24.646ex; height:5.343ex;" alt="{\displaystyle s=4r\cos {\frac {\alpha }{2}}\cos {\frac {\beta }{2}}\cos {\frac {\gamma }{2}}}" loading="lazy"></dd></dl>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s-a=4r\cos {\frac {\alpha }{2}}\sin {\frac {\beta }{2}}\sin {\frac {\gamma }{2}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b1910018847cb4187c26654acd4ee77eb627c41f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:28.206ex; height:5.343ex;" alt="{\displaystyle s-a=4r\cos {\frac {\alpha }{2}}\sin {\frac {\beta }{2}}\sin {\frac {\gamma }{2}}}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Flächeninhalt_und_Umkreisradius">Flächeninhalt und Umkreisradius</h3></div>

Der Flächeninhalt des Dreiecks wird hier mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/545fd099af8541605f7ee55f08225526be88ce57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" loading="lazy"> bezeichnet (nicht, wie heute üblich, mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy">, um eine Verwechselung mit der Dreiecksecke <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

</semantics>

<a href="Satz_des_Heron" title="Satz des Heron">Heronsche Formel:</a>

<msqrt>

<mrow>

<mrow>

</mrow>

</mrow>

<mrow>

<mrow>

</mrow>

</mrow>

<mrow>

<mrow>

</mrow>

</mrow>

</msqrt>

</mrow>

<mfrac>

</mfrac>

</mrow>

<msqrt>

<mrow>

<mrow>

</mrow>

</mrow>

<mrow>

<mrow>

</mrow>

</mrow>

<mrow>

<mrow>

</mrow>

</mrow>

<mrow>

<mrow>

</mrow>

</mrow>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F={\sqrt {s\left(s-a\right)\left(s-b\right)\left(s-c\right)}}={\frac {1}{4}}{\sqrt {\left(a+b+c\right)\left(b+c-a\right)\left(c+a-b\right)\left(a+b-c\right)}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1cff40f9e76f4a6caedc61ec2ad061494109357d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:81.349ex; height:5.176ex;" alt="{\displaystyle F={\sqrt {s\left(s-a\right)\left(s-b\right)\left(s-c\right)}}={\frac {1}{4}}{\sqrt {\left(a+b+c\right)\left(b+c-a\right)\left(c+a-b\right)\left(a+b-c\right)}}}" loading="lazy"></dd></dl>

<mfrac>

</mfrac>

</mrow>

<msqrt>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F={\frac {1}{4}}{\sqrt {2\left(b^{2}c^{2}+c^{2}a^{2}+a^{2}b^{2}\right)-\left(a^{4}+b^{4}+c^{4}\right)}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/dfd7d8e89014c6445ec227181353f4812439c1a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:48.364ex; height:5.176ex;" alt="{\displaystyle F={\frac {1}{4}}{\sqrt {2\left(b^{2}c^{2}+c^{2}a^{2}+a^{2}b^{2}\right)-\left(a^{4}+b^{4}+c^{4}\right)}}}" loading="lazy"></dd></dl>

Weitere Flächenformeln:

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F={\frac {1}{2}}bc\sin \alpha ={\frac {1}{2}}ca\sin \beta ={\frac {1}{2}}ab\sin \gamma }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b112157af09c0eceebb7617060f228b7c04532ad.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:38.472ex; height:5.176ex;" alt="{\displaystyle F={\frac {1}{2}}bc\sin \alpha ={\frac {1}{2}}ca\sin \beta ={\frac {1}{2}}ab\sin \gamma }" loading="lazy"></dd></dl>

<mfrac>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<mfrac>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<mfrac>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F={\frac {1}{2}}ah_{a}={\frac {1}{2}}bh_{b}={\frac {1}{2}}ch_{c}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c4399ece46d68e533b08a4075b3e6e6fc5a34540.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:27.267ex; height:5.176ex;" alt="{\displaystyle F={\frac {1}{2}}ah_{a}={\frac {1}{2}}bh_{b}={\frac {1}{2}}ch_{c}}" loading="lazy">, wobei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h_{a}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle h_{a}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/00531874c3a9e751e1c78d4f84483fcec2e75eba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.441ex; height:2.509ex;" alt="{\displaystyle h_{a}}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h_{b}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle h_{b}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/140aec52ffb1fd0966772704b2fe00827cdefa13.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.277ex; height:2.509ex;" alt="{\displaystyle h_{b}}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h_{c}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle h_{c}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d06972075e10d9390c826454530d3e2a6351dc45.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.283ex; height:2.509ex;" alt="{\displaystyle h_{c}}" loading="lazy"> die Längen der von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

</semantics>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/47136aad860d145f75f3eed3022df827cee94d7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" loading="lazy"> bzw. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4fc55753007cd3c18576f7933f6f089196732029.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" loading="lazy"> ausgehenden Höhen des Dreiecks <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ABC}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle ABC}</annotation>

</semantics>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F=2r^{2}\sin \,\alpha \,\sin \,\beta \,\sin \,\gamma }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cf54fc56018410ebe963dba45ff0a9d863466bb8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.011ex; height:3.176ex;" alt="{\displaystyle F=2r^{2}\sin \,\alpha \,\sin \,\beta \,\sin \,\gamma }" loading="lazy"></dd></dl>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F={\frac {abc}{4r}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8e02db2024cb4f6fc1f1f4ee4796191defd934ff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:8.91ex; height:5.343ex;" alt="{\displaystyle F={\frac {abc}{4r}}}" loading="lazy"></dd></dl>

<msub>

</mrow>

</msub>

<mrow>

<mrow>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

<mrow>

<mrow>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

<mrow>

<mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F=\rho s=\rho _{a}\left(s-a\right)=\rho _{b}\left(s-b\right)=\rho _{c}\left(s-c\right)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/23334167893cfcb4378336e8a78ce99f6dc6faf8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:44.633ex; height:2.843ex;" alt="{\displaystyle F=\rho s=\rho _{a}\left(s-a\right)=\rho _{b}\left(s-b\right)=\rho _{c}\left(s-c\right)}" loading="lazy"></dd></dl>

<msqrt>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F={\sqrt {\rho \rho _{a}\rho _{b}\rho _{c}}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/448f20368abca205d9232128a79409234d18674e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:14.567ex; height:3.009ex;" alt="{\displaystyle F={\sqrt {\rho \rho _{a}\rho _{b}\rho _{c}}}}" loading="lazy"></dd></dl>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F=4\rho r\cos \,{\frac {\alpha }{2}}\,\cos \,{\frac {\beta }{2}}\,\cos \,{\frac {\gamma }{2}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b905a9c2047cf11c02f4dd52f5536804918ac5e7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:28.434ex; height:5.343ex;" alt="{\displaystyle F=4\rho r\cos \,{\frac {\alpha }{2}}\,\cos \,{\frac {\beta }{2}}\,\cos \,{\frac {\gamma }{2}}}" loading="lazy"></dd></dl>

<msup>

</mrow>

</msup>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F=s^{2}\tan \,{\frac {\alpha }{2}}\,\tan \,{\frac {\beta }{2}}\,\tan \,{\frac {\gamma }{2}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2cb2fcf6dddc8afc9680b294cd3c7f48a7364d3a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:27.911ex; height:5.343ex;" alt="{\displaystyle F=s^{2}\tan \,{\frac {\alpha }{2}}\,\tan \,{\frac {\beta }{2}}\,\tan \,{\frac {\gamma }{2}}}" loading="lazy"></dd></dl>

<msup>

</mrow>

</msup>

<msqrt>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mstyle>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F=\rho ^{2}{\sqrt {\dfrac {h_{a}\,h_{b}\,h_{c}}{(h_{a}-2\rho )(h_{b}-2\rho )(h_{c}-2\rho )}}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f90d3ba3252954cde3989e8678e2faf7a363b502.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:38.298ex; height:7.509ex;" alt="{\displaystyle F=\rho ^{2}{\sqrt {\dfrac {h_{a}\,h_{b}\,h_{c}}{(h_{a}-2\rho )(h_{b}-2\rho )(h_{c}-2\rho )}}}}" loading="lazy">, mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dfrac {1}{\rho }}={\dfrac {1}{h_{a}}}+{\dfrac {1}{h_{b}}}+{\dfrac {1}{h_{c}}}}">

<mfrac>

</mfrac>

</mstyle>

</mrow>

<mfrac>

<msub>

</mrow>

</msub>

</mfrac>

</mstyle>

</mrow>

<mfrac>

<msub>

</mrow>

</msub>

</mfrac>

</mstyle>

</mrow>

<mfrac>

<msub>

</mrow>

</msub>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\dfrac {1}{\rho }}={\dfrac {1}{h_{a}}}+{\dfrac {1}{h_{b}}}+{\dfrac {1}{h_{c}}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/313f0eff47c3d7a935fea6a11e7d2c1ff21dd3b3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:20.326ex; height:5.676ex;" alt="{\displaystyle {\dfrac {1}{\rho }}={\dfrac {1}{h_{a}}}+{\dfrac {1}{h_{b}}}+{\dfrac {1}{h_{c}}}}" loading="lazy"></dd></dl>

<msqrt>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mstyle>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F={\sqrt {\dfrac {r\,h_{a}\,h_{b}\,h_{c}}{2}}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/65d2c59e52dc9892f8e396d749d3c3f93b81263d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:17.21ex; height:6.343ex;" alt="{\displaystyle F={\sqrt {\dfrac {r\,h_{a}\,h_{b}\,h_{c}}{2}}}}" loading="lazy"></dd></dl>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

<mrow>

</mrow>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F={\dfrac {\,h_{a}\,h_{b}\,h_{c}}{2\rho \,{(\sin \alpha +\sin \beta +\sin \gamma )}}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f33b324dbec9587339c400f1ff9b9dad6f6c30ea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:29.727ex; height:6.176ex;" alt="{\displaystyle F={\dfrac {\,h_{a}\,h_{b}\,h_{c}}{2\rho \,{(\sin \alpha +\sin \beta +\sin \gamma )}}}}" loading="lazy"></dd></dl>

Erweiterter Sinussatz:

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {a}{\sin \alpha }}={\frac {b}{\sin \beta }}={\frac {c}{\sin \gamma }}=2r={\frac {abc}{2F}}}</annotation>

</semantics>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a=2r\,\sin \alpha }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/109c21c9dfe7128ff5b7f4e15a19eeb5f3f85f16.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.044ex; height:2.176ex;" alt="{\displaystyle a=2r\,\sin \alpha }" loading="lazy"></dd></dl>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b=2r\,\sin \beta }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2f643cdbf10e006ad55c3750bcd7206012a37b0b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.656ex; height:2.509ex;" alt="{\displaystyle b=2r\,\sin \beta }" loading="lazy"></dd></dl>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c=2r\,\sin \gamma }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/66c8ae7a50c576a6f38c8161f79582a5ec43ed40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.596ex; height:2.676ex;" alt="{\displaystyle c=2r\,\sin \gamma }" loading="lazy"></dd></dl>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r={\frac {abc}{4F}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8bebd59edb3ce6524e3f7763493ceebf0095eb90.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:8.217ex; height:5.343ex;" alt="{\displaystyle r={\frac {abc}{4F}}}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="In-_und_Ankreisradien">In- und Ankreisradien</h3></div>

In diesem Abschnitt werden Formeln aufgelistet, in denen der <a href="Inkreis#Inkreis_eines_Dreiecks" title="Inkreis"> Inkreisradius</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \rho }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1f7d439671d1289b6a816e6af7a304be40608d64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }" loading="lazy"> und die <a href="Ankreis" title="Ankreis">Ankreisradien</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho _{a}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \rho _{a}}</annotation>

</semantics>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \rho _{b}}</annotation>

</semantics>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \rho _{c}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/523e8c709cfeaef7ed497ce5be80144fdf2e9b70.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.146ex; height:2.176ex;" alt="{\displaystyle \rho _{c}}" loading="lazy"> des Dreiecks <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ABC}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle ABC}</annotation>

</semantics>

<mrow>

<mrow>

</mrow>

</mrow>

<mfrac>

</mfrac>

</mrow>

<mrow>

<mrow>

</mrow>

</mrow>

<mfrac>

</mfrac>

</mrow>

<mrow>

<mrow>

</mrow>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \rho =\left(s-a\right)\tan {\frac {\alpha }{2}}=\left(s-b\right)\tan {\frac {\beta }{2}}=\left(s-c\right)\tan {\frac {\gamma }{2}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/123af333044eb976dc0e48f1581eaf56d0f8d6ce.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:49.944ex; height:5.343ex;" alt="{\displaystyle \rho =\left(s-a\right)\tan {\frac {\alpha }{2}}=\left(s-b\right)\tan {\frac {\beta }{2}}=\left(s-c\right)\tan {\frac {\gamma }{2}}}" loading="lazy"></dd></dl>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \rho =4r\sin {\frac {\alpha }{2}}\sin {\frac {\beta }{2}}\sin {\frac {\gamma }{2}}=s\tan {\frac {\alpha }{2}}\tan {\frac {\beta }{2}}\tan {\frac {\gamma }{2}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9857afda4eafd324f3213cb408fc05d99c47a43f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:47.172ex; height:5.343ex;" alt="{\displaystyle \rho =4r\sin {\frac {\alpha }{2}}\sin {\frac {\beta }{2}}\sin {\frac {\gamma }{2}}=s\tan {\frac {\alpha }{2}}\tan {\frac {\beta }{2}}\tan {\frac {\gamma }{2}}}" loading="lazy"></dd></dl>

<mrow>

<mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \rho =r\left(\cos \alpha +\cos \beta +\cos \gamma -1\right)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e6f715fb9c42db9054cd8e4bb97246f539e8c769.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.806ex; height:2.843ex;" alt="{\displaystyle \rho =r\left(\cos \alpha +\cos \beta +\cos \gamma -1\right)}" loading="lazy"></dd></dl>

<mfrac>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \rho ={\frac {F}{s}}={\frac {abc}{4rs}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/675eb7b0cc7865d9cc78406795fb8dfd6d03d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:14.113ex; height:5.343ex;" alt="{\displaystyle \rho ={\frac {F}{s}}={\frac {abc}{4rs}}}" loading="lazy"></dd></dl>

<msqrt>

<mfrac>

<mrow>

<mrow>

</mrow>

</mrow>

<mrow>

<mrow>

</mrow>

</mrow>

<mrow>

<mrow>

</mrow>

</mrow>

</mfrac>

</msqrt>

</mrow>

<mfrac>

</mfrac>

</mrow>

<msqrt>

<mfrac>

<mrow>

<mrow>

</mrow>

</mrow>

<mrow>

<mrow>

</mrow>

</mrow>

<mrow>

<mrow>

</mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \rho ={\sqrt {\frac {\left(s-a\right)\left(s-b\right)\left(s-c\right)}{s}}}={\frac {1}{2}}{\sqrt {\frac {\left(b+c-a\right)\left(c+a-b\right)\left(a+b-c\right)}{a+b+c}}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5ee6be8d1a210441f6780900cb46638a2e1c5324.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:69.893ex; height:7.843ex;" alt="{\displaystyle \rho ={\sqrt {\frac {\left(s-a\right)\left(s-b\right)\left(s-c\right)}{s}}}={\frac {1}{2}}{\sqrt {\frac {\left(b+c-a\right)\left(c+a-b\right)\left(a+b-c\right)}{a+b+c}}}}" loading="lazy"></dd></dl>

<mfrac>

<mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \rho ={\frac {a}{\cot {\frac {\beta }{2}}+\cot {\frac {\gamma }{2}}}}={\frac {b}{\cot {\frac {\gamma }{2}}+\cot {\frac {\alpha }{2}}}}={\frac {c}{\cot {\frac {\alpha }{2}}+\cot {\frac {\beta }{2}}}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3f60025d349a8308f78dac9fdf72833529b9b30d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:53.236ex; height:7.343ex;" alt="{\displaystyle \rho ={\frac {a}{\cot {\frac {\beta }{2}}+\cot {\frac {\gamma }{2}}}}={\frac {b}{\cot {\frac {\gamma }{2}}+\cot {\frac {\alpha }{2}}}}={\frac {c}{\cot {\frac {\alpha }{2}}+\cot {\frac {\beta }{2}}}}}" loading="lazy"></dd></dl>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a\cdot b+b\cdot c+c\cdot a=s^{2}+\rho ^{2}+4\cdot \rho \cdot r}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bf89866c508fe1ce27cb987d8732bd0561f447ac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:37.138ex; height:3.176ex;" alt="{\displaystyle a\cdot b+b\cdot c+c\cdot a=s^{2}+\rho ^{2}+4\cdot \rho \cdot r}" loading="lazy"> <a href="#cite_note-1">[1]</a></dd></dl>

Wichtige Ungleichung: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\rho \leq r}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 2\rho \leq r}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d7ea525418062bd1fa337a3e937d2629bfda73c8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.512ex; height:2.676ex;" alt="{\displaystyle 2\rho \leq r}" loading="lazy">; Gleichheit tritt nur dann ein, wenn Dreieck <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ABC}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle ABC}</annotation>

</semantics>

<msub>

</mrow>

</msub>

<mfrac>

</mfrac>

</mrow>

<mrow>

<mrow>

</mrow>

</mrow>

<mfrac>

</mfrac>

</mrow>

<mrow>

<mrow>

</mrow>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \rho _{a}=s\tan {\frac {\alpha }{2}}=\left(s-b\right)\cot {\frac {\gamma }{2}}=\left(s-c\right)\cot {\frac {\beta }{2}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/74243f4ab0fef960ccff4ffe96d65b39e69e9230.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:44.646ex; height:5.343ex;" alt="{\displaystyle \rho _{a}=s\tan {\frac {\alpha }{2}}=\left(s-b\right)\cot {\frac {\gamma }{2}}=\left(s-c\right)\cot {\frac {\beta }{2}}}" loading="lazy"></dd></dl>

<msub>

</mrow>

</msub>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

<mrow>

<mrow>

</mrow>

</mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \rho _{a}=4r\sin {\frac {\alpha }{2}}\cos {\frac {\beta }{2}}\cos {\frac {\gamma }{2}}=\left(s-a\right)\tan {\frac {\alpha }{2}}\cot {\frac {\beta }{2}}\cot {\frac {\gamma }{2}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fae2e5f2043111b9fd0fe71e27fa1a286519e2fc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:54.144ex; height:5.343ex;" alt="{\displaystyle \rho _{a}=4r\sin {\frac {\alpha }{2}}\cos {\frac {\beta }{2}}\cos {\frac {\gamma }{2}}=\left(s-a\right)\tan {\frac {\alpha }{2}}\cot {\frac {\beta }{2}}\cot {\frac {\gamma }{2}}}" loading="lazy"></dd></dl>

<msub>

</mrow>

</msub>

<mrow>

<mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \rho _{a}=r\left(-\cos \alpha +\cos \beta +\cos \gamma +1\right)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ec8ff598946bbe8b846a2207df8e4b7e97f29cf5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.103ex; height:2.843ex;" alt="{\displaystyle \rho _{a}=r\left(-\cos \alpha +\cos \beta +\cos \gamma +1\right)}" loading="lazy"></dd></dl>

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

<mrow>

<mrow>

</mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \rho _{a}={\frac {F}{s-a}}={\frac {abc}{4r\left(s-a\right)}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ecb9a6c0fc9888c379855bc1b989da34623ba433.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:24.902ex; height:6.176ex;" alt="{\displaystyle \rho _{a}={\frac {F}{s-a}}={\frac {abc}{4r\left(s-a\right)}}}" loading="lazy"></dd></dl>

<msub>

</mrow>

</msub>

<msqrt>

<mfrac>

<mrow>

<mrow>

<mrow>

</mrow>

</mrow>

<mrow>

<mrow>

</mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</msqrt>

</mrow>

<mfrac>

</mfrac>

</mrow>

<msqrt>

<mfrac>

<mrow>

<mrow>

</mrow>

</mrow>

<mrow>

<mrow>

</mrow>

</mrow>

<mrow>

<mrow>

</mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \rho _{a}={\sqrt {\frac {s\left(s-b\right)\left(s-c\right)}{s-a}}}={\frac {1}{2}}{\sqrt {\frac {\left(a+b+c\right)\left(c+a-b\right)\left(a+b-c\right)}{b+c-a}}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5de20d0996116d3ba61b7aa8c750cc35581b168e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:65.116ex; height:7.676ex;" alt="{\displaystyle \rho _{a}={\sqrt {\frac {s\left(s-b\right)\left(s-c\right)}{s-a}}}={\frac {1}{2}}{\sqrt {\frac {\left(a+b+c\right)\left(c+a-b\right)\left(a+b-c\right)}{b+c-a}}}}" loading="lazy"></dd></dl>

Die Ankreise sind gleichberechtigt: Jede Formel für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho _{a}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \rho _{a}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cf5bcd5f18e417072a2f15baf57774af83e76672.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.304ex; height:2.176ex;" alt="{\displaystyle \rho _{a}}" loading="lazy"> gilt in analoger Form für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho _{b}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \rho _{b}}</annotation>

</semantics>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \rho _{c}}</annotation>

</semantics>

<mfrac>

</mfrac>

</mrow>

<mfrac>

<msub>

</mrow>

</msub>

</mfrac>

</mrow>

<mfrac>

<msub>

</mrow>

</msub>

</mfrac>

</mrow>

<mfrac>

<msub>

</mrow>

</msub>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{\rho }}={\frac {1}{\rho _{a}}}+{\frac {1}{\rho _{b}}}+{\frac {1}{\rho _{c}}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/65849660605ed119305465c8d5b13d1b569d4767.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:19.915ex; height:5.676ex;" alt="{\displaystyle {\frac {1}{\rho }}={\frac {1}{\rho _{a}}}+{\frac {1}{\rho _{b}}}+{\frac {1}{\rho _{c}}}}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Höhen">Höhen</h3></div>

Die Längen der von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

</semantics>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/47136aad860d145f75f3eed3022df827cee94d7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" loading="lazy"> bzw. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4fc55753007cd3c18576f7933f6f089196732029.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" loading="lazy"> ausgehenden Höhen des Dreiecks <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ABC}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle ABC}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5e55b44cfd965fbdc7a328d5db8a35a619db0971.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.273ex; height:2.176ex;" alt="{\displaystyle ABC}" loading="lazy"> werden mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h_{a}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle h_{a}}</annotation>

</semantics>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle h_{b}}</annotation>

</semantics>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle h_{c}}</annotation>

</semantics>

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<mrow>

<mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle h_{a}=b\sin \gamma =c\sin \beta ={\frac {2F}{a}}=2r\sin \beta \sin \gamma =2r\left(\cos \alpha +\cos \beta \cos \gamma \right)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d688efbcd9860b7fbd8d4e3fc7ffad735de9f904.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:67.808ex; height:5.176ex;" alt="{\displaystyle h_{a}=b\sin \gamma =c\sin \beta ={\frac {2F}{a}}=2r\sin \beta \sin \gamma =2r\left(\cos \alpha +\cos \beta \cos \gamma \right)}" loading="lazy"></dd></dl>

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<mrow>

<mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle h_{b}=c\sin \alpha =a\sin \gamma ={\frac {2F}{b}}=2r\sin \gamma \sin \alpha =2r\left(\cos \beta +\cos \alpha \cos \gamma \right)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f37d2bbbcc94914e8fe55c8ae3e7e6c135cccfce.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:68.187ex; height:5.343ex;" alt="{\displaystyle h_{b}=c\sin \alpha =a\sin \gamma ={\frac {2F}{b}}=2r\sin \gamma \sin \alpha =2r\left(\cos \beta +\cos \alpha \cos \gamma \right)}" loading="lazy"></dd></dl>

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<mrow>

<mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle h_{c}=a\sin \beta =b\sin \alpha ={\frac {2F}{c}}=2r\sin \alpha \sin \beta =2r\left(\cos \gamma +\cos \alpha \cos \beta \right)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4da5c421d1cc29f56d5fc3ad546fe0c205de1a1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:68.323ex; height:5.176ex;" alt="{\displaystyle h_{c}=a\sin \beta =b\sin \alpha ={\frac {2F}{c}}=2r\sin \alpha \sin \beta =2r\left(\cos \gamma +\cos \alpha \cos \beta \right)}" loading="lazy"></dd></dl>

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle h_{a}={\frac {a}{\cot \beta +\cot \gamma }};\;\;\;\;\;h_{b}={\frac {b}{\cot \gamma +\cot \alpha }};\;\;\;\;\;h_{c}={\frac {c}{\cot \alpha +\cot \beta }}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ade245e6f1d669d9ca12e246bcf43f99fa3e6503.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:64.929ex; height:6.009ex;" alt="{\displaystyle h_{a}={\frac {a}{\cot \beta +\cot \gamma }};\;\;\;\;\;h_{b}={\frac {b}{\cot \gamma +\cot \alpha }};\;\;\;\;\;h_{c}={\frac {c}{\cot \alpha +\cot \beta }}}" loading="lazy"></dd></dl>

<mfrac>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<mfrac>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<mfrac>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F={\frac {1}{2}}ah_{a}={\frac {1}{2}}bh_{b}={\frac {1}{2}}ch_{c}}</annotation>

</semantics>

<mfrac>

<msub>

</mrow>

</msub>

</mfrac>

</mrow>

<mfrac>

<msub>

</mrow>

</msub>

</mfrac>

</mrow>

<mfrac>

<msub>

</mrow>

</msub>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

<msub>

</mrow>

</msub>

</mfrac>

</mrow>

<mfrac>

<msub>

</mrow>

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</mfrac>

</mrow>

<mfrac>

<msub>

</mrow>

</msub>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{h_{a}}}+{\frac {1}{h_{b}}}+{\frac {1}{h_{c}}}={\frac {1}{\rho }}={\frac {1}{\rho _{a}}}+{\frac {1}{\rho _{b}}}+{\frac {1}{\rho _{c}}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8b0d8fa4ea77d26e2e223a1f9fa927e0cf012883.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:38.204ex; height:5.676ex;" alt="{\displaystyle {\frac {1}{h_{a}}}+{\frac {1}{h_{b}}}+{\frac {1}{h_{c}}}={\frac {1}{\rho }}={\frac {1}{\rho _{a}}}+{\frac {1}{\rho _{b}}}+{\frac {1}{\rho _{c}}}}" loading="lazy"></dd></dl>

Hat das Dreieck <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ABC}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle ABC}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5e55b44cfd965fbdc7a328d5db8a35a619db0971.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.273ex; height:2.176ex;" alt="{\displaystyle ABC}" loading="lazy"> einen rechten Winkel bei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4fc55753007cd3c18576f7933f6f089196732029.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" loading="lazy"> (ist also <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma =90^{\circ }}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \gamma =90^{\circ }}</annotation>

</semantics>

<msub>

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</msub>

<mfrac>

<mrow>

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</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle h_{c}={\frac {ab}{c}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/eb7d5860452f774d25ff0893d4ca96357fe222b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:8.445ex; height:5.343ex;" alt="{\displaystyle h_{c}={\frac {ab}{c}}}" loading="lazy"></dd></dl>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle h_{a}=b}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/27c903b2c7ce86550f305e74a56cc9f9e75e1db3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.537ex; height:2.509ex;" alt="{\displaystyle h_{a}=b}" loading="lazy"></dd></dl>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle h_{b}=a}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6b15d371c92a6fee23e02e93e337c67393e04822.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.605ex; height:2.509ex;" alt="{\displaystyle h_{b}=a}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Seitenhalbierende">Seitenhalbierende</h3></div>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

</semantics>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/47136aad860d145f75f3eed3022df827cee94d7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" loading="lazy"> bzw. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4fc55753007cd3c18576f7933f6f089196732029.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" loading="lazy"> ausgehenden Seitenhalbierenden des Dreiecks <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ABC}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle ABC}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5e55b44cfd965fbdc7a328d5db8a35a619db0971.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.273ex; height:2.176ex;" alt="{\displaystyle ABC}" loading="lazy"> werden <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{a}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s_{a}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bdf3b2fb584c60084dba330329b2ebc2982d9b88.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.192ex; height:2.009ex;" alt="{\displaystyle s_{a}}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{b}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s_{b}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/de2f6218809aff65f4c3ed8bdf0f7bed2dc393a1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.028ex; height:2.009ex;" alt="{\displaystyle s_{b}}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{c}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s_{c}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/91c4d1b073949e8f032e50ff913afbf43bb5acb7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.035ex; height:2.009ex;" alt="{\displaystyle s_{c}}" loading="lazy"> genannt.

<msub>

</mrow>

</msub>

<mfrac>

</mfrac>

</mrow>

<msqrt>

<msup>

</mrow>

</msup>

<msup>

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<msup>

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<mfrac>

</mfrac>

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<msqrt>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

<msqrt>

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s_{a}={\frac {1}{2}}{\sqrt {2b^{2}+2c^{2}-a^{2}}}={\frac {1}{2}}{\sqrt {b^{2}+c^{2}+2bc\cos \alpha }}={\sqrt {{\frac {a^{2}}{4}}+bc\cos \alpha }}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/289862303972afe17e27017257f12c8f29de36f0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:68.53ex; height:6.176ex;" alt="{\displaystyle s_{a}={\frac {1}{2}}{\sqrt {2b^{2}+2c^{2}-a^{2}}}={\frac {1}{2}}{\sqrt {b^{2}+c^{2}+2bc\cos \alpha }}={\sqrt {{\frac {a^{2}}{4}}+bc\cos \alpha }}}" loading="lazy"></dd></dl>

<msub>

</mrow>

</msub>

<mfrac>

</mfrac>

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<msqrt>

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<mfrac>

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<msqrt>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s_{b}={\frac {1}{2}}{\sqrt {2c^{2}+2a^{2}-b^{2}}}={\frac {1}{2}}{\sqrt {c^{2}+a^{2}+2ca\cos \beta }}={\sqrt {{\frac {b^{2}}{4}}+ca\cos \beta }}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5f4a3fa3157bb899b6b967b614ba18b5787ad827.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:68.519ex; height:6.176ex;" alt="{\displaystyle s_{b}={\frac {1}{2}}{\sqrt {2c^{2}+2a^{2}-b^{2}}}={\frac {1}{2}}{\sqrt {c^{2}+a^{2}+2ca\cos \beta }}={\sqrt {{\frac {b^{2}}{4}}+ca\cos \beta }}}" loading="lazy"></dd></dl>

<msub>

</mrow>

</msub>

<mfrac>

</mfrac>

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<msqrt>

<msup>

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<msup>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s_{c}={\frac {1}{2}}{\sqrt {2a^{2}+2b^{2}-c^{2}}}={\frac {1}{2}}{\sqrt {a^{2}+b^{2}+2ab\cos \gamma }}={\sqrt {{\frac {c^{2}}{4}}+ab\cos \gamma }}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cb28fa025ddd498456e0db9b3f5fdaa99b428c3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:68.368ex; height:6.176ex;" alt="{\displaystyle s_{c}={\frac {1}{2}}{\sqrt {2a^{2}+2b^{2}-c^{2}}}={\frac {1}{2}}{\sqrt {a^{2}+b^{2}+2ab\cos \gamma }}={\sqrt {{\frac {c^{2}}{4}}+ab\cos \gamma }}}" loading="lazy"></dd></dl>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

<mfrac>

</mfrac>

</mrow>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

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</msup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s_{a}^{2}+s_{b}^{2}+s_{c}^{2}={\frac {3}{4}}\left(a^{2}+b^{2}+c^{2}\right)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b610513559381f16639b625b310cd391d6932126.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:31.854ex; height:5.176ex;" alt="{\displaystyle s_{a}^{2}+s_{b}^{2}+s_{c}^{2}={\frac {3}{4}}\left(a^{2}+b^{2}+c^{2}\right)}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Winkelhalbierende">Winkelhalbierende</h3></div>

Wir bezeichnen mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w_{\alpha }}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle w_{\alpha }}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5138271001e8db3f92ad1431b891a2d1434fe150.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.948ex; height:2.009ex;" alt="{\displaystyle w_{\alpha }}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w_{\beta }}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle w_{\beta }}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1b23a4aaa3542828081105d33840517fec3a35db.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.838ex; height:2.343ex;" alt="{\displaystyle w_{\beta }}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w_{\gamma }}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle w_{\gamma }}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/91ed8aac2720d782abc0fef7058a0c7b98a03f36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.789ex; height:2.343ex;" alt="{\displaystyle w_{\gamma }}" loading="lazy"> die Längen der von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

</semantics>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/47136aad860d145f75f3eed3022df827cee94d7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" loading="lazy"> bzw. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4fc55753007cd3c18576f7933f6f089196732029.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" loading="lazy"> ausgehenden Winkelhalbierenden im Dreieck <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ABC}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle ABC}</annotation>

</semantics>

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

<mfrac>

</mfrac>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mfrac>

</mrow>

<mfrac>

<msqrt>

</msqrt>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle w_{\alpha }={\frac {2bc\cos {\frac {\alpha }{2}}}{b+c}}={\frac {2F}{a\cos {\frac {\beta -\gamma }{2}}}}={\frac {\sqrt {bc(b+c-a)(a+b+c)}}{b+c}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5bf920ff7c11a3e7861045cbc6c7889e74e1743b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:58.533ex; height:8.176ex;" alt="{\displaystyle w_{\alpha }={\frac {2bc\cos {\frac {\alpha }{2}}}{b+c}}={\frac {2F}{a\cos {\frac {\beta -\gamma }{2}}}}={\frac {\sqrt {bc(b+c-a)(a+b+c)}}{b+c}}}" loading="lazy"></dd></dl>

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

<mfrac>

</mfrac>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

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<mrow>

<mfrac>

<mrow>

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</mrow>

</mfrac>

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<mfrac>

<msqrt>

</msqrt>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle w_{\beta }={\frac {2ca\cos {\frac {\beta }{2}}}{c+a}}={\frac {2F}{b\cos {\frac {\gamma -\alpha }{2}}}}={\frac {\sqrt {ca(c+a-b)(a+b+c)}}{c+a}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e69561c06ed8800d722956e0eeec762252e155ea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.671ex; width:58.655ex; height:8.676ex;" alt="{\displaystyle w_{\beta }={\frac {2ca\cos {\frac {\beta }{2}}}{c+a}}={\frac {2F}{b\cos {\frac {\gamma -\alpha }{2}}}}={\frac {\sqrt {ca(c+a-b)(a+b+c)}}{c+a}}}" loading="lazy"></dd></dl>

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

<mfrac>

</mfrac>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mfrac>

</mrow>

<mfrac>

<msqrt>

</msqrt>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle w_{\gamma }={\frac {2ab\cos {\frac {\gamma }{2}}}{a+b}}={\frac {2F}{c\cos {\frac {\alpha -\beta }{2}}}}={\frac {\sqrt {ab(a+b-c)(a+b+c)}}{a+b}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1fadac3f992052c3db9b0f8f13d8630213ff9168.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:58.597ex; height:8.343ex;" alt="{\displaystyle w_{\gamma }={\frac {2ab\cos {\frac {\gamma }{2}}}{a+b}}={\frac {2F}{c\cos {\frac {\alpha -\beta }{2}}}}={\frac {\sqrt {ab(a+b-c)(a+b+c)}}{a+b}}}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Allgemeine_Trigonometrie_in_der_Ebene">Allgemeine Trigonometrie in der Ebene</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Periodizität">Periodizität</h3></div>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sin x\quad =\quad \sin(x+2n\pi );\quad n\in \mathbb {Z} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d9a3490c5417b44bf994273df2a39a3d5acb9d16.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.182ex; height:2.843ex;" alt="{\displaystyle \sin x\quad =\quad \sin(x+2n\pi );\quad n\in \mathbb {Z} }" loading="lazy"></dd>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \cos x\quad =\quad \cos(x+2n\pi );\quad n\in \mathbb {Z} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5642c5006b7c71131c6e5a4f640d4c88900fb70c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.693ex; height:2.843ex;" alt="{\displaystyle \cos x\quad =\quad \cos(x+2n\pi );\quad n\in \mathbb {Z} }" loading="lazy"></dd>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \tan x\quad =\quad \tan(x+n\pi );\quad n\in \mathbb {Z} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7c11437247b42bbc96b3fc7c3fde79824f719644.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.028ex; height:2.843ex;" alt="{\displaystyle \tan x\quad =\quad \tan(x+n\pi );\quad n\in \mathbb {Z} }" loading="lazy"></dd>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \cot x\quad =\quad \cot(x+n\pi );\quad n\in \mathbb {Z} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b181225746f33b8d53c59afad3c08ec6798f5dbf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.508ex; height:2.843ex;" alt="{\displaystyle \cot x\quad =\quad \cot(x+n\pi );\quad n\in \mathbb {Z} }" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Gegenseitige_Darstellung">Gegenseitige Darstellung</h3></div>

Die trigonometrischen Funktionen lassen sich ineinander umwandeln oder gegenseitig darstellen. Es gelten folgende Zusammenhänge:

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \tan x={\frac {\sin x}{\cos x}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f024fc11e57e7fbac757210cc79be66aa12ab86b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.839ex; height:5.176ex;" alt="{\displaystyle \tan x={\frac {\sin x}{\cos x}}}" loading="lazy"></dd></dl>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sin ^{2}x+\cos ^{2}x=1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1d1938e6828e597c076248d3ec430e0a7e5f98c8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:18.61ex; height:2.843ex;" alt="{\displaystyle \sin ^{2}x+\cos ^{2}x=1}" loading="lazy">      („<a href="Trigonometrischer_Pythagoras" title="Trigonometrischer Pythagoras">Trigonometrischer Pythagoras</a>“)</dd></dl>

<msup>

</mrow>

</msup>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1+\tan ^{2}x={\frac {1}{\cos ^{2}x}}=\sec ^{2}x}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e450d1ca2a9212ea5780105dbe047441cc29c713.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:28.801ex; height:5.509ex;" alt="{\displaystyle 1+\tan ^{2}x={\frac {1}{\cos ^{2}x}}=\sec ^{2}x}" loading="lazy"></dd></dl>

<msup>

</mrow>

</msup>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1+\cot ^{2}x={\frac {1}{\sin ^{2}x}}=\csc ^{2}x}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4a198c3d1942fcf14c03cfd93f7e19a993bc6c1f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:28.285ex; height:5.676ex;" alt="{\displaystyle 1+\cot ^{2}x={\frac {1}{\sin ^{2}x}}=\csc ^{2}x}" loading="lazy"></dd></dl>

(Siehe auch den Abschnitt <a class="mw-selflink-fragment" href="#Phasenverschiebungen">Phasenverschiebungen</a>.)

Mittels dieser Gleichungen lassen sich die drei vorkommenden Funktionen durch eine der beiden anderen darstellen:

<msqrt>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sin x\;=\;{\sqrt {1-\cos ^{2}x}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d85e0bb5fdcc871ffa1fe0170796e004e3cd3a27.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.17ex; height:3.509ex;" alt="{\displaystyle \sin x\;=\;{\sqrt {1-\cos ^{2}x}}}" loading="lazy">

</td>

<td>für

</td>

<mrow>

<mrow>

</mrow>

</mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\in \left[0,\pi \right[\quad =\quad [0^{\circ },180^{\circ }[}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/65e0fa972b2b6d24fdffe884ee4ce647deb50697.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.822ex; height:2.843ex;" alt="{\displaystyle x\in \left[0,\pi \right[\quad =\quad [0^{\circ },180^{\circ }[}" loading="lazy">

</td></tr>

<tr>

<msqrt>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sin x\;=\;-{\sqrt {1-\cos ^{2}x}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bc5a2c010bc78c9aa4aad11742b82148df89aa94.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.978ex; height:3.509ex;" alt="{\displaystyle \sin x\;=\;-{\sqrt {1-\cos ^{2}x}}}" loading="lazy">

</td>

<td>für

</td>

<mrow>

<mrow>

</mrow>

</mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\in \left[\pi ,2\pi \right[\quad =\quad [180^{\circ },360^{\circ }[}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d65e73d16662793f59430657b5d8c383d58047e7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.479ex; height:2.843ex;" alt="{\displaystyle x\in \left[\pi ,2\pi \right[\quad =\quad [180^{\circ },360^{\circ }[}" loading="lazy">

</td></tr>

<tr>

<mfrac>

<mrow>

</mrow>

<msqrt>

<msup>

</mrow>

</msup>

</msqrt>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sin x\;=\;{\frac {\tan x}{\sqrt {1+\tan ^{2}x}}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f7e6eb49085ea516007327af0629f263e6db9ef8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:22.255ex; height:6.509ex;" alt="{\displaystyle \sin x\;=\;{\frac {\tan x}{\sqrt {1+\tan ^{2}x}}}}" loading="lazy">

</td>

<td>für

</td>

<mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

<mrow>

<mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\in \left[0,{\frac {\pi }{2}}\right[\;\cup \;\left]{\frac {3\pi }{2}},2\pi \right[\quad =\quad [0^{\circ },90^{\circ }[\;\cup \;]270^{\circ },360^{\circ }[}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ac3f92c0f2af860ff485f97d43f23613e86774da.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:53.835ex; height:6.176ex;" alt="{\displaystyle x\in \left[0,{\frac {\pi }{2}}\right[\;\cup \;\left]{\frac {3\pi }{2}},2\pi \right[\quad =\quad [0^{\circ },90^{\circ }[\;\cup \;]270^{\circ },360^{\circ }[}" loading="lazy">

</td></tr>

<tr>

<mfrac>

<mrow>

</mrow>

<msqrt>

<msup>

</mrow>

</msup>

</msqrt>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sin x\;=\;-{\frac {\tan x}{\sqrt {1+\tan ^{2}x}}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6a0ccc751ba784fe27448eef42e335e827d83a52.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:24.063ex; height:6.509ex;" alt="{\displaystyle \sin x\;=\;-{\frac {\tan x}{\sqrt {1+\tan ^{2}x}}}}" loading="lazy">

</td>

<td>für

</td>

<mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\in \left]{\frac {\pi }{2}},{\frac {3\pi }{2}}\right[\quad =\quad ]90^{\circ },270^{\circ }[}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cc2b5907a1bc2124087735dbfce5637d5a74dcd4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:30.505ex; height:6.176ex;" alt="{\displaystyle x\in \left]{\frac {\pi }{2}},{\frac {3\pi }{2}}\right[\quad =\quad ]90^{\circ },270^{\circ }[}" loading="lazy">

</td></tr>

<tr>

<msqrt>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \cos x\;=\;{\sqrt {1-\sin ^{2}x}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d5748e9c8211d007424ca63df4033edb05038dc3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.17ex; height:3.509ex;" alt="{\displaystyle \cos x\;=\;{\sqrt {1-\sin ^{2}x}}}" loading="lazy">

</td>

<td>für

</td>

<mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

<mrow>

<mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\in \left[0,{\frac {\pi }{2}}\right[\;\cup \;\left[{\frac {3\pi }{2}},2\pi \right[\quad =\quad [0^{\circ },90^{\circ }[\;\cup \;[270^{\circ },360^{\circ }[}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/80977e6de05d5811dc94187b48fe6bb14448ff32.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:53.835ex; height:6.176ex;" alt="{\displaystyle x\in \left[0,{\frac {\pi }{2}}\right[\;\cup \;\left[{\frac {3\pi }{2}},2\pi \right[\quad =\quad [0^{\circ },90^{\circ }[\;\cup \;[270^{\circ },360^{\circ }[}" loading="lazy">

</td></tr>

<tr>

<msqrt>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \cos x\;=\;-{\sqrt {1-\sin ^{2}x}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/781ff392510c670fc17598490dd18dabca376564.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.978ex; height:3.509ex;" alt="{\displaystyle \cos x\;=\;-{\sqrt {1-\sin ^{2}x}}}" loading="lazy">

</td>

<td>für

</td>

<mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\in \left[{\frac {\pi }{2}},{\frac {3\pi }{2}}\right[\quad =\quad [90^{\circ },270^{\circ }[}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/dd4b181bba2db1e25be3922d7d19992c85406540.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:31.15ex; height:6.176ex;" alt="{\displaystyle x\in \left[{\frac {\pi }{2}},{\frac {3\pi }{2}}\right[\quad =\quad [90^{\circ },270^{\circ }[}" loading="lazy">

</td></tr>

<tr>

<mfrac>

<msqrt>

<msup>

</mrow>

</msup>

</msqrt>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \cos x={\frac {1}{\sqrt {1+\tan ^{2}x}}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3bc168525a025584ed4a4c0fce012e42a88ad515.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:21.22ex; height:6.509ex;" alt="{\displaystyle \cos x={\frac {1}{\sqrt {1+\tan ^{2}x}}}}" loading="lazy">

</td>

<td>für

</td>

<mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

<mrow>

<mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

</semantics>

</td></tr>

<tr>

<mfrac>

<msqrt>

<msup>

</mrow>

</msup>

</msqrt>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \cos x=-{\frac {1}{\sqrt {1+\tan ^{2}x}}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/47188bf4897ef9bc6f346ba78a5898dbbba1c2d6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:23.028ex; height:6.509ex;" alt="{\displaystyle \cos x=-{\frac {1}{\sqrt {1+\tan ^{2}x}}}}" loading="lazy">

</td>

<td>für

</td>

<mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\in \left]{\frac {\pi }{2}},{\frac {3\pi }{2}}\right[\quad =\quad ]90^{\circ },270^{\circ }[}</annotation>

</semantics>

</td></tr>

<tr>

<mfrac>

<msqrt>

<msup>

</mrow>

</msup>

</msqrt>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \tan x={\frac {\sqrt {1-\cos ^{2}x}}{\cos x}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7cc24e54d73b6b18d9d0271545c5bcfbb1f4ea03.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:21.22ex; height:6.176ex;" alt="{\displaystyle \tan x={\frac {\sqrt {1-\cos ^{2}x}}{\cos x}}}" loading="lazy">

</td>

<td>für

</td>

<mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

<mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\in \left[0,{\frac {\pi }{2}}\right[\;\cup \;\left]{\frac {\pi }{2}},\pi \right[\quad =\quad [0^{\circ },90^{\circ }[\;\cup \;]90^{\circ },180^{\circ }[}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2c17abe7fc5ec6ba7cc8e641f49b22ba3e063b63.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:50.088ex; height:4.843ex;" alt="{\displaystyle x\in \left[0,{\frac {\pi }{2}}\right[\;\cup \;\left]{\frac {\pi }{2}},\pi \right[\quad =\quad [0^{\circ },90^{\circ }[\;\cup \;]90^{\circ },180^{\circ }[}" loading="lazy">

</td></tr>

<tr>

<mfrac>

<msqrt>

<msup>

</mrow>

</msup>

</msqrt>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \tan x=-{\frac {\sqrt {1-\cos ^{2}x}}{\cos x}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2d71c255efe3cf596a319435942a7294fd53a0a4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:23.028ex; height:6.176ex;" alt="{\displaystyle \tan x=-{\frac {\sqrt {1-\cos ^{2}x}}{\cos x}}}" loading="lazy">

</td>

<td>für

</td>

<mrow>

<mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

<mrow>

<mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\in \left[\pi ,{\frac {3\pi }{2}}\right[\;\cup \;\left]{\frac {3\pi }{2}},2\pi \right[\quad =\quad [180^{\circ },270^{\circ }[\;\cup \;]270^{\circ },360^{\circ }[}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1ef7ace2e13e827d94d551099dc2d14a8e786fb6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:58.915ex; height:6.176ex;" alt="{\displaystyle x\in \left[\pi ,{\frac {3\pi }{2}}\right[\;\cup \;\left]{\frac {3\pi }{2}},2\pi \right[\quad =\quad [180^{\circ },270^{\circ }[\;\cup \;]270^{\circ },360^{\circ }[}" loading="lazy">

</td></tr>

<tr>

<mfrac>

<mrow>

</mrow>

<msqrt>

<msup>

</mrow>

</msup>

</msqrt>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \tan x={\frac {\sin x}{\sqrt {1-\sin ^{2}x}}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e7c4e83e6392786a79bad2a7d2e04ad7195f3795.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:20.964ex; height:6.509ex;" alt="{\displaystyle \tan x={\frac {\sin x}{\sqrt {1-\sin ^{2}x}}}}" loading="lazy">

</td>

<td>für

</td>

<mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

<mrow>

<mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

</semantics>

</td></tr>

<tr>

<mfrac>

<mrow>

</mrow>

<msqrt>

<msup>

</mrow>

</msup>

</msqrt>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \tan x=-{\frac {\sin x}{\sqrt {1-\sin ^{2}x}}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bfe7a4df7a48325f62a74503a2ba4e9c0dba2979.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:22.772ex; height:6.509ex;" alt="{\displaystyle \tan x=-{\frac {\sin x}{\sqrt {1-\sin ^{2}x}}}}" loading="lazy">

</td>

<td>für

</td>

<mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\in \left]{\frac {\pi }{2}},{\frac {3\pi }{2}}\right[\quad =\quad ]90^{\circ },270^{\circ }[}</annotation>

</semantics>

</td></tr></tbody></table>

<div class="mw-heading mw-heading3"><h3 id="Vorzeichen_der_Winkelfunktionen">Vorzeichen der Winkelfunktionen</h3></div>

</mrow>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sin x>0\quad {\text{für}}\quad x\in \left]0^{\circ },180^{\circ }\right[}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fdac0f290fd61f2d4f5b9a9696bf3169b7cd0918.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:29.775ex; height:3.343ex;" alt="{\displaystyle \sin x>0\quad {\text{für}}\quad x\in \left]0^{\circ },180^{\circ }\right[}" loading="lazy"></dd></dl>

</mrow>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sin x<0\quad {\text{für}}\quad x\in \left]180^{\circ },360^{\circ }\right[}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/170b6c1d4a0f5062cb5c97c5c1157b4e2535e625.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:32.1ex; height:3.343ex;" alt="{\displaystyle \sin x<0\quad {\text{für}}\quad x\in \left]180^{\circ },360^{\circ }\right[}" loading="lazy"></dd></dl>

<dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos x>0\quad {\text{für}}\quad x\in \left[0^{\circ },90^{\circ }\right[\cup \left]270^{\circ },360^{\circ }\right]}">

</mrow>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \cos x>0\quad {\text{für}}\quad x\in \left[0^{\circ },90^{\circ }\right[\cup \left]270^{\circ },360^{\circ }\right]}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/af94605331a13a2721b46d99396163d49faf800d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:42.861ex; height:3.343ex;" alt="{\displaystyle \cos x>0\quad {\text{für}}\quad x\in \left[0^{\circ },90^{\circ }\right[\cup \left]270^{\circ },360^{\circ }\right]}" loading="lazy"></dd></dl>

</mrow>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \cos x<0\quad {\text{für}}\quad x\in \left]90^{\circ },270^{\circ }\right[}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/518e3b67bc5d734254ae99debc89eb0aee804f3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:31.193ex; height:3.343ex;" alt="{\displaystyle \cos x<0\quad {\text{für}}\quad x\in \left]90^{\circ },270^{\circ }\right[}" loading="lazy"></dd></dl>

<dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tan x>0\quad {\text{für}}\quad x\in \left]0^{\circ },90^{\circ }\right[\cup \left]180^{\circ },270^{\circ }\right[}">

</mrow>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \tan x>0\quad {\text{für}}\quad x\in \left]0^{\circ },90^{\circ }\right[\cup \left]180^{\circ },270^{\circ }\right[}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/28fa7ed5c5b9ee66db9fa769dd73c51984ab1557.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:43.11ex; height:3.343ex;" alt="{\displaystyle \tan x>0\quad {\text{für}}\quad x\in \left]0^{\circ },90^{\circ }\right[\cup \left]180^{\circ },270^{\circ }\right[}" loading="lazy"></dd></dl>

</mrow>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \tan x<0\quad {\text{für}}\quad x\in \left]90^{\circ },180^{\circ }\right[\cup \left]270^{\circ },360^{\circ }\right[}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/330c10e3c7a41f96ff875db1ce209871c195f4b8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:45.435ex; height:3.343ex;" alt="{\displaystyle \tan x<0\quad {\text{für}}\quad x\in \left]90^{\circ },180^{\circ }\right[\cup \left]270^{\circ },360^{\circ }\right[}" loading="lazy"></dd></dl>

Die <a href="Vorzeichen_(Zahl)" title="Vorzeichen (Zahl)">Vorzeichen</a> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cot }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \cot }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c5bce4c1822b49c4e7ecd7b9aa07f2fcb1706817.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.099ex; height:2.009ex;" alt="{\displaystyle \cot }" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sec }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sec }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3dd93d9784c55e8ef40f81993114bd3c3d01084c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.981ex; height:1.676ex;" alt="{\displaystyle \sec }" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \csc }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \csc }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/399542cf3e9a873371132ffebe579187d48962ae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.981ex; height:1.676ex;" alt="{\displaystyle \csc }" loading="lazy"> stimmen überein mit denen ihrer Kehrwertfunktionen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tan }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \tan }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b57c91704d9ab0366a6436869e2968491efc5155.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.36ex; height:2.009ex;" alt="{\displaystyle \tan }" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \cos }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e473a3de151d75296f141f9f482fe59d582a7509.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.111ex; height:1.676ex;" alt="{\displaystyle \cos }" loading="lazy"> bzw. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sin }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ee55beec18afd710e7ab767964b915b020c65093.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.856ex; height:2.176ex;" alt="{\displaystyle \sin }" loading="lazy">.

<div class="mw-heading mw-heading3"><h3 id="Wichtige_Funktionswerte">Wichtige Funktionswerte</h3></div>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation>

</semantics>

</th>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation>

</semantics>

</th>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sin \alpha }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/071763dba322940a74766aaf79f4569c5954dcf6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.73ex; height:2.176ex;" alt="{\displaystyle \sin \alpha }" loading="lazy">

</th>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \cos \alpha }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/277355f4e11bfdd8864cdb3eada54c1f9f736d0b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.986ex; height:1.676ex;" alt="{\displaystyle \cos \alpha }" loading="lazy">

</th>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \tan \alpha }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1c622cda1e123d1a43bffbc0b8c1f57530cfc4e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.234ex; height:2.009ex;" alt="{\displaystyle \tan \alpha }" loading="lazy">

</th>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \cot \alpha }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/501fdd5fa1c1ed45cbfa50933e8ec3136070e6b5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.974ex; height:2.009ex;" alt="{\displaystyle \cot \alpha }" loading="lazy">

</th></tr>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0^{\circ }}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ffd0e1e92cf5770c2bfbb1de8b4b7bf904c9deef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.217ex; height:2.343ex;" alt="{\displaystyle 0^{\circ }}" loading="lazy">

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \,0}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7d8f9502c23a3583421ba7a0735330be129b2447.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \,0}" loading="lazy">

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \,0}</annotation>

</semantics>

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \,1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4af1ced4a2eda6aedb8bea3cc0738ef8a7b79452.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \,1}" loading="lazy">

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \,0}</annotation>

</semantics>

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \pm \infty }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c586ae37f8efec026b8a4ea3f6a5253576c2c4e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.132ex; height:2.176ex;" alt="{\displaystyle \pm \infty }" loading="lazy">

</td></tr>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 15^{\circ }}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a6b2ba390041accdbd09de910f720bd6e873d137.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.379ex; height:2.343ex;" alt="{\displaystyle 15^{\circ }}" loading="lazy">

</td>

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {\pi }{12}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c4003cb335c111eff28d24f990d71ce744bdbce2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:2.48ex; height:3.176ex;" alt="{\displaystyle {\tfrac {\pi }{12}}}" loading="lazy">

</td>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<msqrt>

</msqrt>

</mrow>

<msqrt>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{4}}({\sqrt {6}}-{\sqrt {2}})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/94170142bb0318c3f62a3de4c838ab5e6117e1d7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:12.504ex; height:3.509ex;" alt="{\displaystyle {\tfrac {1}{4}}({\sqrt {6}}-{\sqrt {2}})}" loading="lazy">

</td>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<msqrt>

</msqrt>

</mrow>

<msqrt>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{4}}({\sqrt {6}}+{\sqrt {2}})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3a456da8059e2a0e125bb309b52724ec105c921c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:12.504ex; height:3.509ex;" alt="{\displaystyle {\tfrac {1}{4}}({\sqrt {6}}+{\sqrt {2}})}" loading="lazy">

</td>

<msqrt>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 2-{\sqrt {3}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/df3c5bc3d7b3e1148963e3279582cc8f07a4b789.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.101ex; height:2.843ex;" alt="{\displaystyle 2-{\sqrt {3}}}" loading="lazy">

</td>

<msqrt>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 2+{\sqrt {3}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/442e3ac1b93fc34af6ec2955a62d030e4f818df5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.101ex; height:2.843ex;" alt="{\displaystyle 2+{\sqrt {3}}}" loading="lazy">

</td></tr>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 18^{\circ }}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/16f8ae67f2d56e4095fdf0d10ebc72183313de82.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.379ex; height:2.343ex;" alt="{\displaystyle 18^{\circ }}" loading="lazy">

</td>

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {\pi }{10}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/92d4412f525ba517b7c31c7056cfa691f4162e09.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:2.48ex; height:3.343ex;" alt="{\displaystyle {\tfrac {\pi }{10}}}" loading="lazy">

</td>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<mrow>

<mrow>

<msqrt>

</msqrt>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{4}}\left({\sqrt {5}}-1\right)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ac14cb52b40374b7bdc630767d704a3cb65e7e69.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:11.276ex; height:3.509ex;" alt="{\displaystyle {\tfrac {1}{4}}\left({\sqrt {5}}-1\right)}" loading="lazy">

</td>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<msqrt>

<msqrt>

</msqrt>

</mrow>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{4}}{\sqrt {10+2{\sqrt {5}}}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/25396f9653504beff444cf52b5b217ff3a1a5ae2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:13.408ex; height:4.843ex;" alt="{\displaystyle {\tfrac {1}{4}}{\sqrt {10+2{\sqrt {5}}}}}" loading="lazy">

</td>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<msqrt>

<msqrt>

</msqrt>

</mrow>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{5}}{\sqrt {25-10{\sqrt {5}}}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fac2a8ecec55f80ae4bc7757e2ca9b622f8a5638.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:14.57ex; height:4.843ex;" alt="{\displaystyle {\tfrac {1}{5}}{\sqrt {25-10{\sqrt {5}}}}}" loading="lazy">

</td>

<msqrt>

<msqrt>

</msqrt>

</mrow>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\sqrt {5+2{\sqrt {5}}}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/05bdf8d97798c643d8de3cf1ca55d3357815229e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:10.587ex; height:4.843ex;" alt="{\displaystyle {\sqrt {5+2{\sqrt {5}}}}}" loading="lazy">

</td></tr>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 30^{\circ }}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/42f29df9c101d2a8dae3f1552342cfe4c3adb76c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.379ex; height:2.343ex;" alt="{\displaystyle 30^{\circ }}" loading="lazy">

</td>

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {\pi }{6}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/75bc23c5dc0aac6e2066ae76ee4a5cf52fd3e19b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:1.778ex; height:3.343ex;" alt="{\displaystyle {\tfrac {\pi }{6}}}" loading="lazy">

</td>

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{2}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/edef8290613648790a8ac1a95c2fb7c3972aea2f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:1.658ex; height:3.509ex;" alt="{\displaystyle {\tfrac {1}{2}}}" loading="lazy">

</td>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<msqrt>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{2}}{\sqrt {3}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/69ed0b8e310a3c4fd60d8eba24dc63f326b5cf19.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:4.756ex; height:3.509ex;" alt="{\displaystyle {\tfrac {1}{2}}{\sqrt {3}}}" loading="lazy">

</td>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<msqrt>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{3}}{\sqrt {3}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1501ede357a50bf359be0ef745b9b3a5bd0df2b4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:4.756ex; height:3.676ex;" alt="{\displaystyle {\tfrac {1}{3}}{\sqrt {3}}}" loading="lazy">

</td>

<msqrt>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\sqrt {3}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3b19c09494138b5082459afac7f9a8d99c546fcd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.098ex; height:2.843ex;" alt="{\displaystyle {\sqrt {3}}}" loading="lazy">

</td></tr>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 36^{\circ }}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/34fb8091def5cc9caece92b514b07fa743cf2fc3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.379ex; height:2.343ex;" alt="{\displaystyle 36^{\circ }}" loading="lazy">

</td>

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {\pi }{5}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3dc8ac67580c4e2439889672d4b14d9b5ad67720.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:1.778ex; height:3.343ex;" alt="{\displaystyle {\tfrac {\pi }{5}}}" loading="lazy">

</td>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<msqrt>

<msqrt>

</msqrt>

</mrow>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{4}}{\sqrt {10-2{\sqrt {5}}}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/34eaf04b764379d9da2dd47604f6d40b1e220cef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:13.408ex; height:4.843ex;" alt="{\displaystyle {\tfrac {1}{4}}{\sqrt {10-2{\sqrt {5}}}}}" loading="lazy">

</td>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<mrow>

<mrow>

<msqrt>

</msqrt>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{4}}\left(1+{\sqrt {5}}\right)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9a55716e6401a3f1e5d5b60c5e224246659a869b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:11.276ex; height:3.509ex;" alt="{\displaystyle {\tfrac {1}{4}}\left(1+{\sqrt {5}}\right)}" loading="lazy">

</td>

<msqrt>

<msqrt>

</msqrt>

</mrow>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\sqrt {5-2{\sqrt {5}}}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/30681903de7b5edf35330e0df3a112470e8f8419.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:10.587ex; height:4.843ex;" alt="{\displaystyle {\sqrt {5-2{\sqrt {5}}}}}" loading="lazy">

</td>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<msqrt>

<msqrt>

</msqrt>

</mrow>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{5}}{\sqrt {25+10{\sqrt {5}}}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ec109f4bb5d2b0171923b08be1765dc640c3f86d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:14.57ex; height:4.843ex;" alt="{\displaystyle {\tfrac {1}{5}}{\sqrt {25+10{\sqrt {5}}}}}" loading="lazy">

</td></tr>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 45^{\circ }}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c28223ddedeb94a84bb15474cc64b5ce436cbe50.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.379ex; height:2.343ex;" alt="{\displaystyle 45^{\circ }}" loading="lazy">

</td>

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {\pi }{4}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a1f065c03576345b1b536c3cd513af53e8c35c5f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:1.778ex; height:3.176ex;" alt="{\displaystyle {\tfrac {\pi }{4}}}" loading="lazy">

</td>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<msqrt>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{2}}{\sqrt {2}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3fe7d9a258302af7674f63d30a212d746e62b29e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:4.756ex; height:3.509ex;" alt="{\displaystyle {\tfrac {1}{2}}{\sqrt {2}}}" loading="lazy">

</td>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<msqrt>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{2}}{\sqrt {2}}}</annotation>

</semantics>

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1\,}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bfd1e7984fe6e1b79a26404a8138a6c6ee41a476.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle 1\,}" loading="lazy">

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1\,}</annotation>

</semantics>

</td></tr>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 54^{\circ }}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/47c792d186c1239c4e574044a944723cb58dba48.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.379ex; height:2.343ex;" alt="{\displaystyle 54^{\circ }}" loading="lazy">

</td>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {3\pi }{10}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/32545c8405bd937079e45f70e79fcc5eebff2c6e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:2.6ex; height:3.676ex;" alt="{\displaystyle {\tfrac {3\pi }{10}}}" loading="lazy">

</td>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<mrow>

<mrow>

<msqrt>

</msqrt>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{4}}\left(1+{\sqrt {5}}\right)}</annotation>

</semantics>

</td>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<msqrt>

<msqrt>

</msqrt>

</mrow>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{4}}{\sqrt {10-2{\sqrt {5}}}}}</annotation>

</semantics>

</td>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<msqrt>

<msqrt>

</msqrt>

</mrow>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{5}}{\sqrt {25+10{\sqrt {5}}}}}</annotation>

</semantics>

</td>

<msqrt>

<msqrt>

</msqrt>

</mrow>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\sqrt {5-2{\sqrt {5}}}}}</annotation>

</semantics>

</td></tr>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 60^{\circ }}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/08c42292485b447b7f627a7accd90d5b439c11d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.379ex; height:2.343ex;" alt="{\displaystyle 60^{\circ }}" loading="lazy">

</td>

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {\pi }{3}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/35e0953cd4cd6fa5a3f4b473c8bd0cf32bbb5c7e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:1.778ex; height:3.343ex;" alt="{\displaystyle {\tfrac {\pi }{3}}}" loading="lazy">

</td>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<msqrt>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{2}}{\sqrt {3}}}</annotation>

</semantics>

</td>

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{2}}}</annotation>

</semantics>

</td>

<msqrt>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\sqrt {3}}}</annotation>

</semantics>

</td>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<msqrt>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{3}}{\sqrt {3}}}</annotation>

</semantics>

</td></tr>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 72^{\circ }}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e0cfa1907f003a92fb16d6a32aa7a4f5da99f112.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.379ex; height:2.343ex;" alt="{\displaystyle 72^{\circ }}" loading="lazy">

</td>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {2\pi }{5}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5508e6906bf83215949906383498fa5b3402b730.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:2.6ex; height:3.676ex;" alt="{\displaystyle {\tfrac {2\pi }{5}}}" loading="lazy">

</td>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<msqrt>

<msqrt>

</msqrt>

</mrow>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{4}}{\sqrt {10+2{\sqrt {5}}}}}</annotation>

</semantics>

</td>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<mrow>

<mrow>

<msqrt>

</msqrt>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{4}}\left({\sqrt {5}}-1\right)}</annotation>

</semantics>

</td>

<msqrt>

<msqrt>

</msqrt>

</mrow>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\sqrt {5+2{\sqrt {5}}}}}</annotation>

</semantics>

</td>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<msqrt>

<msqrt>

</msqrt>

</mrow>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{5}}{\sqrt {25-10{\sqrt {5}}}}}</annotation>

</semantics>

</td></tr>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 75^{\circ }}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/565c6cb6b28eeab72c86cdb73940a1ad0d2d1d26.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.379ex; height:2.343ex;" alt="{\displaystyle 75^{\circ }}" loading="lazy">

</td>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {5\pi }{12}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/400cec5f3cfa4c977c82ec1002e208ea770c244b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:2.6ex; height:3.509ex;" alt="{\displaystyle {\tfrac {5\pi }{12}}}" loading="lazy">

</td>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<msqrt>

</msqrt>

</mrow>

<msqrt>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{4}}({\sqrt {6}}+{\sqrt {2}})}</annotation>

</semantics>

</td>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<msqrt>

</msqrt>

</mrow>

<msqrt>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{4}}({\sqrt {6}}-{\sqrt {2}})}</annotation>

</semantics>

</td>

<msqrt>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 2+{\sqrt {3}}}</annotation>

</semantics>

</td>

<msqrt>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 2-{\sqrt {3}}}</annotation>

</semantics>

</td></tr>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 90^{\circ }}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c326d317eddef3ad3e6625e018a708e290a039f6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.379ex; height:2.343ex;" alt="{\displaystyle 90^{\circ }}" loading="lazy">

</td>

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {\pi }{2}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b4e31a202557dfbf326b44ebcc914ba3ab08fff1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:1.778ex; height:3.176ex;" alt="{\displaystyle {\tfrac {\pi }{2}}}" loading="lazy">

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \,1}</annotation>

</semantics>

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \,0}</annotation>

</semantics>

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \pm \infty }</annotation>

</semantics>

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \,0}</annotation>

</semantics>

</td></tr>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 108^{\circ }}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fd905b3d50a310d3eee16e9024ceb780ee15e3c7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.542ex; height:2.343ex;" alt="{\displaystyle 108^{\circ }}" loading="lazy">

</td>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {3\pi }{5}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e0f2a248c0fda2e4b70860d198f3882259564ae2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:2.6ex; height:3.676ex;" alt="{\displaystyle {\tfrac {3\pi }{5}}}" loading="lazy">

</td>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<msqrt>

<msqrt>

</msqrt>

</mrow>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{4}}{\sqrt {10+2{\sqrt {5}}}}}</annotation>

</semantics>

</td>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<mrow>

<mrow>

<msqrt>

</msqrt>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{4}}\left(1-{\sqrt {5}}\right)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/aa85234034357d11d872183e80f3e197f4526b82.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:11.276ex; height:3.509ex;" alt="{\displaystyle {\tfrac {1}{4}}\left(1-{\sqrt {5}}\right)}" loading="lazy">

</td>

<msqrt>

<msqrt>

</msqrt>

</mrow>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -{\sqrt {5+2{\sqrt {5}}}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/72cde02043a244bcb2448d3175a20ed047615774.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:12.396ex; height:4.843ex;" alt="{\displaystyle -{\sqrt {5+2{\sqrt {5}}}}}" loading="lazy">

</td>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<msqrt>

<msqrt>

</msqrt>

</mrow>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -{\tfrac {1}{5}}{\sqrt {25-10{\sqrt {5}}}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1020e3258b0edca4c044479accc28eee805f0dd8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:16.379ex; height:4.843ex;" alt="{\displaystyle -{\tfrac {1}{5}}{\sqrt {25-10{\sqrt {5}}}}}" loading="lazy">

</td></tr>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 120^{\circ }}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/00c3c12a803847f3a6e77045c89a59de1855c453.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.542ex; height:2.343ex;" alt="{\displaystyle 120^{\circ }}" loading="lazy">

</td>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {2\pi }{3}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d347f5b40db81e556675c27ed6da65079059c5d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:2.6ex; height:3.676ex;" alt="{\displaystyle {\tfrac {2\pi }{3}}}" loading="lazy">

</td>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<msqrt>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{2}}{\sqrt {3}}}</annotation>

</semantics>

</td>

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -{\tfrac {1}{2}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b0f99130274f09ef6b857419bf5c2529a6423597.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:3.466ex; height:3.509ex;" alt="{\displaystyle -{\tfrac {1}{2}}}" loading="lazy">

</td>

<msqrt>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -{\sqrt {3}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4c7cc8150f4cf1328d10ea1883d66f073136647b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.906ex; height:2.843ex;" alt="{\displaystyle -{\sqrt {3}}}" loading="lazy">

</td>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<msqrt>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -{\tfrac {1}{3}}{\sqrt {3}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/11abf634134e925b139bde18d163d97e8e9bfd92.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:6.565ex; height:3.676ex;" alt="{\displaystyle -{\tfrac {1}{3}}{\sqrt {3}}}" loading="lazy">

</td></tr>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 135^{\circ }}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f560757baeebf2413d9c813ce362119a391af1e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.542ex; height:2.343ex;" alt="{\displaystyle 135^{\circ }}" loading="lazy">

</td>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {3\pi }{4}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5d07bd3db593a831f431ecfac005cc32102ea760.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:2.6ex; height:3.509ex;" alt="{\displaystyle {\tfrac {3\pi }{4}}}" loading="lazy">

</td>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<msqrt>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{2}}{\sqrt {2}}}</annotation>

</semantics>

</td>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<msqrt>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -{\tfrac {1}{2}}{\sqrt {2}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e32c9a144b868a156390034357a1185dc581b062.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:6.565ex; height:3.509ex;" alt="{\displaystyle -{\tfrac {1}{2}}{\sqrt {2}}}" loading="lazy">

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -1\,}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f821a3bcf09229fc33b355f35f6a91e7b1952c04.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:3.358ex; height:2.343ex;" alt="{\displaystyle -1\,}" loading="lazy">

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -1\,}</annotation>

</semantics>

</td></tr>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 180^{\circ }}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c5d0431ce231935522dc0cb52df7f2b406cdadc3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.542ex; height:2.343ex;" alt="{\displaystyle 180^{\circ }}" loading="lazy">

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \pi \,}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b64589f24cc8e145d021afa17d6564d55ea5a95c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.719ex; height:1.676ex;" alt="{\displaystyle \pi \,}" loading="lazy">

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \,0}</annotation>

</semantics>

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \,-1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ed955aaf03fe246e5dca2d8840e37adf17d22f25.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:3.358ex; height:2.343ex;" alt="{\displaystyle \,-1}" loading="lazy">

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \,0}</annotation>

</semantics>

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \pm \infty }</annotation>

</semantics>

</td></tr>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 270^{\circ }}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1d46a7b7ced26e658ef8ae2a8feaf5fe3113193f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.542ex; height:2.343ex;" alt="{\displaystyle 270^{\circ }}" loading="lazy">

</td>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {3\pi }{2}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1d3931916b7162c3b40be4327140f391b5d495b2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:2.6ex; height:3.509ex;" alt="{\displaystyle {\tfrac {3\pi }{2}}}" loading="lazy">

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \,-1}</annotation>

</semantics>

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \,0}</annotation>

</semantics>

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \pm \infty }</annotation>

</semantics>

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \,0}</annotation>

</semantics>

</td></tr>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 360^{\circ }}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cb451712d2b1f32ec132e9cf56eca04a64cad60e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.542ex; height:2.343ex;" alt="{\displaystyle 360^{\circ }}" loading="lazy">

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 2\pi }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/73efd1f6493490b058097060a572606d2c550a06.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.494ex; height:2.176ex;" alt="{\displaystyle 2\pi }" loading="lazy">

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \,0}</annotation>

</semantics>

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \,1}</annotation>

</semantics>

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \,0}</annotation>

</semantics>

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \pm \infty }</annotation>

</semantics>

</td></tr></tbody></table>

Mit Hilfe der Additionstheoreme sind noch viele weitere Werte durch algebraische Ausdrücke (ggfs. mit verschachtelten Quadratwurzeln) darstellbar, zum Beispiel alle ganzzahligen Vielfachen von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3^{\circ }}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 3^{\circ }}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a07e3862e33e06b4a4204f7b4e36ba0da4658e0c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.217ex; height:2.343ex;" alt="{\displaystyle 3^{\circ }}" loading="lazy"> durch die Subtraktionen: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 18^{\circ }-15^{\circ }}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 18^{\circ }-15^{\circ }}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/20fe084af9ac4eb781c191f7622db0878f9a5eb9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.599ex; height:2.509ex;" alt="{\displaystyle 18^{\circ }-15^{\circ }}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 36^{\circ }-30^{\circ }}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 36^{\circ }-30^{\circ }}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/07a28623b309402e04ae031bd7f48ef1d66c7d58.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.599ex; height:2.509ex;" alt="{\displaystyle 36^{\circ }-30^{\circ }}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 45^{\circ }-36^{\circ }}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 45^{\circ }-36^{\circ }}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/51fe192b91cfb1a6b730f4b9b1450aba407301dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.599ex; height:2.509ex;" alt="{\displaystyle 45^{\circ }-36^{\circ }}" loading="lazy"> usw.<a href="#cite_note-2">[2]</a>

<div class="mw-heading mw-heading3"><h3 id="Symmetrien">Symmetrien</h3></div>

Die trigonometrischen Funktionen haben einfache Symmetrien:

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\sin(-x)&=-\sin(x)\\\cos(-x)&=+\cos(x)\\\tan(-x)&=-\tan(x)\\\cot(-x)&=-\cot(x)\\\sec(-x)&=+\sec(x)\\\csc(-x)&=-\csc(x)\\\end{aligned}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/49eabf0318c42fce7203c07f58ce6c48fa1b05a1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.671ex; width:20.851ex; height:18.509ex;" alt="{\displaystyle {\begin{aligned}\sin(-x)&=-\sin(x)\\\cos(-x)&=+\cos(x)\\\tan(-x)&=-\tan(x)\\\cot(-x)&=-\cot(x)\\\sec(-x)&=+\sec(x)\\\csc(-x)&=-\csc(x)\\\end{aligned}}}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Phasenverschiebungen">Phasenverschiebungen</h3></div>

<mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sin \left(x+{\frac {\pi }{2}}\right)=\cos x\;\quad {\text{bzw.}}\quad \sin \left(x+90^{\circ }\right)=\cos x\;}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d6f15b519e7cfd7df36e90e6e7a54915ff7a4632.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:51.009ex; height:4.843ex;" alt="{\displaystyle \sin \left(x+{\frac {\pi }{2}}\right)=\cos x\;\quad {\text{bzw.}}\quad \sin \left(x+90^{\circ }\right)=\cos x\;}" loading="lazy"></dd>

<mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \cos \left(x+{\frac {\pi }{2}}\right)=-\sin x\;\quad {\text{bzw.}}\quad \cos \left(x+90^{\circ }\right)=-\sin x\;}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1aeef04cf4e11b3f5b6033fdbbc9966595f83faa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:55.399ex; height:4.843ex;" alt="{\displaystyle \cos \left(x+{\frac {\pi }{2}}\right)=-\sin x\;\quad {\text{bzw.}}\quad \cos \left(x+90^{\circ }\right)=-\sin x\;}" loading="lazy"></dd></dl>

<mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \tan \left(x+{\frac {\pi }{2}}\right)=-\cot x\;\quad {\text{bzw.}}\quad \tan \left(x+90^{\circ }\right)=-\cot x\;}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5ffa7095211675280c890d2eb511c3da38edcdeb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:56.384ex; height:4.843ex;" alt="{\displaystyle \tan \left(x+{\frac {\pi }{2}}\right)=-\cot x\;\quad {\text{bzw.}}\quad \tan \left(x+90^{\circ }\right)=-\cot x\;}" loading="lazy"></dd></dl>

<mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \cot \left(x+{\frac {\pi }{2}}\right)=-\tan x\;\quad {\text{bzw.}}\quad \cot \left(x+90^{\circ }\right)=-\tan x\;}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a8245fcdf9b7221c7ffab7f8952e793af329a1aa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:56.384ex; height:4.843ex;" alt="{\displaystyle \cot \left(x+{\frac {\pi }{2}}\right)=-\tan x\;\quad {\text{bzw.}}\quad \cot \left(x+90^{\circ }\right)=-\tan x\;}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Rückführung_auf_spitze_Winkel">Rückführung auf spitze Winkel</h3></div>

<mrow>

<mrow>

</mrow>

</mrow>

</mrow>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sin x\ \;=\;\;\;\sin \left(\pi -x\right)\,\quad {\text{bzw.}}\quad \sin x\ =\;\;\;\sin \left(180^{\circ }-x\right)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7d6d53f3be5eb04c7598c7973f71f4ff7393cbcc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:54.632ex; height:2.843ex;" alt="{\displaystyle \sin x\ \;=\;\;\;\sin \left(\pi -x\right)\,\quad {\text{bzw.}}\quad \sin x\ =\;\;\;\sin \left(180^{\circ }-x\right)}" loading="lazy"></dd>

<mrow>

<mrow>

</mrow>

</mrow>

</mrow>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \cos x\ \,=-\cos \left(\pi -x\right)\quad {\text{bzw.}}\quad \cos x\ =-\cos \left(180^{\circ }-x\right)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2cd3b6ceeb5f97594b26c76257c83501088f28d4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:55.528ex; height:2.843ex;" alt="{\displaystyle \cos x\ \,=-\cos \left(\pi -x\right)\quad {\text{bzw.}}\quad \cos x\ =-\cos \left(180^{\circ }-x\right)}" loading="lazy"></dd>

<mrow>

<mrow>

</mrow>

</mrow>

</mrow>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \tan x\ =-\tan \left(\pi -x\right)\quad {\text{bzw.}}\quad \tan x\ =-\tan \left(180^{\circ }-x\right)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b73e1edccfb614e51ce9bca609dec6f801208453.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:56.135ex; height:2.843ex;" alt="{\displaystyle \tan x\ =-\tan \left(\pi -x\right)\quad {\text{bzw.}}\quad \tan x\ =-\tan \left(180^{\circ }-x\right)}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Darstellung_durch_den_Tangens_des_halben_Winkels">Darstellung durch den Tangens des halben Winkels</h3></div>

Mit der Bezeichnung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t=\tan {\tfrac {x}{2}}}">

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t=\tan {\tfrac {x}{2}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ef970dda45f18d891ea64d38b3b8273f0400426d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:9.461ex; height:3.176ex;" alt="{\displaystyle t=\tan {\tfrac {x}{2}}}" loading="lazy">

gelten die folgenden Beziehungen für beliebiges <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

<mfrac>

<mrow>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sin x={\frac {2t}{1+t^{2}}},}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0fed3024bf3d63914829586010a4e0c2539330f8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:15.051ex; height:5.676ex;" alt="{\displaystyle \sin x={\frac {2t}{1+t^{2}}},}" loading="lazy">

</td>

<td>

</td>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \cos x={\frac {1-t^{2}}{1+t^{2}}},}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7f5e18db282b53fc7e04f9c4eb02389b1891d52e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:15.306ex; height:6.176ex;" alt="{\displaystyle \cos x={\frac {1-t^{2}}{1+t^{2}}},}" loading="lazy">

</td></tr>

<tr>

<mfrac>

<mrow>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \tan x={\frac {2t}{1-t^{2}}},}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/45ab39b98d6f41b622c8fc0276fcc240c4b43df2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:15.555ex; height:5.676ex;" alt="{\displaystyle \tan x={\frac {2t}{1-t^{2}}},}" loading="lazy">

</td>

<td>

</td>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \cot x={\frac {1-t^{2}}{2t}},}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ff6100e12924c779c8012e3fb629c9b255dc13c8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:15.294ex; height:5.676ex;" alt="{\displaystyle \cot x={\frac {1-t^{2}}{2t}},}" loading="lazy">

</td></tr>

<tr>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sec x={\frac {1+t^{2}}{1-t^{2}}},}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7104bffe4643feefe6840abe404438a71ebdb7b5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:15.176ex; height:6.176ex;" alt="{\displaystyle \sec x={\frac {1+t^{2}}{1-t^{2}}},}" loading="lazy">

</td>

<td>

</td>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \csc x={\frac {1+t^{2}}{2t}}.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5688edf9cacf222cbd46a828ffa4783cc51cfe72.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:15.176ex; height:5.676ex;" alt="{\displaystyle \csc x={\frac {1+t^{2}}{2t}}.}" loading="lazy">

</td></tr></tbody></table>

<div class="mw-heading mw-heading3"><h3 id="Additionstheoreme">Additionstheoreme</h3></div>

Für <a href="Sinus_und_Kosinus" title="Sinus und Kosinus">Sinus und Kosinus</a> lassen sich die Additionstheoreme aus der Verkettung zweier Drehungen um den Winkel <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b8a6208ec717213d4317e666f1ae872e00620a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" loading="lazy"> herleiten. Das ist elementargeometrisch möglich; sehr viel einfacher ist das koordinatenweise Ablesen der Formeln aus dem <a href="Drehmatrix" title="Drehmatrix">Produkt zweier Drehmatrizen der Ebene <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{2}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e150115ab9f63023215109595b76686a1ff890fd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{2}}" loading="lazy"></a>. Alternativ folgen die Additionstheoreme aus der Anwendung der <a href="Eulersche_Formel#Eulersche_Formel" title="Eulersche Formel">Eulerschen Formel</a> auf die Beziehung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle e^{i(x+y)}=e^{ix}\cdot e^{iy}}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle e^{i(x+y)}=e^{ix}\cdot e^{iy}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bf8784c2845d219aa33fc7a84a75213ddddeb241.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:16.5ex; height:2.676ex;" alt="{\displaystyle \textstyle e^{i(x+y)}=e^{ix}\cdot e^{iy}}" loading="lazy">. Die Ergebnisse für das Doppelvorzeichen ergeben sich durch Anwendung der <a class="mw-selflink-fragment" href="#Symmetrien">Symmetrien</a>.<a href="#cite_note-3">[3]</a>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sin(x\pm y)=\sin x\cdot \cos y\pm \cos x\cdot \sin y}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6586aa07fe77e9ccd8617dbd587397a0ba025fe2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:37.74ex; height:2.843ex;" alt="{\displaystyle \sin(x\pm y)=\sin x\cdot \cos y\pm \cos x\cdot \sin y}" loading="lazy"><a href="#cite_note-forster_anal1-4">[4]</a></dd></dl>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \cos(x\pm y)=\cos x\cdot \cos y\mp \sin x\cdot \sin y}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1a52e5b827da7c8844079ed60c12a14318599303.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:37.995ex; height:2.843ex;" alt="{\displaystyle \cos(x\pm y)=\cos x\cdot \cos y\mp \sin x\cdot \sin y}" loading="lazy"><a href="#cite_note-forster_anal1-4">[4]</a></dd></dl>

Geometrische Herleitungen sind in Figur 1 und Figur 2 für Winkel <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b79333175c8b3f0840bfb4ec41b8072c83ea88d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \beta }</annotation>

</semantics>

Zu Figur 1:

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sin(\alpha +\beta )=\sin \alpha \cdot \cos \beta +\cos \alpha \cdot \sin \beta }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2bca99e0341886603f6ab710fa8b1cd6df754082.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:38.743ex; height:2.843ex;" alt="{\displaystyle \sin(\alpha +\beta )=\sin \alpha \cdot \cos \beta +\cos \alpha \cdot \sin \beta }" loading="lazy"></dd>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \cos(\alpha +\beta )=\cos \alpha \cdot \cos \beta -\sin \alpha \cdot \sin \beta }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8d3aff26f93fac9f03b64c54e11a13dbd2984ca7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:38.999ex; height:2.843ex;" alt="{\displaystyle \cos(\alpha +\beta )=\cos \alpha \cdot \cos \beta -\sin \alpha \cdot \sin \beta }" loading="lazy"></dd></dl>

Zu Figur 2:

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sin(\alpha -\beta )=\sin \alpha \cdot \cos \beta -\cos \alpha \cdot \sin \beta }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d91c8efff01d232a09e399e2393eecfa8ce8d35e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:38.743ex; height:2.843ex;" alt="{\displaystyle \sin(\alpha -\beta )=\sin \alpha \cdot \cos \beta -\cos \alpha \cdot \sin \beta }" loading="lazy"></dd>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \cos(\alpha -\beta )=\cos \alpha \cdot \cos \beta +\sin \alpha \cdot \sin \beta }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/90eaeb20ca8842a067c667ddfd58b19015bcb760.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:38.999ex; height:2.843ex;" alt="{\displaystyle \cos(\alpha -\beta )=\cos \alpha \cdot \cos \beta +\sin \alpha \cdot \sin \beta }" loading="lazy"></dd></dl>

Durch Erweiterung mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle {1 \over \cos x\cos y}}">

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle {1 \over \cos x\cos y}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1db7489d2eb01eb1dc4e9f6d77666aa67c13c5bb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:8.154ex; height:3.676ex;" alt="{\displaystyle \textstyle {1 \over \cos x\cos y}}" loading="lazy"> bzw. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle {1 \over \sin x\sin y}}">

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle {1 \over \sin x\sin y}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/85372887a7805cd1c4693133bad7362a4b16602b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:7.793ex; height:3.843ex;" alt="{\displaystyle \textstyle {1 \over \sin x\sin y}}" loading="lazy"> und Vereinfachung des Doppelbruchs erhält man

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \tan(x\pm y)={\frac {\sin(x\pm y)}{\cos(x\pm y)}}={\frac {\tan x\pm \tan y}{1\mp \tan x\;\tan y}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2c07598e198624b2952942ddbd9ba8cdf4ff93ba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:43.623ex; height:6.509ex;" alt="{\displaystyle \tan(x\pm y)={\frac {\sin(x\pm y)}{\cos(x\pm y)}}={\frac {\tan x\pm \tan y}{1\mp \tan x\;\tan y}}}" loading="lazy">,</dd></dl>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \cot(x\pm y)={\frac {\cos(x\pm y)}{\sin(x\pm y)}}={\frac {\cot x\cot y\mp 1}{\cot y\pm \cot x}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c4ffafd223fa4565d7f72778bc8aacaa83ca7119.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:42.198ex; height:6.509ex;" alt="{\displaystyle \cot(x\pm y)={\frac {\cos(x\pm y)}{\sin(x\pm y)}}={\frac {\cot x\cot y\mp 1}{\cot y\pm \cot x}}}" loading="lazy">.</dd></dl>

Für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x=y}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/409a91214d63eabe46ec10ff3cbba689ab687366.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.584ex; height:2.009ex;" alt="{\displaystyle x=y}" loading="lazy"> folgen hieraus die <a class="mw-selflink-fragment" href="#Doppelwinkelfunktionen">Doppelwinkelfunktionen</a>, für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=\pi /2}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y=\pi /2}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4dcd5ad6ff49e351c5c37f112a6ba624df0bf19c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.911ex; height:2.843ex;" alt="{\displaystyle y=\pi /2}" loading="lazy"> die <a class="mw-selflink-fragment" href="#Phasenverschiebungen">Phasenverschiebungen</a>.

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sin(x+y)\cdot \sin(x-y)=\cos ^{2}y-\cos ^{2}x=\sin ^{2}x-\sin ^{2}y}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a117161ac4a44022dae917d573db51744d5f6479.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:56.207ex; height:3.176ex;" alt="{\displaystyle \sin(x+y)\cdot \sin(x-y)=\cos ^{2}y-\cos ^{2}x=\sin ^{2}x-\sin ^{2}y}" loading="lazy"></dd></dl>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \cos(x+y)\cdot \cos(x-y)=\cos ^{2}y-\sin ^{2}x=\cos ^{2}x-\sin ^{2}y}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3b940a0d1f7cb9846f913cc4935de4516614d83c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:56.718ex; height:3.176ex;" alt="{\displaystyle \cos(x+y)\cdot \cos(x-y)=\cos ^{2}y-\sin ^{2}x=\cos ^{2}x-\sin ^{2}y}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Additionstheoreme_für_Arkusfunktionen">Additionstheoreme für Arkusfunktionen</h3></div>

Für die <a href="Arkusfunktion" title="Arkusfunktion">Arkusfunktionen</a> gelten folgende Additionstheoreme<a href="#cite_note-6">[6]</a>

<th>Summanden

</th>

<th>Summenformel

</th>

<th>Gültigkeitsbereich

</th></tr>

<mi>arcsin</mi>

<mi>arcsin</mi>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \arcsin x+\arcsin y=}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/45a50a60ed0c857ea47d1e7be8540ab188bd09f5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:20.477ex; height:2.509ex;" alt="{\displaystyle \arcsin x+\arcsin y=}" loading="lazy">

</td>

<mi>arcsin</mi>

<mrow>

<mrow>

<msqrt>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

<msqrt>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \arcsin \left(x{\sqrt {1-y^{2}}}+y{\sqrt {1-x^{2}}}\right)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cedcdf7b32f8b3966eeced5743a72fa78bbad0dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:31.961ex; height:6.176ex;" alt="{\displaystyle \arcsin \left(x{\sqrt {1-y^{2}}}+y{\sqrt {1-x^{2}}}\right)}" loading="lazy">

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle xy\leq 0}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/15d4477ac83d6ae1b50731864c0303c06ad7858d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.746ex; height:2.509ex;" alt="{\displaystyle xy\leq 0}" loading="lazy"> oder <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{2}+y^{2}\leq 1}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x^{2}+y^{2}\leq 1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e7d9d28b54f7d979fbab09df2e3842a4135bda29.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.7ex; height:3.009ex;" alt="{\displaystyle x^{2}+y^{2}\leq 1}" loading="lazy">

</td></tr>

<td>

</td>

<mi>arcsin</mi>

<mrow>

<mrow>

<msqrt>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

<msqrt>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \pi -\arcsin \left(x{\sqrt {1-y^{2}}}+y{\sqrt {1-x^{2}}}\right)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/178e39c91f3db764bc1d9d46f6fb08ec6245e768.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:36.133ex; height:6.176ex;" alt="{\displaystyle \pi -\arcsin \left(x{\sqrt {1-y^{2}}}+y{\sqrt {1-x^{2}}}\right)}" loading="lazy">

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x>0}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/80d24be5f0eb4a9173da6038badc8659546021d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;" alt="{\displaystyle x>0}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y>0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y>0}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c973e3cbfee5d7ab9ca2348b578b6ec19a8c019a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.416ex; height:2.509ex;" alt="{\displaystyle y>0}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{2}+y^{2}>1}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x^{2}+y^{2}>1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cf9b45aa7f66fcde0136bc04b12a910039123911.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.7ex; height:3.009ex;" alt="{\displaystyle x^{2}+y^{2}>1}" loading="lazy">

</td></tr>

<td>

</td>

<mi>arcsin</mi>

<mrow>

<mrow>

<msqrt>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

<msqrt>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -\pi -\arcsin \left(x{\sqrt {1-y^{2}}}+y{\sqrt {1-x^{2}}}\right)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9a0c38d6e1256ed16392ddec98fb88c8daff5657.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:37.941ex; height:6.176ex;" alt="{\displaystyle -\pi -\arcsin \left(x{\sqrt {1-y^{2}}}+y{\sqrt {1-x^{2}}}\right)}" loading="lazy">

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x<0}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1a4dbbf970b2d2863dcab589eafe006f08e727d7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;" alt="{\displaystyle x<0}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y<0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y<0}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/dd6a57e5eb282c81c2bb6f5e313012fa77bc08a4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.416ex; height:2.509ex;" alt="{\displaystyle y<0}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{2}+y^{2}>1}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x^{2}+y^{2}>1}</annotation>

</semantics>

</td></tr>

<mi>arcsin</mi>

<mi>arcsin</mi>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \arcsin x-\arcsin y=}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/88acab736311af0e25c6443b5045fb7fa200ae64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:20.477ex; height:2.509ex;" alt="{\displaystyle \arcsin x-\arcsin y=}" loading="lazy">

</td>

<mi>arcsin</mi>

<mrow>

<mrow>

<msqrt>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

<msqrt>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \arcsin \left(x{\sqrt {1-y^{2}}}-y{\sqrt {1-x^{2}}}\right)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fff4362b6f0c5166fe04f06fd2641d7f8cac86c4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:31.961ex; height:6.176ex;" alt="{\displaystyle \arcsin \left(x{\sqrt {1-y^{2}}}-y{\sqrt {1-x^{2}}}\right)}" loading="lazy">

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle xy\geq 0}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d7d706d1523aed20dc910962ade635dd7ccabe5b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.746ex; height:2.509ex;" alt="{\displaystyle xy\geq 0}" loading="lazy"> oder <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{2}+y^{2}\leq 1}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x^{2}+y^{2}\leq 1}</annotation>

</semantics>

</td></tr>

<td>

</td>

<mi>arcsin</mi>

<mrow>

<mrow>

<msqrt>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

<msqrt>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \pi -\arcsin \left(x{\sqrt {1-y^{2}}}-y{\sqrt {1-x^{2}}}\right)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6ac1ae50a8b23e74caff9a6c78885fe541f18691.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:36.133ex; height:6.176ex;" alt="{\displaystyle \pi -\arcsin \left(x{\sqrt {1-y^{2}}}-y{\sqrt {1-x^{2}}}\right)}" loading="lazy">

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x>0}</annotation>

</semantics>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y<0}</annotation>

</semantics>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x^{2}+y^{2}>1}</annotation>

</semantics>

</td></tr>

<td>

</td>

<mi>arcsin</mi>

<mrow>

<mrow>

<msqrt>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

<msqrt>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -\pi -\arcsin \left(x{\sqrt {1-y^{2}}}-y{\sqrt {1-x^{2}}}\right)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/866a4601f4d821411912038ece1677ae391b80b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:37.941ex; height:6.176ex;" alt="{\displaystyle -\pi -\arcsin \left(x{\sqrt {1-y^{2}}}-y{\sqrt {1-x^{2}}}\right)}" loading="lazy">

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x<0}</annotation>

</semantics>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y>0}</annotation>

</semantics>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x^{2}+y^{2}>1}</annotation>

</semantics>

</td></tr>

<mi>arccos</mi>

<mi>arccos</mi>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \arccos x+\arccos y=}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/277318e793ab2c28a2472e40e77592701b2973b8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:20.988ex; height:2.343ex;" alt="{\displaystyle \arccos x+\arccos y=}" loading="lazy">

</td>

<mi>arccos</mi>

<mrow>

<mrow>

<msqrt>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

<msqrt>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \arccos \left(xy-{\sqrt {1-x^{2}}}{\sqrt {1-y^{2}}}\right)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/43c45fdd541fd792640ceffe09e2fd973a2ea16d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:32.216ex; height:6.176ex;" alt="{\displaystyle \arccos \left(xy-{\sqrt {1-x^{2}}}{\sqrt {1-y^{2}}}\right)}" loading="lazy">

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x+y\geq 0}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/90b99efa1928e7536fafa9443e9a39580de3f75a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.586ex; height:2.509ex;" alt="{\displaystyle x+y\geq 0}" loading="lazy">

</td></tr>

<td>

</td>

<mi>arccos</mi>

<mrow>

<mrow>

<msqrt>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

<msqrt>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 2\pi -\arccos \left(xy-{\sqrt {1-x^{2}}}{\sqrt {1-y^{2}}}\right)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8a4584c5dfd7707362783d0fdf2662b2d45b171d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:37.551ex; height:6.176ex;" alt="{\displaystyle 2\pi -\arccos \left(xy-{\sqrt {1-x^{2}}}{\sqrt {1-y^{2}}}\right)}" loading="lazy">

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x+y<0}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a040cadbf2719f4e54c31f5fecac531a3fa525e2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.586ex; height:2.509ex;" alt="{\displaystyle x+y<0}" loading="lazy">

</td></tr>

<mi>arccos</mi>

<mi>arccos</mi>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \arccos x-\arccos y=}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a39c26f1a3690d659a40850e4814758e64f281cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:20.988ex; height:2.343ex;" alt="{\displaystyle \arccos x-\arccos y=}" loading="lazy">

</td>

<mi>arccos</mi>

<mrow>

<mrow>

<msqrt>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

<msqrt>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -\arccos \left(xy+{\sqrt {1-x^{2}}}{\sqrt {1-y^{2}}}\right)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ae51f9679fac86324985c36c3ee45f55ac54f6eb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:34.411ex; height:6.176ex;" alt="{\displaystyle -\arccos \left(xy+{\sqrt {1-x^{2}}}{\sqrt {1-y^{2}}}\right)}" loading="lazy">

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\geq y}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/aded94d634c48071188bf96a76a4d3b7dfb28470.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.584ex; height:2.343ex;" alt="{\displaystyle x\geq y}" loading="lazy">

</td></tr>

<td>

</td>

<mi>arccos</mi>

<mrow>

<mrow>

<msqrt>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

<msqrt>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \arccos \left(xy+{\sqrt {1-x^{2}}}{\sqrt {1-y^{2}}}\right)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f15689822d011bc32894e89572f3be4cf81e8aec.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:32.216ex; height:6.176ex;" alt="{\displaystyle \arccos \left(xy+{\sqrt {1-x^{2}}}{\sqrt {1-y^{2}}}\right)}" loading="lazy">

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x<y}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/aeb239de6fee56ea8b6a65f7858d95b87632069f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.584ex; height:2.176ex;" alt="{\displaystyle x<y}" loading="lazy">

</td></tr>

<mi>arctan</mi>

<mi>arctan</mi>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \arctan x+\arctan y=}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f02ac44f9ed2c8689695df5f4435e4d604c82d1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:21.485ex; height:2.343ex;" alt="{\displaystyle \arctan x+\arctan y=}" loading="lazy">

</td>

<mi>arctan</mi>

<mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \arctan \left({\frac {x+y}{1-xy}}\right)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b050e3c95185937f421e7a1121a204b3ac86ea9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:17.211ex; height:6.176ex;" alt="{\displaystyle \arctan \left({\frac {x+y}{1-xy}}\right)}" loading="lazy">

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle xy<1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/52a37f9da078cc9727bc6b0b6c93725260ce8460.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.746ex; height:2.509ex;" alt="{\displaystyle xy<1}" loading="lazy">

</td></tr>

<td>

</td>

<mi>arctan</mi>

<mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \pi +\arctan \left({\frac {x+y}{1-xy}}\right)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/aa61a90f6ad63f7dc9da1584538b7ceec94b5150.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:21.384ex; height:6.176ex;" alt="{\displaystyle \pi +\arctan \left({\frac {x+y}{1-xy}}\right)}" loading="lazy">

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x>0}</annotation>

</semantics>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle xy>1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4f8224f0c76cdb965091f0f8e29856ff2dba79f2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.746ex; height:2.509ex;" alt="{\displaystyle xy>1}" loading="lazy">

</td></tr>

<td>

</td>

<mi>arctan</mi>

<mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -\pi +\arctan \left({\frac {x+y}{1-xy}}\right)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2c4d158a9c314dc37d7d1124bb43eb11dfa45839.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:23.192ex; height:6.176ex;" alt="{\displaystyle -\pi +\arctan \left({\frac {x+y}{1-xy}}\right)}" loading="lazy">

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x<0}</annotation>

</semantics>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle xy>1}</annotation>

</semantics>

</td></tr>

<mi>arctan</mi>

<mi>arctan</mi>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \arctan x-\arctan y=}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/aa4d906731c5604dcdaa1f6b649c81f5a6544784.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:21.485ex; height:2.343ex;" alt="{\displaystyle \arctan x-\arctan y=}" loading="lazy">

</td>

<mi>arctan</mi>

<mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \arctan \left({\frac {x-y}{1+xy}}\right)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9bf4e14e81c7c9d7d08dd80f90f97fe28a2f91fc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:17.211ex; height:6.176ex;" alt="{\displaystyle \arctan \left({\frac {x-y}{1+xy}}\right)}" loading="lazy">

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle xy>-1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e862ee5f0be94031c6f9388645009e00c901a51a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.554ex; height:2.509ex;" alt="{\displaystyle xy>-1}" loading="lazy">

</td></tr>

<td>

</td>

<mi>arctan</mi>

<mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \pi +\arctan \left({\frac {x-y}{1+xy}}\right)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/78f15c4794a99a7e0d0a7e7978e633fe5f76d905.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:21.384ex; height:6.176ex;" alt="{\displaystyle \pi +\arctan \left({\frac {x-y}{1+xy}}\right)}" loading="lazy">

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x>0}</annotation>

</semantics>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle xy<-1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1ba3b505305f5b164424fe098b422caf30f43335.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.554ex; height:2.509ex;" alt="{\displaystyle xy<-1}" loading="lazy">

</td></tr>

<td>

</td>

<mi>arctan</mi>

<mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -\pi +\arctan \left({\frac {x-y}{1+xy}}\right)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b54fed02d80490819815b2823a67fc11912675e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:23.192ex; height:6.176ex;" alt="{\displaystyle -\pi +\arctan \left({\frac {x-y}{1+xy}}\right)}" loading="lazy">

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x<0}</annotation>

</semantics>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle xy<-1}</annotation>

</semantics>

</td></tr></tbody></table>

<div class="mw-heading mw-heading3"><h3 id="Doppelwinkelfunktionen">Doppelwinkelfunktionen</h3></div>

<mfrac>

<mrow>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sin(2x)=2\sin x\cdot \;\cos x={\frac {2\tan x}{1+\tan ^{2}x}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2dbebe873a4e61ecf6ef23480ef6d4fed1bea0b3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:37.598ex; height:5.676ex;" alt="{\displaystyle \sin(2x)=2\sin x\cdot \;\cos x={\frac {2\tan x}{1+\tan ^{2}x}}}" loading="lazy"></dd></dl>

Eine geometrische Herleitung ist in Figur 3 für Winkel <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b79333175c8b3f0840bfb4ec41b8072c83ea88d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \beta }</annotation>

</semantics>

Zu Figur 3:

<dl><dd>Aus der Berechnung der Flächeninhalte der beiden grauen Dreiecke ergibt sich <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{2}}\cdot 2\sin \alpha \cdot 2\cos \alpha ={\frac {1}{2}}\cdot 2\cdot \sin(2\alpha )}">

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{2}}\cdot 2\sin \alpha \cdot 2\cos \alpha ={\frac {1}{2}}\cdot 2\cdot \sin(2\alpha )}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/046226a9de6336249f1d80b2c5c0f9ba60c1f6f9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:35.105ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{2}}\cdot 2\sin \alpha \cdot 2\cos \alpha ={\frac {1}{2}}\cdot 2\cdot \sin(2\alpha )}" loading="lazy">. Hieraus folgt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\sin \alpha \cdot \cos \alpha =\sin(2\alpha )}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 2\sin \alpha \cdot \cos \alpha =\sin(2\alpha )}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/dc47e322dc99a9d88a3961b81739cb44d9228ccd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.358ex; height:2.843ex;" alt="{\displaystyle 2\sin \alpha \cdot \cos \alpha =\sin(2\alpha )}" loading="lazy">.</dd></dl>

Weitere Beziehungen:

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \cos(2x)=\cos ^{2}x-\sin ^{2}x=1-2\sin ^{2}x=2\cos ^{2}x-1={\frac {1-\tan ^{2}x}{1+\tan ^{2}x}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/37a3ba745dfb5da99ca97f0b98be5ac91ef84985.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:67.739ex; height:6.176ex;" alt="{\displaystyle \cos(2x)=\cos ^{2}x-\sin ^{2}x=1-2\sin ^{2}x=2\cos ^{2}x-1={\frac {1-\tan ^{2}x}{1+\tan ^{2}x}}}" loading="lazy"></dd></dl>

<mfrac>

<mrow>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \tan(2x)={\frac {2\tan x}{1-\tan ^{2}x}}={\frac {2}{\cot x-\tan x}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2cc333f1010df6b985309c126f59c59f5697b1c8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:38.397ex; height:5.676ex;" alt="{\displaystyle \tan(2x)={\frac {2\tan x}{1-\tan ^{2}x}}={\frac {2}{\cot x-\tan x}}}" loading="lazy"></dd></dl>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \cot(2x)={\frac {\cot ^{2}x-1}{2\cot x}}={\frac {\cot x-\tan x}{2}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/39d9abccd9ffc7690cd93300763f3f3092293644.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:37.877ex; height:5.676ex;" alt="{\displaystyle \cot(2x)={\frac {\cot ^{2}x-1}{2\cot x}}={\frac {\cot x-\tan x}{2}}}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Winkelfunktionen_für_weitere_Vielfache">Winkelfunktionen für weitere Vielfache</h3></div>

Die Formeln für Vielfache berechnen sich normalerweise über die komplexen Zahlen aus der Euler-Formel

<mrow>

<mrow>

</mrow>

</mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

<mrow>

<mrow>

</mrow>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle z=r\left(\cos \phi +i\sin \phi \right)\iff z^{n}=r^{n}\left(\cos \phi +i\sin \phi \right)^{n}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5a8760cdba84ce61631bfc7bb7a27c071e180f30.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:52.114ex; height:3.009ex;" alt="{\displaystyle z=r\left(\cos \phi +i\sin \phi \right)\iff z^{n}=r^{n}\left(\cos \phi +i\sin \phi \right)^{n}}" loading="lazy">

und der DeMoivre-Formel

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<mrow>

<mrow>

<mrow>

<mrow>

</mrow>

</mrow>

<mrow>

<mrow>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle z^{n}=r^{n}\left(\cos \left(n\phi \right)+i\sin \left(n\phi \right)\right)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/437da3cf908500f01e89b270975a0b8179095159.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.046ex; height:2.843ex;" alt="{\displaystyle z^{n}=r^{n}\left(\cos \left(n\phi \right)+i\sin \left(n\phi \right)\right)}" loading="lazy">.

Damit ergibt sich

<mrow>

<mrow>

</mrow>

</mrow>

<mrow>

<mrow>

</mrow>

</mrow>

<msup>

<mrow>

<mrow>

</mrow>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \cos \left(n\phi \right)+i\sin \left(n\phi \right)=\left(\cos \phi +i\sin \phi \right)^{n}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/513b6fb6582557faffea01cc3e63cf312b869615.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:38.844ex; height:3.009ex;" alt="{\displaystyle \cos \left(n\phi \right)+i\sin \left(n\phi \right)=\left(\cos \phi +i\sin \phi \right)^{n}}" loading="lazy">.

Zerlegung in Real- und Imaginärteil liefert dann die Formeln für

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \cos }</annotation>

</semantics>

und

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sin }</annotation>

</semantics>

bzw. die allgemeine Reihendarstellung.

Die Formel für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos(nx)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \cos(nx)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a930a1025613b667060f1659b66d739d62576232.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.645ex; height:2.843ex;" alt="{\displaystyle \cos(nx)}" loading="lazy"> steht über <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{n}(\cos x)=\cos(nx)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle T_{n}(\cos x)=\cos(nx)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e522492b2f2834b76fcbfbc985a7e1c6746c601f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.956ex; height:2.843ex;" alt="{\displaystyle T_{n}(\cos x)=\cos(nx)}" loading="lazy"><a href="#cite_note-8">[8]</a> mit den <a href="Tschebyschow-Polynom" title="Tschebyschow-Polynom">Tschebyschow-Polynomen</a> in Beziehung.

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sin(3x)=3\sin x-4\sin ^{3}x\,}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/07369f7fa2292f8f867d3f4b4c04639b04839552.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.781ex; height:3.176ex;" alt="{\displaystyle \sin(3x)=3\sin x-4\sin ^{3}x\,}" loading="lazy"><a href="#cite_note-9">[9]</a>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle =\;\sin x\left(4\cos ^{2}x-1\right)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/aaa9c56168861254f7e3e9641dac72aea1f206c1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:21.622ex; height:3.343ex;" alt="{\displaystyle =\;\sin x\left(4\cos ^{2}x-1\right)}" loading="lazy"></dd></dl></dd></dl>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sin(4x)=8\sin x\;\cos ^{3}x-4\sin x\;\cos x}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3298ad7a0def6798c456bc5219adc72769449691.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:38.114ex; height:3.176ex;" alt="{\displaystyle \sin(4x)=8\sin x\;\cos ^{3}x-4\sin x\;\cos x}" loading="lazy"><a href="#cite_note-10">[10]</a>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle =\;\sin x\left(8\cos ^{3}x-4\cos x\right)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/77d4cf8c1d4f708af973d0416ba02a285239bab4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:26.837ex; height:3.343ex;" alt="{\displaystyle =\;\sin x\left(8\cos ^{3}x-4\cos x\right)}" loading="lazy"></dd></dl></dd></dl>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sin(5x)=5\sin x-20\sin ^{3}x+16\sin ^{5}x\;}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e5a09c7f8d29983469fd7f1d805f67d0ff64e29f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:39.381ex; height:3.176ex;" alt="{\displaystyle \sin(5x)=5\sin x-20\sin ^{3}x+16\sin ^{5}x\;}" loading="lazy"><a href="#cite_note-11">[11]</a>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle =\;\sin x\left(16\cos ^{4}x-12\cos ^{2}x+1\right)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d5a04622cf8d8e7d716edf39cccf39b23e7e5b5f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:34.219ex; height:3.343ex;" alt="{\displaystyle =\;\sin x\left(16\cos ^{4}x-12\cos ^{2}x+1\right)}" loading="lazy"></dd></dl></dd></dl>

<msup>

</mrow>

</msup>

<mrow>

</mrow>

</mfrac>

</mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<mrow>

</mrow>

</mfrac>

</mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sin(nx)=n\;\sin x\;\cos ^{n-1}x-{n \choose 3}\sin ^{3}x\;\cos ^{n-3}x+{n \choose 5}\sin ^{5}x\;\cos ^{n-5}x\;-\ldots +\ldots }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/19f6d779684e12f60c869b6ca974911d8231d93c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:84.137ex; height:6.176ex;" alt="{\displaystyle \sin(nx)=n\;\sin x\;\cos ^{n-1}x-{n \choose 3}\sin ^{3}x\;\cos ^{n-3}x+{n \choose 5}\sin ^{5}x\;\cos ^{n-5}x\;-\ldots +\ldots }" loading="lazy"><a href="#cite_note-12">[12]</a><a href="#cite_note-Bronstein2.5.2.1.3-13">[13]</a>

</mrow>

<mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</munderover>

<msup>

</mrow>

</msup>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle =\;\sum _{j=0}^{\left\lfloor {\frac {n-1}{2}}\right\rfloor }(-1)^{j}{n \choose 2j+1}\sin ^{2j+1}x\;\cos ^{n-2j-1}x}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/11f0026721a244470c37f13d287a09c16d96ab65.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:44.16ex; height:9.176ex;" alt="{\displaystyle =\;\sum _{j=0}^{\left\lfloor {\frac {n-1}{2}}\right\rfloor }(-1)^{j}{n \choose 2j+1}\sin ^{2j+1}x\;\cos ^{n-2j-1}x}" loading="lazy"></dd>

</mrow>

<mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</munderover>

<msup>

</mrow>

</msup>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle =\;\sin x\sum _{k=0}^{\left\lfloor {\frac {n-1}{2}}\right\rfloor }(-1)^{k}{n-k-1 \choose k}2^{n-2k-1}\cos ^{n-2k-1}x}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/938e628b0688a2ef7f3097e926c5a7868e4d5923.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:50.804ex; height:9.009ex;" alt="{\displaystyle =\;\sin x\sum _{k=0}^{\left\lfloor {\frac {n-1}{2}}\right\rfloor }(-1)^{k}{n-k-1 \choose k}2^{n-2k-1}\cos ^{n-2k-1}x}" loading="lazy"></dd></dl></dd></dl>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sin(nx)\;\sin(x)={\frac {\cos((n-1)\,x)-\cos((n+1)\,x)}{2}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/351f228bffec930d320e1fac8a81fbb6d0a9105b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:48.879ex; height:5.676ex;" alt="{\displaystyle \sin(nx)\;\sin(x)={\frac {\cos((n-1)\,x)-\cos((n+1)\,x)}{2}}}" loading="lazy"></dd></dl>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \cos(3x)=4\cos ^{3}x-3\cos x\,}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e6adcf1cdf2f52a4dfaa35e00b09d997e97f8f7b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.548ex; height:3.176ex;" alt="{\displaystyle \cos(3x)=4\cos ^{3}x-3\cos x\,}" loading="lazy"><a href="#cite_note-14">[14]</a></dd></dl>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \cos(4x)=8\cos ^{4}x-8\cos ^{2}x+1\,}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/401d0c814232b3a047ee41e3e2da9e48b3acb4bd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.605ex; height:3.176ex;" alt="{\displaystyle \cos(4x)=8\cos ^{4}x-8\cos ^{2}x+1\,}" loading="lazy"><a href="#cite_note-15">[15]</a></dd></dl>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \cos(5x)=16\cos ^{5}x-20\cos ^{3}x+5\cos x\,}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/654f65892ad5763726a2d59aac3a4fa110e678f5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.145ex; height:3.176ex;" alt="{\displaystyle \cos(5x)=16\cos ^{5}x-20\cos ^{3}x+5\cos x\,}" loading="lazy"><a href="#cite_note-16">[16]</a></dd></dl>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \cos(6x)=32\cos ^{6}x-48\cos ^{4}x+18\cos ^{2}x-1\,}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/43af6ba8f12d695a3e5762a79a0e709a8d3b29cb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:46.364ex; height:3.176ex;" alt="{\displaystyle \cos(6x)=32\cos ^{6}x-48\cos ^{4}x+18\cos ^{2}x-1\,}" loading="lazy"><a href="#cite_note-17">[17]</a></dd></dl>

<msup>

</mrow>

</msup>

<mrow>

</mrow>

</mfrac>

</mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<mrow>

</mrow>

</mfrac>

</mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \cos(nx)=\cos ^{n}x-{n \choose 2}\sin ^{2}x\;\cos ^{n-2}x+{n \choose 4}\sin ^{4}x\;\cos ^{n-4}x\;-\ldots +\ldots }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ef1d6ce445b17ba82d5fce24c4e782836ebe740d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:74.26ex; height:6.176ex;" alt="{\displaystyle \cos(nx)=\cos ^{n}x-{n \choose 2}\sin ^{2}x\;\cos ^{n-2}x+{n \choose 4}\sin ^{4}x\;\cos ^{n-4}x\;-\ldots +\ldots }" loading="lazy"><a href="#cite_note-Bronstein2.5.2.1.3-13">[13]</a><a href="#cite_note-18">[18]</a>

</mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</munderover>

<msup>

</mrow>

</msup>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle =\;\sum _{j=0}^{\left\lfloor {\frac {n}{2}}\right\rfloor }(-1)^{j}{n \choose 2j}\sin ^{2j}x\;\cos ^{n-2j}x}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/75ac809a8d2cf83e0977b6f08e90c54dfec5da80.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:34.25ex; height:9.176ex;" alt="{\displaystyle =\;\sum _{j=0}^{\left\lfloor {\frac {n}{2}}\right\rfloor }(-1)^{j}{n \choose 2j}\sin ^{2j}x\;\cos ^{n-2j}x}" loading="lazy"></dd></dl></dd></dl>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \cos(nx)\;\cos(x)={\frac {\cos((n-1)\,x)+\cos((n+1)\,x)}{2}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/84bf36bb1234c00f533757e64e6089844b62573c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:49.39ex; height:5.676ex;" alt="{\displaystyle \cos(nx)\;\cos(x)={\frac {\cos((n-1)\,x)+\cos((n+1)\,x)}{2}}}" loading="lazy"></dd></dl>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \tan(3x)={\frac {3\tan x-\tan ^{3}x}{1-3\tan ^{2}x}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/82fa74b1e061e40a6ec3927afa434f3ed22f18c1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:27.193ex; height:6.176ex;" alt="{\displaystyle \tan(3x)={\frac {3\tan x-\tan ^{3}x}{1-3\tan ^{2}x}}}" loading="lazy"><a href="#cite_note-Bronstein2.5.2.1.3-13">[13]</a></dd></dl>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \tan(4x)={\frac {4\tan x-4\tan ^{3}x}{1-6\tan ^{2}x+\tan ^{4}x}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4474fad189f0b4c965a61b9b3ffe6836659087bb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:32.25ex; height:6.176ex;" alt="{\displaystyle \tan(4x)={\frac {4\tan x-4\tan ^{3}x}{1-6\tan ^{2}x+\tan ^{4}x}}}" loading="lazy"><a href="#cite_note-Bronstein2.5.2.1.3-13">[13]</a></dd></dl>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \cot(3x)={\frac {\cot ^{3}x-3\cot x}{3\cot ^{2}x-1}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7b61d0b1c995e3b835901bcf114e0d26f67f195e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:26.412ex; height:6.176ex;" alt="{\displaystyle \cot(3x)={\frac {\cot ^{3}x-3\cot x}{3\cot ^{2}x-1}}}" loading="lazy"><a href="#cite_note-Bronstein2.5.2.1.3-13">[13]</a></dd></dl>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \cot(4x)={\frac {\cot ^{4}x-6\cot ^{2}x+1}{4\cot ^{3}x-4\cot x}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5989fc85830b4459560650545d9eaae3d36ed217.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:31.469ex; height:6.176ex;" alt="{\displaystyle \cot(4x)={\frac {\cot ^{4}x-6\cot ^{2}x+1}{4\cot ^{3}x-4\cot x}}}" loading="lazy"><a href="#cite_note-Bronstein2.5.2.1.3-13">[13]</a></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Halbwinkelformeln">Halbwinkelformeln</h3></div>

Zur Berechnung des Funktionswertes des halben Arguments dienen die Halbwinkelformeln<a href="#cite_note-Bronstein2.5.2.1.3-13">[13]</a>, welche sich mittels Substitution aus den Doppelwinkelformeln herleiten lassen:

<mfrac>

</mfrac>

</mrow>

<msqrt>

<mfrac>

<mrow>

</mrow>

</mfrac>

</msqrt>

</mrow>

</mrow>

<mrow>

<mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sin {\frac {x}{2}}={\sqrt {\frac {1-\cos x}{2}}}\quad {\text{für}}\quad x\in \left[0,2\pi \right]}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c05b1bef0e3ec010ffd9aeb20f6b5fcd5facc1c9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:38.338ex; height:6.176ex;" alt="{\displaystyle \sin {\frac {x}{2}}={\sqrt {\frac {1-\cos x}{2}}}\quad {\text{für}}\quad x\in \left[0,2\pi \right]}" loading="lazy"></dd></dl>

<mfrac>

</mfrac>

</mrow>

<msqrt>

<mfrac>

<mrow>

</mrow>

</mfrac>

</msqrt>

</mrow>

</mrow>

<mrow>

<mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \cos {\frac {x}{2}}={\sqrt {\frac {1+\cos x}{2}}}\quad {\text{für}}\quad x\in \left[-\pi ,\pi \right]}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6f10157e35e43a63caa173267e34c4729ab69247.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:39.409ex; height:6.176ex;" alt="{\displaystyle \cos {\frac {x}{2}}={\sqrt {\frac {1+\cos x}{2}}}\quad {\text{für}}\quad x\in \left[-\pi ,\pi \right]}" loading="lazy"></dd></dl>

<mfrac>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \tan {\frac {x}{2}}={\frac {1-\cos x}{\sin x}}={\frac {\sin x}{1+\cos x}}\quad {\text{für}}\quad x\in \mathbb {R} \setminus \pi (2\mathbb {Z} +1)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d1206e76f352434427277fcadb7527bc2aee4534.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:57.029ex; height:5.343ex;" alt="{\displaystyle \tan {\frac {x}{2}}={\frac {1-\cos x}{\sin x}}={\frac {\sin x}{1+\cos x}}\quad {\text{für}}\quad x\in \mathbb {R} \setminus \pi (2\mathbb {Z} +1)}" loading="lazy"></dd></dl>

<mfrac>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \cot {\frac {x}{2}}={\frac {1+\cos x}{\sin x}}={\frac {\sin x}{1-\cos x}}\quad {\text{für}}\quad x\in \mathbb {R} \setminus 2\pi \mathbb {Z} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e797c70eb459b1f0ed192588def04f336c36e2fd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:50.956ex; height:5.343ex;" alt="{\displaystyle \cot {\frac {x}{2}}={\frac {1+\cos x}{\sin x}}={\frac {\sin x}{1-\cos x}}\quad {\text{für}}\quad x\in \mathbb {R} \setminus 2\pi \mathbb {Z} }" loading="lazy"></dd></dl>

Eine geometrische Herleitung der dritten Formel ist in Figur 4 für Winkel <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b79333175c8b3f0840bfb4ec41b8072c83ea88d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \beta }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7ed48a5e36207156fb792fa79d29925d2f7901e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }" loading="lazy"> zwischen 0° und 90° veranschaulicht.<a href="#cite_note-19">[19]</a> Aus der Berechnung der Flächeninhalte der beiden grauen Dreiecke ergibt sich unmittelbar <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tan \left({\frac {\alpha }{2}}\right)={\frac {\sin \alpha }{1+\cos \alpha }}={\frac {1-\cos \alpha }{\sin \alpha }}}">

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \tan \left({\frac {\alpha }{2}}\right)={\frac {\sin \alpha }{1+\cos \alpha }}={\frac {1-\cos \alpha }{\sin \alpha }}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/43ed97bf3a90d32b38599e9080e2ac55bd927fcc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:34.305ex; height:5.343ex;" alt="{\displaystyle \tan \left({\frac {\alpha }{2}}\right)={\frac {\sin \alpha }{1+\cos \alpha }}={\frac {1-\cos \alpha }{\sin \alpha }}}" loading="lazy">.

Außerdem gilt:

<mfrac>

</mfrac>

</mrow>

<mfrac>

<mrow>

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<msup>

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</mfrac>

</mrow>

</mrow>

<mrow>

<mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<mfrac>

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</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \tan {\frac {x}{2}}={\frac {\tan x}{1+{\sqrt {1+\tan ^{2}x}}}}\quad {\text{für}}\quad x\in \left]-{\tfrac {\pi }{2}},{\tfrac {\pi }{2}}\right[}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6b5e598d507ce768e3efd5bfb8bf974e60b004bb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:46.5ex; height:6.509ex;" alt="{\displaystyle \tan {\frac {x}{2}}={\frac {\tan x}{1+{\sqrt {1+\tan ^{2}x}}}}\quad {\text{für}}\quad x\in \left]-{\tfrac {\pi }{2}},{\tfrac {\pi }{2}}\right[}" loading="lazy"></dd></dl>

<mfrac>

</mfrac>

</mrow>

<msqrt>

<msup>

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</msup>

</msqrt>

</mrow>

</mrow>

<mrow>

<mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \cot {\frac {x}{2}}=\cot x+{\sqrt {1+\cot ^{2}x}}\quad {\text{für}}\quad x\in \left]0,\pi \right[}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ab72611c37657b6aedab9984b9976053a5213dab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:45.282ex; height:4.676ex;" alt="{\displaystyle \cot {\frac {x}{2}}=\cot x+{\sqrt {1+\cot ^{2}x}}\quad {\text{für}}\quad x\in \left]0,\pi \right[}" loading="lazy"></dd></dl>

Siehe auch: <a href="Halbwinkelsatz" title="Halbwinkelsatz">Halbwinkelsatz</a>

<div class="mw-heading mw-heading3"><h3 id="Summen_zweier_trigonometrischer_Funktionen_(Identitäten)">Summen zweier trigonometrischer Funktionen (Identitäten)</h3></div>

Aus den <a class="mw-selflink-fragment" href="#Additionstheoreme">Additionstheoremen</a> lassen sich Identitäten ableiten, mit deren Hilfe die Summe zweier trigonometrischer Funktionen als Produkt dargestellt werden kann:<a href="#cite_note-Bronstein2.5.2.1.3-13">[13]</a>

<mtr>

<mtd>

</mtd>

<mtd>

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<mrow>

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</mfrac>

</mrow>

<mfrac>

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<mtr>

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<mtr>

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<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

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</mfrac>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\sin x+\sin y&=2\sin {\frac {x+y}{2}}\cos {\frac {x-y}{2}}\\\sin x-\sin y&=2\cos {\frac {x+y}{2}}\sin {\frac {x-y}{2}}\\\cos x+\cos y&=2\cos {\frac {x+y}{2}}\cos {\frac {x-y}{2}}\\\cos x-\cos y&=-2\sin {\frac {x+y}{2}}\sin {\frac {x-y}{2}}\end{aligned}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/720905bdbd9494bd0ba8edf6e18cca42c1d5f621.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -10.005ex; width:38.726ex; height:21.176ex;" alt="{\displaystyle {\begin{aligned}\sin x+\sin y&=2\sin {\frac {x+y}{2}}\cos {\frac {x-y}{2}}\\\sin x-\sin y&=2\cos {\frac {x+y}{2}}\sin {\frac {x-y}{2}}\\\cos x+\cos y&=2\cos {\frac {x+y}{2}}\cos {\frac {x-y}{2}}\\\cos x-\cos y&=-2\sin {\frac {x+y}{2}}\sin {\frac {x-y}{2}}\end{aligned}}}" loading="lazy"></dd></dl>

<mrow>

<mtr>

<mtd>

<mfrac>

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<mrow>

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</mfrac>

</mstyle>

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<mtr>

<mtd>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mstyle>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left.{\begin{matrix}\tan x+\tan y={\dfrac {\sin(x+y)}{\cos x\cos y}}\\[1em]\tan x-\tan y={\dfrac {\sin(x-y)}{\cos x\cos y}}\end{matrix}}\right\}\Rightarrow \tan x\pm \tan y={\frac {\sin(x\pm y)}{\cos x\cos y}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c82f5271400544871570c9e8e755362c0c27390c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.838ex; width:59.92ex; height:14.843ex;" alt="{\displaystyle \left.{\begin{matrix}\tan x+\tan y={\dfrac {\sin(x+y)}{\cos x\cos y}}\\[1em]\tan x-\tan y={\dfrac {\sin(x-y)}{\cos x\cos y}}\end{matrix}}\right\}\Rightarrow \tan x\pm \tan y={\frac {\sin(x\pm y)}{\cos x\cos y}}}" loading="lazy"></dd></dl>

<mrow>

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<mtd>

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<annotation encoding="application/x-tex">{\displaystyle \left.{\begin{matrix}\cot x+\cot y={\dfrac {\sin(y+x)}{\sin x\sin y}}\\[1em]\cot x-\cot y={\dfrac {\sin(y-x)}{\sin x\sin y}}\end{matrix}}\right\}\Rightarrow \cot x\pm \cot y={\frac {\sin(y\pm x)}{\sin x\sin y}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/324f139ea94aaeceb07ee500aa35eaa16d0755c8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.838ex; width:58.879ex; height:14.843ex;" alt="{\displaystyle \left.{\begin{matrix}\cot x+\cot y={\dfrac {\sin(y+x)}{\sin x\sin y}}\\[1em]\cot x-\cot y={\dfrac {\sin(y-x)}{\sin x\sin y}}\end{matrix}}\right\}\Rightarrow \cot x\pm \cot y={\frac {\sin(y\pm x)}{\sin x\sin y}}}" loading="lazy"></dd></dl>

Daraus ergeben sich noch Spezialfälle:

<mtr>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\cos x+\sin x&={\sqrt {2}}\cdot \sin \left(x+{\frac {\pi }{4}}\right)={\sqrt {2}}\cdot \cos \left(x-{\frac {\pi }{4}}\right)\\\cos x-\sin x&={\sqrt {2}}\cdot \cos \left(x+{\frac {\pi }{4}}\right)=-{\sqrt {2}}\cdot \sin \left(x-{\frac {\pi }{4}}\right)\end{aligned}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d1af7f07018432bb3c0bdf45affa51f0504259aa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.338ex; width:54.746ex; height:9.843ex;" alt="{\displaystyle {\begin{aligned}\cos x+\sin x&={\sqrt {2}}\cdot \sin \left(x+{\frac {\pi }{4}}\right)={\sqrt {2}}\cdot \cos \left(x-{\frac {\pi }{4}}\right)\\\cos x-\sin x&={\sqrt {2}}\cdot \cos \left(x+{\frac {\pi }{4}}\right)=-{\sqrt {2}}\cdot \sin \left(x-{\frac {\pi }{4}}\right)\end{aligned}}}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Produkte_der_Winkelfunktionen">Produkte der Winkelfunktionen</h3></div>

Produkte der trigonometrischen Funktionen lassen sich mit folgenden Formeln berechnen:<a href="#cite_note-Bronstein2.5.2.1.3-13">[13]</a>

<mfrac>

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<annotation encoding="application/x-tex">{\displaystyle \sin x\;\sin y={\frac {1}{2}}{\Big (}\cos(x-y)-\cos(x+y){\Big )}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fbdf85ce68f09d28cd51e5b2b38099e44f798f1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:41.595ex; height:5.176ex;" alt="{\displaystyle \sin x\;\sin y={\frac {1}{2}}{\Big (}\cos(x-y)-\cos(x+y){\Big )}}" loading="lazy"></dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle \cos x\;\cos y={\frac {1}{2}}{\Big (}\cos(x-y)+\cos(x+y){\Big )}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b245641b8e0f75516322c49bf3edad0434a206cc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:42.106ex; height:5.176ex;" alt="{\displaystyle \cos x\;\cos y={\frac {1}{2}}{\Big (}\cos(x-y)+\cos(x+y){\Big )}}" loading="lazy"></dd></dl>

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</mrow>

</mrow>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sin x\;\cos y={\frac {1}{2}}{\Big (}\sin(x-y)+\sin(x+y){\Big )}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a70353bdec37afab686c482cbdffb2cfe0f8f10f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:41.339ex; height:5.176ex;" alt="{\displaystyle \sin x\;\cos y={\frac {1}{2}}{\Big (}\sin(x-y)+\sin(x+y){\Big )}}" loading="lazy"></dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle \tan x\;\tan y={\frac {\tan x+\tan y}{\cot x+\cot y}}=-{\frac {\tan x-\tan y}{\cot x-\cot y}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9f264236ebb81fc31a499d5ecf60be754ac05c9a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:46.326ex; height:5.676ex;" alt="{\displaystyle \tan x\;\tan y={\frac {\tan x+\tan y}{\cot x+\cot y}}=-{\frac {\tan x-\tan y}{\cot x-\cot y}}}" loading="lazy"></dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle \cot x\;\cot y={\frac {\cot x+\cot y}{\tan x+\tan y}}=-{\frac {\cot x-\cot y}{\tan x-\tan y}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/27c57817c66ced9696131d1a88600c338664fd46.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:45.806ex; height:5.676ex;" alt="{\displaystyle \cot x\;\cot y={\frac {\cot x+\cot y}{\tan x+\tan y}}=-{\frac {\cot x-\cot y}{\tan x-\tan y}}}" loading="lazy"></dd></dl>

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</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \tan x\;\cot y={\frac {\tan x+\cot y}{\cot x+\tan y}}=-{\frac {\tan x-\cot y}{\cot x-\tan y}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f71971336fa0490cec07645e42bbdad9d907766a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:45.546ex; height:5.676ex;" alt="{\displaystyle \tan x\;\cot y={\frac {\tan x+\cot y}{\cot x+\tan y}}=-{\frac {\tan x-\cot y}{\cot x-\tan y}}}" loading="lazy"></dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle \sin x\;\sin y\;\sin z={\frac {1}{4}}{\Big (}\sin(x+y-z)+\sin(y+z-x)+\sin(z+x-y)-\sin(x+y+z){\Big )}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fdfa4793e98f7471e76ce0d9cfcf5e2f3c7a61c2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:87.823ex; height:5.176ex;" alt="{\displaystyle \sin x\;\sin y\;\sin z={\frac {1}{4}}{\Big (}\sin(x+y-z)+\sin(y+z-x)+\sin(z+x-y)-\sin(x+y+z){\Big )}}" loading="lazy"></dd></dl>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \cos x\;\cos y\;\cos z={\frac {1}{4}}{\Big (}\cos(x+y-z)+\cos(y+z-x)+\cos(z+x-y)+\cos(x+y+z){\Big )}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a931948e89d11c857f10d40e2477490d0aeb0006.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:89.611ex; height:5.176ex;" alt="{\displaystyle \cos x\;\cos y\;\cos z={\frac {1}{4}}{\Big (}\cos(x+y-z)+\cos(y+z-x)+\cos(z+x-y)+\cos(x+y+z){\Big )}}" loading="lazy"></dd></dl>

<mfrac>

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</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sin x\;\sin y\;\cos z={\frac {1}{4}}{\Big (}-\cos(x+y-z)+\cos(y+z-x)+\cos(z+x-y)-\cos(x+y+z){\Big )}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b0762f0e9b8a98c0adc77431f32fa639b44d723b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:91.554ex; height:5.176ex;" alt="{\displaystyle \sin x\;\sin y\;\cos z={\frac {1}{4}}{\Big (}-\cos(x+y-z)+\cos(y+z-x)+\cos(z+x-y)-\cos(x+y+z){\Big )}}" loading="lazy"></dd></dl>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sin x\;\cos y\;\cos z={\frac {1}{4}}{\Big (}\sin(x+y-z)-\sin(y+z-x)+\sin(z+x-y)+\sin(x+y+z){\Big )}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/668ada9f6d993b5369975ec4444c24d65f185e56.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:88.334ex; height:5.176ex;" alt="{\displaystyle \sin x\;\cos y\;\cos z={\frac {1}{4}}{\Big (}\sin(x+y-z)-\sin(y+z-x)+\sin(z+x-y)+\sin(x+y+z){\Big )}}" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle \prod _{m=1}^{n}\cos(x_{m})={\frac {1}{2^{n}}}\sum _{k_{1}=1}^{2}\cdots \sum _{k_{n}=1}^{2}\left[\exp \left({\text{i}}\sum _{\nu =1}^{n}(-1)^{k_{\nu }}x_{\nu }\right)\right]={\frac {1}{2^{n-1}}}\sum _{k_{2}=1}^{2}\cdots \sum _{k_{n}=1}^{2}\left[\cos \left(x_{1}+\sum _{\nu =2}^{n}(-1)^{k_{\nu }}x_{\nu }\right)\right]}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/857634cd97a722c95294a0a74547c8f081596355.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:101.313ex; height:7.676ex;" alt="{\displaystyle \prod _{m=1}^{n}\cos(x_{m})={\frac {1}{2^{n}}}\sum _{k_{1}=1}^{2}\cdots \sum _{k_{n}=1}^{2}\left[\exp \left({\text{i}}\sum _{\nu =1}^{n}(-1)^{k_{\nu }}x_{\nu }\right)\right]={\frac {1}{2^{n-1}}}\sum _{k_{2}=1}^{2}\cdots \sum _{k_{n}=1}^{2}\left[\cos \left(x_{1}+\sum _{\nu =2}^{n}(-1)^{k_{\nu }}x_{\nu }\right)\right]}" loading="lazy"></dd>

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<mtd>

<mtext>gerade</mtext>

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</mtd>

<mtd>

<mtext>ungerade</mtext>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle \prod _{m=1}^{n}\sin(x_{m})={\frac {1}{(2{\text{i}})^{n}}}\sum _{k_{1}=1}^{2}\cdots \sum _{k_{n}=1}^{2}\left[\prod _{\mu =1}^{n}(-1)^{k_{\mu }}\cdot \exp \left({\text{i}}\sum _{\nu =1}^{n}(-1)^{k_{\nu }}x_{\nu }\right)\right]={\frac {1}{2^{n-1}}}\sum _{k_{2}=1}^{2}\cdots \sum _{k_{n}=1}^{2}\left[\prod _{\mu =2}^{n}(-1)^{k_{\mu }}\cdot {\begin{cases}\displaystyle (-1)^{n/2}\cdot \cos \left(x_{1}+\sum _{\nu =2}^{n}(-1)^{k_{\nu }}x_{\nu }\right)&{\text{gerade}}\;n\\\displaystyle (-1)^{(n-1)/2}\cdot \sin \left(x_{1}+\sum _{\nu =2}^{n}(-1)^{k_{\nu }}x_{\nu }\right)&{\text{ungerade}}\;n\end{cases}}\right]}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/20ec1339958b378bdfab93d89ae5a470552a1c12.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.838ex; width:155.925ex; height:14.843ex;" alt="{\displaystyle \prod _{m=1}^{n}\sin(x_{m})={\frac {1}{(2{\text{i}})^{n}}}\sum _{k_{1}=1}^{2}\cdots \sum _{k_{n}=1}^{2}\left[\prod _{\mu =1}^{n}(-1)^{k_{\mu }}\cdot \exp \left({\text{i}}\sum _{\nu =1}^{n}(-1)^{k_{\nu }}x_{\nu }\right)\right]={\frac {1}{2^{n-1}}}\sum _{k_{2}=1}^{2}\cdots \sum _{k_{n}=1}^{2}\left[\prod _{\mu =2}^{n}(-1)^{k_{\mu }}\cdot {\begin{cases}\displaystyle (-1)^{n/2}\cdot \cos \left(x_{1}+\sum _{\nu =2}^{n}(-1)^{k_{\nu }}x_{\nu }\right)&{\text{gerade}}\;n\\\displaystyle (-1)^{(n-1)/2}\cdot \sin \left(x_{1}+\sum _{\nu =2}^{n}(-1)^{k_{\nu }}x_{\nu }\right)&{\text{ungerade}}\;n\end{cases}}\right]}" loading="lazy"></dd></dl>

Aus der <a class="mw-selflink-fragment" href="#Doppelwinkelfunktionen">Doppelwinkelfunktion</a> für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin(2x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sin(2x)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0e2822c6491b347058bf997b431f1a87ff412a44.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.157ex; height:2.843ex;" alt="{\displaystyle \sin(2x)}" loading="lazy"> folgt außerdem:

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sin x\;\cos x={\frac {1}{2}}\sin(2x)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4e7b7c92f4973ad7ac2a6de20189a496f96d472b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:23.074ex; height:5.176ex;" alt="{\displaystyle \sin x\;\cos x={\frac {1}{2}}\sin(2x)}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Potenzen_der_Winkelfunktionen">Potenzen der Winkelfunktionen</h3></div>

<div class="mw-heading mw-heading4"><h4 id="Sinus">Sinus</h4></div>

<msup>

</mrow>

</msup>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sin ^{2}x={\frac {1}{2}}\ {\Big (}1-\cos(2x){\Big )}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5fe1eb0db7c2d60fb7f8d740d2803e88c18ade4c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:25.495ex; height:5.176ex;" alt="{\displaystyle \sin ^{2}x={\frac {1}{2}}\ {\Big (}1-\cos(2x){\Big )}}" loading="lazy"><a href="#cite_note-Bronstein2.5.2.1.3-13">[13]</a><a href="#cite_note-20">[20]</a></dd></dl>

<msup>

</mrow>

</msup>

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</mfrac>

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</mrow>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sin ^{3}x={\frac {1}{4}}\ {\Big (}3\,\sin x-\sin(3x){\Big )}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/03888a7e5d05a4fa1a55388cb4cbebbcffcc8060.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:30.586ex; height:5.176ex;" alt="{\displaystyle \sin ^{3}x={\frac {1}{4}}\ {\Big (}3\,\sin x-\sin(3x){\Big )}}" loading="lazy"><a href="#cite_note-Bronstein2.5.2.1.3-13">[13]</a><a href="#cite_note-21">[21]</a></dd></dl>

<msup>

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<annotation encoding="application/x-tex">{\displaystyle \sin ^{4}x={\frac {1}{8}}\ {\Big (}3-4\,\cos(2x)+\cos(4x){\Big )}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0486b0d3f608dbd4f8e8175841a3893fc1b8fe29.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:37.685ex; height:5.176ex;" alt="{\displaystyle \sin ^{4}x={\frac {1}{8}}\ {\Big (}3-4\,\cos(2x)+\cos(4x){\Big )}}" loading="lazy"><a href="#cite_note-Bronstein2.5.2.1.3-13">[13]</a><a href="#cite_note-22">[22]</a></dd></dl>

<msup>

</mrow>

</msup>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sin ^{5}x={\frac {1}{16}}\ {\Big (}10\,\sin x-5\,\sin(3x)+\sin(5x){\Big )}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3e3c742bb425a373df188ec22f447ad8fb88bd59.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:44.845ex; height:5.176ex;" alt="{\displaystyle \sin ^{5}x={\frac {1}{16}}\ {\Big (}10\,\sin x-5\,\sin(3x)+\sin(5x){\Big )}}" loading="lazy"><a href="#cite_note-23">[23]</a></dd></dl>

<msup>

</mrow>

</msup>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sin ^{6}x={\frac {1}{32}}\ {\Big (}10-15\,\cos(2x)+6\,\cos(4x)-\cos(6x){\Big )}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e8c19b66c5e0a5862fb44a4746ad5cf4208ea5ed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:53.362ex; height:5.176ex;" alt="{\displaystyle \sin ^{6}x={\frac {1}{32}}\ {\Big (}10-15\,\cos(2x)+6\,\cos(4x)-\cos(6x){\Big )}}" loading="lazy"><a href="#cite_note-24">[24]</a></dd></dl>

<msup>

</mrow>

</msup>

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

</mrow>

</mrow>

</munderover>

<mrow>

</mrow>

</mfrac>

</mrow>

<mrow>

<mrow>

<mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sin ^{n}x={\frac {1}{2^{n}}}\,\sum _{k=0}^{n}{n \choose k}\,\cos \left((n-2k)\left(x-{\frac {\pi }{2}}\right)\right)\ ;\quad n\in \mathbb {N} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ed7155c080d0b70ba2a803cfacb74a2c48443584.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:55.869ex; height:7.009ex;" alt="{\displaystyle \sin ^{n}x={\frac {1}{2^{n}}}\,\sum _{k=0}^{n}{n \choose k}\,\cos \left((n-2k)\left(x-{\frac {\pi }{2}}\right)\right)\ ;\quad n\in \mathbb {N} }" loading="lazy"></dd></dl>

<msup>

</mrow>

</msup>

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

<mrow>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mfrac>

</mrow>

<mfrac>

<msup>

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</msup>

</mfrac>

</mrow>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</munderover>

<msup>

<mfrac>

</mfrac>

</mrow>

</mrow>

</msup>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</mrow>

<mtext> gerade </mtext>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sin ^{n}x={\frac {1}{2^{n}}}\,{n \choose {\frac {n}{2}}}+{\frac {1}{2^{n-1}}}\sum _{k=0}^{{\frac {n}{2}}-1}(-1)^{{\frac {n}{2}}-k}\,{n \choose k}\,\cos {((n-2k)x)};\quad n\in \mathbb {N} {\text{ und }}n{\text{ gerade }}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c9563ad90f6a8943e47efd46e3655fbe60d8d9d6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:83.295ex; height:8.509ex;" alt="{\displaystyle \sin ^{n}x={\frac {1}{2^{n}}}\,{n \choose {\frac {n}{2}}}+{\frac {1}{2^{n-1}}}\sum _{k=0}^{{\frac {n}{2}}-1}(-1)^{{\frac {n}{2}}-k}\,{n \choose k}\,\cos {((n-2k)x)};\quad n\in \mathbb {N} {\text{ und }}n{\text{ gerade }}}" loading="lazy"></dd></dl>

<msup>

</mrow>

</msup>

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

</mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</munderover>

<msup>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</msup>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</mrow>

<mtext> ungerade</mtext>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sin ^{n}x={\frac {1}{2^{n-1}}}\,\sum _{k=0}^{\frac {n-1}{2}}(-1)^{{\frac {n-1}{2}}-k}\,{n \choose k}\,\sin {((n-2k)x)};\quad n\in \mathbb {N} {\text{ und }}n{\text{ ungerade}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0eb3328288183e462d3665259431d79446d3b443.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:75.065ex; height:8.843ex;" alt="{\displaystyle \sin ^{n}x={\frac {1}{2^{n-1}}}\,\sum _{k=0}^{\frac {n-1}{2}}(-1)^{{\frac {n-1}{2}}-k}\,{n \choose k}\,\sin {((n-2k)x)};\quad n\in \mathbb {N} {\text{ und }}n{\text{ ungerade}}}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading4"><h4 id="Kosinus">Kosinus</h4></div>

<msup>

</mrow>

</msup>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \cos ^{2}x={\frac {1}{2}}\ {\Big (}1+\cos(2x){\Big )}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8ce1bce60e3ce89b72d63d9520c1c6fdfb01a0ba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:25.751ex; height:5.176ex;" alt="{\displaystyle \cos ^{2}x={\frac {1}{2}}\ {\Big (}1+\cos(2x){\Big )}}" loading="lazy"><a href="#cite_note-Bronstein2.5.2.1.3-13">[13]</a><a href="#cite_note-25">[25]</a></dd></dl>

<msup>

</mrow>

</msup>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \cos ^{3}x={\frac {1}{4}}\ {\Big (}3\,\cos x+\cos(3x){\Big )}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/61dce131c936a103d315f71180f93d3d034d300a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:31.353ex; height:5.176ex;" alt="{\displaystyle \cos ^{3}x={\frac {1}{4}}\ {\Big (}3\,\cos x+\cos(3x){\Big )}}" loading="lazy"><a href="#cite_note-Bronstein2.5.2.1.3-13">[13]</a><a href="#cite_note-26">[26]</a></dd></dl>

<msup>

</mrow>

</msup>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \cos ^{4}x={\frac {1}{8}}\ {\Big (}3+4\,\cos(2x)+\cos(4x){\Big )}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b2acba9d1ccf7ab25b3c464ee2435db78affdc7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:37.94ex; height:5.176ex;" alt="{\displaystyle \cos ^{4}x={\frac {1}{8}}\ {\Big (}3+4\,\cos(2x)+\cos(4x){\Big )}}" loading="lazy"><a href="#cite_note-Bronstein2.5.2.1.3-13">[13]</a><a href="#cite_note-27">[27]</a></dd></dl>

<msup>

</mrow>

</msup>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \cos ^{5}x={\frac {1}{16}}\ {\Big (}10\,\cos x+5\,\cos(3x)+\cos(5x){\Big )}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bad0eff417901c19314dfd5281c6a2e67fc7ea99.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:45.867ex; height:5.176ex;" alt="{\displaystyle \cos ^{5}x={\frac {1}{16}}\ {\Big (}10\,\cos x+5\,\cos(3x)+\cos(5x){\Big )}}" loading="lazy"><a href="#cite_note-28">[28]</a></dd></dl>

<msup>

</mrow>

</msup>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \cos ^{6}x={\frac {1}{32}}\ {\Big (}10+15\,\cos(2x)+6\,\cos(4x)+\cos(6x){\Big )}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a0e2f94aa463a71f88ee8ee4f982a5d935553bfc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:53.617ex; height:5.176ex;" alt="{\displaystyle \cos ^{6}x={\frac {1}{32}}\ {\Big (}10+15\,\cos(2x)+6\,\cos(4x)+\cos(6x){\Big )}}" loading="lazy"><a href="#cite_note-29">[29]</a></dd></dl>

<msup>

</mrow>

</msup>

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

</mrow>

</mrow>

</munderover>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \cos ^{n}x={\frac {1}{2^{n}}}\,\sum _{k=0}^{n}{n \choose k}\,\cos((n-2k)x);\quad n\in \mathbb {N} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0747fe412464be2dd2f102c1f935963b7385c474.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:46.406ex; height:7.009ex;" alt="{\displaystyle \cos ^{n}x={\frac {1}{2^{n}}}\,\sum _{k=0}^{n}{n \choose k}\,\cos((n-2k)x);\quad n\in \mathbb {N} }" loading="lazy"></dd></dl>

<msup>

</mrow>

</msup>

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

<mrow>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mfrac>

</mrow>

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</munderover>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</mrow>

<mtext> gerade </mtext>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \cos ^{n}x={\frac {1}{2^{n}}}\,{n \choose {\frac {n}{2}}}+{\frac {1}{2^{n-1}}}\sum _{k=0}^{{\frac {n}{2}}-1}{n \choose k}\,\cos {((n-2k)x)};\quad n\in \mathbb {N} {\text{ und }}n{\text{ gerade }}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/948ee4681c13e1a87c46510f1056924428ab2322.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:74.766ex; height:8.509ex;" alt="{\displaystyle \cos ^{n}x={\frac {1}{2^{n}}}\,{n \choose {\frac {n}{2}}}+{\frac {1}{2^{n-1}}}\sum _{k=0}^{{\frac {n}{2}}-1}{n \choose k}\,\cos {((n-2k)x)};\quad n\in \mathbb {N} {\text{ und }}n{\text{ gerade }}}" loading="lazy"></dd></dl>

<msup>

</mrow>

</msup>

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

</mrow>

<mfrac>

<mrow>

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</mfrac>

</mrow>

</munderover>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</mrow>

<mtext> ungerade</mtext>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \cos ^{n}x={\frac {1}{2^{n-1}}}\,\sum _{k=0}^{\frac {n-1}{2}}{n \choose k}\,\cos {((n-2k)x)};\quad n\in \mathbb {N} {\text{ und }}n{\text{ ungerade}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f09f6e84cb62249d3c20139fb902c6c3508a89c0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:65.087ex; height:8.843ex;" alt="{\displaystyle \cos ^{n}x={\frac {1}{2^{n-1}}}\,\sum _{k=0}^{\frac {n-1}{2}}{n \choose k}\,\cos {((n-2k)x)};\quad n\in \mathbb {N} {\text{ und }}n{\text{ ungerade}}}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading4"><h4 id="Tangens">Tangens</h4></div>

<msup>

</mrow>

</msup>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \tan ^{2}x={\frac {1-\cos(2x)}{1+\cos(2x)}}=\sec ^{2}(x)-1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8f240daafd1c60088ccd30555950244e55e5365a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:35.756ex; height:6.509ex;" alt="{\displaystyle \tan ^{2}x={\frac {1-\cos(2x)}{1+\cos(2x)}}=\sec ^{2}(x)-1}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Umrechnung_in_andere_trigonometrische_Funktionen">Umrechnung in andere trigonometrische Funktionen</h3></div>

<mi>arccos</mi>

<mi>arcsin</mi>

<msqrt>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sin(\arccos x)=\cos(\arcsin x)={\sqrt {1-x^{2}}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/eb3d925c7516a151e19c1214aeb9b27c83447d44.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.106ex; height:3.509ex;" alt="{\displaystyle \sin(\arccos x)=\cos(\arcsin x)={\sqrt {1-x^{2}}}}" loading="lazy"></dd></dl>

<mi>arctan</mi>

<mi>arccot</mi>

<mfrac>

<msqrt>

<msup>

</mrow>

</msup>

</msqrt>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sin(\arctan x)=\cos(\operatorname {arccot} x)={\frac {x}{\sqrt {1+x^{2}}}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/390c2be0f9a72cf04719626c37c60259e63dd0e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:41.435ex; height:6.009ex;" alt="{\displaystyle \sin(\arctan x)=\cos(\operatorname {arccot} x)={\frac {x}{\sqrt {1+x^{2}}}}}" loading="lazy"></dd></dl>

<mi>arccot</mi>

<mi>arctan</mi>

<mfrac>

<msqrt>

<msup>

</mrow>

</msup>

</msqrt>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sin(\operatorname {arccot} x)=\cos(\arctan x)={\frac {1}{\sqrt {1+x^{2}}}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e7b0f80a248c79c1bcb02755a611e206c2a723b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:41.435ex; height:6.509ex;" alt="{\displaystyle \sin(\operatorname {arccot} x)=\cos(\arctan x)={\frac {1}{\sqrt {1+x^{2}}}}}" loading="lazy"></dd></dl>

<mi>arcsin</mi>

<mi>arccos</mi>

<mfrac>

<msqrt>

<msup>

</mrow>

</msup>

</msqrt>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \tan(\arcsin x)=\cot(\arccos x)={\frac {x}{\sqrt {1-x^{2}}}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2c56c5d7ce7b22c46deb6223478d8a21195d3326.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:41.435ex; height:6.009ex;" alt="{\displaystyle \tan(\arcsin x)=\cot(\arccos x)={\frac {x}{\sqrt {1-x^{2}}}}}" loading="lazy"></dd></dl>

<mi>arccos</mi>

<mi>arcsin</mi>

<mfrac>

<msqrt>

<msup>

</mrow>

</msup>

</msqrt>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \tan(\arccos x)=\cot(\arcsin x)={\frac {\sqrt {1-x^{2}}}{x}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/24299451f1275ac355d38f6b76d6b7a0d5a0f121.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:41.435ex; height:6.176ex;" alt="{\displaystyle \tan(\arccos x)=\cot(\arcsin x)={\frac {\sqrt {1-x^{2}}}{x}}}" loading="lazy"></dd></dl>

<mi>arccot</mi>

<mi>arctan</mi>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \tan(\operatorname {arccot} x)=\cot(\arctan x)={\frac {1}{x}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9332541067936f15f11b173aefc580b162394618.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:34.546ex; height:5.176ex;" alt="{\displaystyle \tan(\operatorname {arccot} x)=\cot(\arctan x)={\frac {1}{x}}}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Weitere_Formeln_für_den_Fall_α_+_β_+_γ_=_180°">Weitere Formeln für den Fall α + β + γ = 180°</h3></div>

Die folgenden Formeln gelten für beliebige ebene Dreiecke und folgen nach längeren Termumformungen aus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha +\beta +\gamma =180^{\circ }}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha +\beta +\gamma =180^{\circ }}</annotation>

</semantics>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mtd>

<mtd>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\tan \alpha +\tan \beta +\tan \gamma &=\tan \alpha \tan \beta \tan \gamma \\\cot {\frac {\alpha }{2}}+\cot {\frac {\beta }{2}}+\cot {\frac {\gamma }{2}}&=\cot {\frac {\alpha }{2}}\cot {\frac {\beta }{2}}\cot {\frac {\gamma }{2}}\end{aligned}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/99499d302eadb4df23ec0c527dac0d8670d19f79.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.671ex; width:44.405ex; height:8.509ex;" alt="{\displaystyle {\begin{aligned}\tan \alpha +\tan \beta +\tan \gamma &=\tan \alpha \tan \beta \tan \gamma \\\cot {\frac {\alpha }{2}}+\cot {\frac {\beta }{2}}+\cot {\frac {\gamma }{2}}&=\cot {\frac {\alpha }{2}}\cot {\frac {\beta }{2}}\cot {\frac {\gamma }{2}}\end{aligned}}}" loading="lazy"></dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\cot \beta \cot \gamma +\cot \gamma \cot \alpha +\cot \alpha \cot \beta &=1\\\tan {\frac {\beta }{2}}\tan {\frac {\gamma }{2}}+\tan {\frac {\gamma }{2}}\tan {\frac {\alpha }{2}}+\tan {\frac {\alpha }{2}}\tan {\frac {\beta }{2}}&=1\end{aligned}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e9937abc6220db35f3c85c922706e8ca6e3877e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.671ex; width:47.516ex; height:8.509ex;" alt="{\displaystyle {\begin{aligned}\cot \beta \cot \gamma +\cot \gamma \cot \alpha +\cot \alpha \cot \beta &=1\\\tan {\frac {\beta }{2}}\tan {\frac {\gamma }{2}}+\tan {\frac {\gamma }{2}}\tan {\frac {\alpha }{2}}+\tan {\frac {\alpha }{2}}\tan {\frac {\beta }{2}}&=1\end{aligned}}}" loading="lazy"></dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\sin \alpha +\sin \beta +\sin \gamma &=4\cos {\frac {\alpha }{2}}\cos {\frac {\beta }{2}}\cos {\frac {\gamma }{2}}\\-\sin \alpha +\sin \beta +\sin \gamma &=4\cos {\frac {\alpha }{2}}\sin {\frac {\beta }{2}}\sin {\frac {\gamma }{2}}\end{aligned}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2709e6f233e149a2e769849cd4fe19f8296de880.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.838ex; width:44.945ex; height:10.843ex;" alt="{\displaystyle {\begin{aligned}\sin \alpha +\sin \beta +\sin \gamma &=4\cos {\frac {\alpha }{2}}\cos {\frac {\beta }{2}}\cos {\frac {\gamma }{2}}\\-\sin \alpha +\sin \beta +\sin \gamma &=4\cos {\frac {\alpha }{2}}\sin {\frac {\beta }{2}}\sin {\frac {\gamma }{2}}\end{aligned}}}" loading="lazy"></dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\cos \alpha +\cos \beta +\cos \gamma &=4\sin {\frac {\alpha }{2}}\sin {\frac {\beta }{2}}\sin {\frac {\gamma }{2}}+1\\-\cos \alpha +\cos \beta +\cos \gamma &=4\sin {\frac {\alpha }{2}}\cos {\frac {\beta }{2}}\cos {\frac {\gamma }{2}}-1\end{aligned}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cdd3051b57f186d94cff5c2bbb6be692a96297f3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.838ex; width:49.459ex; height:10.843ex;" alt="{\displaystyle {\begin{aligned}\cos \alpha +\cos \beta +\cos \gamma &=4\sin {\frac {\alpha }{2}}\sin {\frac {\beta }{2}}\sin {\frac {\gamma }{2}}+1\\-\cos \alpha +\cos \beta +\cos \gamma &=4\sin {\frac {\alpha }{2}}\cos {\frac {\beta }{2}}\cos {\frac {\gamma }{2}}-1\end{aligned}}}" loading="lazy"></dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\sin 2\alpha +\sin 2\beta +\sin 2\gamma &=4\sin \alpha \sin \beta \sin \gamma \\-\sin 2\alpha +\sin 2\beta +\sin 2\gamma &=4\sin \alpha \cos \beta \cos \gamma \end{aligned}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a4fe75bc5d4f8fd7d27b28f6b7b56f08176f7a03.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:45.668ex; height:5.843ex;" alt="{\displaystyle {\begin{aligned}\sin 2\alpha +\sin 2\beta +\sin 2\gamma &=4\sin \alpha \sin \beta \sin \gamma \\-\sin 2\alpha +\sin 2\beta +\sin 2\gamma &=4\sin \alpha \cos \beta \cos \gamma \end{aligned}}}" loading="lazy"></dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\cos 2\alpha +\cos 2\beta +\cos 2\gamma &=-4\cos \alpha \cos \beta \cos \gamma -1\\-\cos 2\alpha +\cos 2\beta +\cos 2\gamma &=-4\cos \alpha \sin \beta \sin \gamma +1\end{aligned}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fe108fd47c28144e826a03bab8cf2ec824d85e24.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:52.501ex; height:5.843ex;" alt="{\displaystyle {\begin{aligned}\cos 2\alpha +\cos 2\beta +\cos 2\gamma &=-4\cos \alpha \cos \beta \cos \gamma -1\\-\cos 2\alpha +\cos 2\beta +\cos 2\gamma &=-4\cos \alpha \sin \beta \sin \gamma +1\end{aligned}}}" loading="lazy"></dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\sin ^{2}\alpha +\sin ^{2}\beta +\sin ^{2}\gamma &=2\cos \alpha \cos \beta \cos \gamma +2\\-\sin ^{2}\alpha +\sin ^{2}\beta +\sin ^{2}\gamma &=2\cos \alpha \sin \beta \sin \gamma \end{aligned}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1e6ae99058f5b69d0947b8eff2af619f790ff08e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:49.602ex; height:6.176ex;" alt="{\displaystyle {\begin{aligned}\sin ^{2}\alpha +\sin ^{2}\beta +\sin ^{2}\gamma &=2\cos \alpha \cos \beta \cos \gamma +2\\-\sin ^{2}\alpha +\sin ^{2}\beta +\sin ^{2}\gamma &=2\cos \alpha \sin \beta \sin \gamma \end{aligned}}}" loading="lazy"></dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\cos ^{2}\alpha +\cos ^{2}\beta +\cos ^{2}\gamma &=-2\cos \alpha \cos \beta \cos \gamma +1\\-\cos ^{2}\alpha +\cos ^{2}\beta +\cos ^{2}\gamma &=-2\cos \alpha \sin \beta \sin \gamma +1\end{aligned}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/874c31ed18c415141d638730f942d68ea872911c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:52.177ex; height:6.176ex;" alt="{\displaystyle {\begin{aligned}\cos ^{2}\alpha +\cos ^{2}\beta +\cos ^{2}\gamma &=-2\cos \alpha \cos \beta \cos \gamma +1\\-\cos ^{2}\alpha +\cos ^{2}\beta +\cos ^{2}\gamma &=-2\cos \alpha \sin \beta \sin \gamma +1\end{aligned}}}" loading="lazy"></dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\sin ^{2}2\alpha +\sin ^{2}2\beta +\sin ^{2}2\gamma &=-2\cos 2\alpha \cos 2\beta \cos 2\gamma +2\\-\sin ^{2}2\alpha +\sin ^{2}2\beta +\sin ^{2}2\gamma &=-2\cos 2\alpha \sin 2\beta \sin 2\gamma \end{aligned}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d15da4ad27accafc2848be6b1502e3ff94c16c92.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:58.385ex; height:6.176ex;" alt="{\displaystyle {\begin{aligned}\sin ^{2}2\alpha +\sin ^{2}2\beta +\sin ^{2}2\gamma &=-2\cos 2\alpha \cos 2\beta \cos 2\gamma +2\\-\sin ^{2}2\alpha +\sin ^{2}2\beta +\sin ^{2}2\gamma &=-2\cos 2\alpha \sin 2\beta \sin 2\gamma \end{aligned}}}" loading="lazy"></dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\cos ^{2}2\alpha +\cos ^{2}2\beta +\cos ^{2}2\gamma &=2\cos 2\alpha \cos 2\beta \cos 2\gamma +1\\-\cos ^{2}2\alpha +\cos ^{2}2\beta +\cos ^{2}2\gamma &=2\cos 2\alpha \sin 2\beta \sin 2\gamma +1\end{aligned}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4721bb819e8987046d2b6cd071bbbc6c4fa57c75.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:57.343ex; height:6.176ex;" alt="{\displaystyle {\begin{aligned}\cos ^{2}2\alpha +\cos ^{2}2\beta +\cos ^{2}2\gamma &=2\cos 2\alpha \cos 2\beta \cos 2\gamma +1\\-\cos ^{2}2\alpha +\cos ^{2}2\beta +\cos ^{2}2\gamma &=2\cos 2\alpha \sin 2\beta \sin 2\gamma +1\end{aligned}}}" loading="lazy"></dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\sin ^{2}{\frac {\alpha }{2}}+\sin ^{2}{\frac {\beta }{2}}+\sin ^{2}{\frac {\gamma }{2}}&=-2\sin {\frac {\alpha }{2}}\sin {\frac {\beta }{2}}\sin {\frac {\gamma }{2}}+1\\-\sin ^{2}{\frac {\alpha }{2}}+\sin ^{2}{\frac {\beta }{2}}+\sin ^{2}{\frac {\gamma }{2}}&=-2\sin {\frac {\alpha }{2}}\cos {\frac {\beta }{2}}\cos {\frac {\gamma }{2}}+1\end{aligned}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f7380165eb01de9b9139d65cf55f037c968e3f3a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.838ex; width:56.171ex; height:10.843ex;" alt="{\displaystyle {\begin{aligned}\sin ^{2}{\frac {\alpha }{2}}+\sin ^{2}{\frac {\beta }{2}}+\sin ^{2}{\frac {\gamma }{2}}&=-2\sin {\frac {\alpha }{2}}\sin {\frac {\beta }{2}}\sin {\frac {\gamma }{2}}+1\\-\sin ^{2}{\frac {\alpha }{2}}+\sin ^{2}{\frac {\beta }{2}}+\sin ^{2}{\frac {\gamma }{2}}&=-2\sin {\frac {\alpha }{2}}\cos {\frac {\beta }{2}}\cos {\frac {\gamma }{2}}+1\end{aligned}}}" loading="lazy"></dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\cos ^{2}{\frac {\alpha }{2}}+\cos ^{2}{\frac {\beta }{2}}+\cos ^{2}{\frac {\gamma }{2}}&=2\sin {\frac {\alpha }{2}}\sin {\frac {\beta }{2}}\sin {\frac {\gamma }{2}}+2\\-\cos ^{2}{\frac {\alpha }{2}}+\cos ^{2}{\frac {\beta }{2}}+\cos ^{2}{\frac {\gamma }{2}}&=2\sin {\frac {\alpha }{2}}\cos {\frac {\beta }{2}}\cos {\frac {\gamma }{2}}\end{aligned}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2301dcdeaaa97e61ae6c200022bfded8b7d455e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.838ex; width:54.619ex; height:10.843ex;" alt="{\displaystyle {\begin{aligned}\cos ^{2}{\frac {\alpha }{2}}+\cos ^{2}{\frac {\beta }{2}}+\cos ^{2}{\frac {\gamma }{2}}&=2\sin {\frac {\alpha }{2}}\sin {\frac {\beta }{2}}\sin {\frac {\gamma }{2}}+2\\-\cos ^{2}{\frac {\alpha }{2}}+\cos ^{2}{\frac {\beta }{2}}+\cos ^{2}{\frac {\gamma }{2}}&=2\sin {\frac {\alpha }{2}}\cos {\frac {\beta }{2}}\cos {\frac {\gamma }{2}}\end{aligned}}}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Sinusoid_und_Linearkombination_mit_gleicher_Phase">Sinusoid und Linearkombination mit gleicher Phase</h3></div>

<dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}a\sin \alpha +b\cos \alpha =&{\begin{cases}{\sqrt {a^{2}+b^{2}}}\sin \left(\alpha +\arctan \left({\tfrac {b}{a}}\right)\right)&{\text{, für alle }}a>0\\{\sqrt {a^{2}+b^{2}}}\cos \left(\alpha -\arctan \left({\tfrac {a}{b}}\right)\right)&{\text{, für alle }}b>0\end{cases}}\end{aligned}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}a\sin \alpha +b\cos \alpha =&{\begin{cases}{\sqrt {a^{2}+b^{2}}}\sin \left(\alpha +\arctan \left({\tfrac {b}{a}}\right)\right)&{\text{, für alle }}a>0\\{\sqrt {a^{2}+b^{2}}}\cos \left(\alpha -\arctan \left({\tfrac {a}{b}}\right)\right)&{\text{, für alle }}b>0\end{cases}}\end{aligned}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/08b4b9d1ee876c5d8a75fcd2807e7be8a4501c93.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:69.36ex; height:9.343ex;" alt="{\displaystyle {\begin{aligned}a\sin \alpha +b\cos \alpha =&{\begin{cases}{\sqrt {a^{2}+b^{2}}}\sin \left(\alpha +\arctan \left({\tfrac {b}{a}}\right)\right)&{\text{, für alle }}a>0\\{\sqrt {a^{2}+b^{2}}}\cos \left(\alpha -\arctan \left({\tfrac {a}{b}}\right)\right)&{\text{, für alle }}b>0\end{cases}}\end{aligned}}}" loading="lazy"></dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}a\cos \alpha +b\sin \alpha =\operatorname {sgn}(a){\sqrt {a^{2}+b^{2}}}\cos \left(\alpha +\arctan \left(-{\tfrac {b}{a}}\right)\right)\end{aligned}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/74ce17383bb44184817be083a504ec7632af4189.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:58.676ex; height:4.843ex;" alt="{\displaystyle {\begin{aligned}a\cos \alpha +b\sin \alpha =\operatorname {sgn} (a){\sqrt {a^{2}+b^{2}}}\cos \left(\alpha +\arctan \left(-{\tfrac {b}{a}}\right)\right)\end{aligned}}}" loading="lazy"><a href="#cite_note-30">[30]</a></dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle a\sin(x+\alpha )+b\sin(x+\beta )={\sqrt {a^{2}+b^{2}+2ab\cos(\alpha -\beta )}}\cdot \sin(x+\delta ),}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a366244bdf45de7731ac858703b8350e627c247d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:68.337ex; height:4.843ex;" alt="{\displaystyle a\sin(x+\alpha )+b\sin(x+\beta )={\sqrt {a^{2}+b^{2}+2ab\cos(\alpha -\beta )}}\cdot \sin(x+\delta ),}" loading="lazy"></dd></dl>

wobei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta =\operatorname {atan2} (a\sin \alpha +b\sin \beta ,a\cos \alpha +b\cos \beta ).}">

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<annotation encoding="application/x-tex">{\displaystyle \delta =\operatorname {atan2} (a\sin \alpha +b\sin \beta ,a\cos \alpha +b\cos \beta ).}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/af1753293541bad0dd84f3a303c7b3961f10fefc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:44.127ex; height:2.843ex;" alt="{\displaystyle \delta =\operatorname {atan2} (a\sin \alpha +b\sin \beta ,a\cos \alpha +b\cos \beta ).}" loading="lazy">

Allgemeiner ist

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<annotation encoding="application/x-tex">{\displaystyle \sum _{i}a_{i}\sin(x+\delta _{i})=a\sin(x+\delta ),}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6436361f9869baa8e2dfb11f06aa19247d9e087a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:32.072ex; height:5.509ex;" alt="{\displaystyle \sum _{i}a_{i}\sin(x+\delta _{i})=a\sin(x+\delta ),}" loading="lazy"></dd></dl>

wobei

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<annotation encoding="application/x-tex">{\displaystyle a^{2}=\sum _{i,j}a_{i}a_{j}\cos(\delta _{i}-\delta _{j})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/17d757c684ae3fdce8dca0b3b1b9bdf44555cf11.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:25.216ex; height:5.843ex;" alt="{\displaystyle a^{2}=\sum _{i,j}a_{i}a_{j}\cos(\delta _{i}-\delta _{j})}" loading="lazy"></dd></dl>

und

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<annotation encoding="application/x-tex">{\displaystyle \delta =\operatorname {atan2} \left(\sum _{i}a_{i}\sin \delta _{i},\sum _{i}a_{i}\cos \delta _{i}\right).}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8ca1f584829a4adb251629bc8b258f1edf5191ce.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:37.916ex; height:7.509ex;" alt="{\displaystyle \delta =\operatorname {atan2} \left(\sum _{i}a_{i}\sin \delta _{i},\sum _{i}a_{i}\cos \delta _{i}\right).}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Ableitungen_und_Stammfunktionen">Ableitungen und Stammfunktionen</h3></div>

Siehe Formelsammlung <a href="Tabelle_von_Ableitungs-_und_Stammfunktionen#Trigonometrische_Funktionen_und_Hyperbelfunktionen" title="Tabelle von Ableitungs- und Stammfunktionen">Ableitungen und Stammfunktionen</a>

<div class="mw-heading mw-heading3"><h3 id="Bestimmte_Integrale">Bestimmte Integrale</h3></div>

Die Lösungen der nachfolgenden bestimmten Integrale stehen im Zusammenhang mit der <a href="Eulersche_Betafunktion" title="Eulersche Betafunktion">Euler’schen Betafunktion</a>, welche weiterhin mit der <a href="Gammafunktion" title="Gammafunktion">Gammafunktion</a> verknüpft ist. Das zweite Integral ist z. B. in der Physik bei der Berechnung von Kräften zwischen <a href="Magnet" title="Magnet">zylinderförmigen Dauermagneten</a> unter Verwendung der sogenannten <a href="Multipolentwicklung" title="Multipolentwicklung">Multipol-Entwicklung</a> hilfreich.

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<annotation encoding="application/x-tex">{\displaystyle \int _{0}^{\pi /2}\cos ^{\nu _{1}}\varphi \sin ^{\nu _{2}}\varphi \;{\text{d}}\varphi ={\frac {1}{2}}\cdot \operatorname {B} \left({\frac {\nu _{1}+1}{2}},{\frac {\nu _{2}+1}{2}}\right)\;,\quad {\text{Re}}\left({\frac {\nu _{j}+1}{2}}\right)>0}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b8f1c58f4ea4db61a908f18c7b095cb7c355165d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:71.719ex; height:6.509ex;" alt="{\displaystyle \int _{0}^{\pi /2}\cos ^{\nu _{1}}\varphi \sin ^{\nu _{2}}\varphi \;{\text{d}}\varphi ={\frac {1}{2}}\cdot \operatorname {B} \left({\frac {\nu _{1}+1}{2}},{\frac {\nu _{2}+1}{2}}\right)\;,\quad {\text{Re}}\left({\frac {\nu _{j}+1}{2}}\right)>0}" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle \int _{0}^{\pi }\cos ^{\nu _{1}}\varphi \sin ^{\nu _{2}}\varphi \;{\text{d}}\varphi =\operatorname {B} \left({\frac {\nu _{1}+1}{2}},{\frac {\nu _{2}+1}{2}}\right)\cdot {\frac {1+(-1)^{\nu _{1}}}{2}},\quad \nu _{j}=0,1,2,3,\dots }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f572661ae42bb425cef4419c741508c0db9c7f39.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:77.882ex; height:6.343ex;" alt="{\displaystyle \int _{0}^{\pi }\cos ^{\nu _{1}}\varphi \sin ^{\nu _{2}}\varphi \;{\text{d}}\varphi =\operatorname {B} \left({\frac {\nu _{1}+1}{2}},{\frac {\nu _{2}+1}{2}}\right)\cdot {\frac {1+(-1)^{\nu _{1}}}{2}},\quad \nu _{j}=0,1,2,3,\dots }" loading="lazy"></dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle \int _{0}^{2\pi }\cos ^{\nu _{1}}\varphi \sin ^{\nu _{2}}\varphi \;{\text{d}}\varphi =2\cdot \operatorname {B} \left({\frac {\nu _{1}+1}{2}},{\frac {\nu _{2}+1}{2}}\right)\cdot {\frac {1+(-1)^{\nu _{1}}}{2}}\cdot {\frac {1+(-1)^{\nu _{2}}}{2}},\quad \nu _{j}=0,1,2,3,\dots }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d17dbfec38be99510629e98df1cd9fb8f6516bde.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:94.719ex; height:6.343ex;" alt="{\displaystyle \int _{0}^{2\pi }\cos ^{\nu _{1}}\varphi \sin ^{\nu _{2}}\varphi \;{\text{d}}\varphi =2\cdot \operatorname {B} \left({\frac {\nu _{1}+1}{2}},{\frac {\nu _{2}+1}{2}}\right)\cdot {\frac {1+(-1)^{\nu _{1}}}{2}}\cdot {\frac {1+(-1)^{\nu _{2}}}{2}},\quad \nu _{j}=0,1,2,3,\dots }" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Reihenentwicklung">Reihenentwicklung</h3></div>

Wie auch sonst in der <a href="Analysis" title="Analysis">Analysis</a> werden alle Winkel im <a href="Bogenma%C3%9F" class="mw-redirect" title="Bogenmaß">Bogenmaß</a> angegeben.

Man kann zeigen, dass der Kosinus die <a href="Differentialrechnung" title="Differentialrechnung">Ableitung</a> des Sinus darstellt und die Ableitung des Kosinus der negative Sinus ist. Hat man diese Ableitungen, kann man die <a href="Taylorreihe" title="Taylorreihe">Taylorreihe</a> entwickeln (am einfachsten mit dem Entwicklungspunkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=0}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/953917eaf52f2e1baad54c8c9e3d6f9bb3710cdc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;" alt="{\displaystyle x=0}" loading="lazy">) und zeigen, dass die folgenden Identitäten für alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

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<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> aus den <a href="Reelle_Zahlen" class="mw-redirect" title="Reelle Zahlen">reellen Zahlen</a> gelten. Mit diesen Reihen werden die trigonometrischen Funktionen für <a href="Komplexe_Zahl" title="Komplexe Zahl">komplexe</a> Argumente definiert (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{n}}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2f568bf6d34e97b9fdda0dc7e276d6c4501d2045.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.982ex; height:2.509ex;" alt="{\displaystyle B_{n}}" loading="lazy"> bzw. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta _{n}}">

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<annotation encoding="application/x-tex">{\displaystyle \beta _{n}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6f6cd4f61e83a945551749f7384429eeee37b2b1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.534ex; height:2.509ex;" alt="{\displaystyle \beta _{n}}" loading="lazy"> bezeichnet dabei die <a href="Bernoulli-Zahl" title="Bernoulli-Zahl">Bernoulli-Zahlen</a>):

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\sin x&=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n+1)!}}x^{2n+1}\\&=x-{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}-{\frac {x^{7}}{7!}}\pm \cdots \;,\qquad |x|<\infty \end{aligned}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/09c6a6662292ba76906959205400b479fb862116.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.838ex; width:48.254ex; height:12.843ex;" alt="{\displaystyle {\begin{aligned}\sin x&=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n+1)!}}x^{2n+1}\\&=x-{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}-{\frac {x^{7}}{7!}}\pm \cdots \;,\qquad |x|<\infty \end{aligned}}}" loading="lazy"></dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\cos x&=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n)!}}x^{2n}\\&=1-{\frac {x^{2}}{2!}}+{\frac {x^{4}}{4!}}-{\frac {x^{6}}{6!}}\pm \cdots \;,\qquad |x|<\infty \end{aligned}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5dfc7edf91f13dc125faf05d17849b55e40a0cab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.838ex; width:48.342ex; height:12.843ex;" alt="{\displaystyle {\begin{aligned}\cos x&=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n)!}}x^{2n}\\&=1-{\frac {x^{2}}{2!}}+{\frac {x^{4}}{4!}}-{\frac {x^{6}}{6!}}\pm \cdots \;,\qquad |x|<\infty \end{aligned}}}" loading="lazy"></dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\tan x&=\sum _{n=1}^{\infty }(-1)^{n}{\frac {2^{2n}(1-2^{2n})\beta _{2n}}{(2n)!}}x^{2n-1}=\sum _{n=1}^{\infty }{\frac {(-1)^{n-1}2^{2n}(2^{2n}-1)B_{2n}}{(2n)!}}x^{2n-1}\\&=x+{\frac {1}{3}}x^{3}+{\frac {2}{15}}x^{5}+{\frac {17}{315}}x^{7}+{\frac {62}{2835}}x^{9}+\,\cdots \qquad |x|<{\tfrac {\pi }{2}}\end{aligned}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cfafb49683fa8218e083163121e41ac03f5f1402.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:77.429ex; height:12.509ex;" alt="{\displaystyle {\begin{aligned}\tan x&=\sum _{n=1}^{\infty }(-1)^{n}{\frac {2^{2n}(1-2^{2n})\beta _{2n}}{(2n)!}}x^{2n-1}=\sum _{n=1}^{\infty }{\frac {(-1)^{n-1}2^{2n}(2^{2n}-1)B_{2n}}{(2n)!}}x^{2n-1}\\&=x+{\frac {1}{3}}x^{3}+{\frac {2}{15}}x^{5}+{\frac {17}{315}}x^{7}+{\frac {62}{2835}}x^{9}+\,\cdots \qquad |x|<{\tfrac {\pi }{2}}\end{aligned}}}" loading="lazy"><a href="#cite_note-31">[31]</a></dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\cot x&={\frac {1}{x}}-\sum _{n=1}^{\infty }{\frac {\left(-1\right)^{n-1}2^{2n}\beta _{2n}}{(2n)!}}x^{2n-1}={\frac {1}{x}}-\sum _{n=1}^{\infty }{\frac {(-1)^{n-1}2^{2n}B_{2n}}{(2n)!}}x^{2n-1}\\&={\frac {1}{x}}-{\frac {1}{3}}x-{\frac {1}{45}}x^{3}-{\frac {2}{945}}x^{5}-{\frac {1}{4725}}x^{7}-\,\cdots ,\qquad 0<|x|<\pi \end{aligned}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1647f2deb72c21217b280340b39277e636f8fa92.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.838ex; width:71.639ex; height:12.843ex;" alt="{\displaystyle {\begin{aligned}\cot x&={\frac {1}{x}}-\sum _{n=1}^{\infty }{\frac {\left(-1\right)^{n-1}2^{2n}\beta _{2n}}{(2n)!}}x^{2n-1}={\frac {1}{x}}-\sum _{n=1}^{\infty }{\frac {(-1)^{n-1}2^{2n}B_{2n}}{(2n)!}}x^{2n-1}\\&={\frac {1}{x}}-{\frac {1}{3}}x-{\frac {1}{45}}x^{3}-{\frac {2}{945}}x^{5}-{\frac {1}{4725}}x^{7}-\,\cdots ,\qquad 0<|x|<\pi \end{aligned}}}" loading="lazy"><a href="#cite_note-32">[32]</a></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Produktentwicklung">Produktentwicklung</h3></div>

</mrow>

</mrow>

</munderover>

<mrow>

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<msup>

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</msup>

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<msup>

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<msup>

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</msup>

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</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sin(x)=x\prod _{k=1}^{\infty }\left(1-{\frac {x^{2}}{k^{2}\pi ^{2}}}\right)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/babb7ccd3c5a73b532fde2dd81644275d9178892.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:27.08ex; height:6.843ex;" alt="{\displaystyle \sin(x)=x\prod _{k=1}^{\infty }\left(1-{\frac {x^{2}}{k^{2}\pi ^{2}}}\right)}" loading="lazy"></dd></dl>

</mrow>

</mrow>

</munderover>

<mrow>

<mrow>

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<mrow>

<msup>

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</msup>

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<mrow>

<msup>

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<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \cos(x)=\prod _{k=1}^{\infty }\left(1-{\frac {4x^{2}}{(2k-1)^{2}\pi ^{2}}}\right)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/eee41de17dcf51a45517a0359c375d0f82fd7bd5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:32.594ex; height:6.843ex;" alt="{\displaystyle \cos(x)=\prod _{k=1}^{\infty }\left(1-{\frac {4x^{2}}{(2k-1)^{2}\pi ^{2}}}\right)}" loading="lazy"></dd>

</mrow>

</mrow>

</munderover>

<mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sin(x)=\prod _{n=-\infty }^{\infty }\left({\frac {x+n\pi }{{\frac {\pi }{2}}+n\pi }}\right)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0c6114aef9bc6d486b907f4321a1fb7bdb062657.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:26.529ex; height:7.509ex;" alt="{\displaystyle \sin(x)=\prod _{n=-\infty }^{\infty }\left({\frac {x+n\pi }{{\frac {\pi }{2}}+n\pi }}\right)}" loading="lazy"></dd>

</mrow>

</mrow>

</munderover>

<mrow>

<mfrac>

<mrow>

<mfrac>

</mfrac>

</mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \cos(x)=\prod _{n=-\infty }^{\infty }\left({\frac {x+n\pi +{\frac {\pi }{2}}}{{\frac {\pi }{2}}+n\pi }}\right)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bf11993c9768e45c7a5b3d343e2cce10ebfd71df.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:30.955ex; height:7.509ex;" alt="{\displaystyle \cos(x)=\prod _{n=-\infty }^{\infty }\left({\frac {x+n\pi +{\frac {\pi }{2}}}{{\frac {\pi }{2}}+n\pi }}\right)}" loading="lazy"></dd>

</mrow>

</mrow>

</munderover>

<mrow>

<mfrac>

<mrow>

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<mrow>

<mfrac>

</mfrac>

</mrow>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \tan(x)=\prod _{n=-\infty }^{\infty }\left({\frac {x+n\pi }{x+n\pi +{\frac {\pi }{2}}}}\right)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/92dc105991696a839dc00fe30c8222607c460edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:31.203ex; height:7.509ex;" alt="{\displaystyle \tan(x)=\prod _{n=-\infty }^{\infty }\left({\frac {x+n\pi }{x+n\pi +{\frac {\pi }{2}}}}\right)}" loading="lazy"></dd>

</mrow>

</mrow>

</munderover>

<mrow>

<mfrac>

<mrow>

<mfrac>

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</mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \csc(x)=\prod _{n=-\infty }^{\infty }\left({\frac {{\frac {\pi }{2}}+n\pi }{x+n\pi }}\right)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/38cf26717eb22cdab9f9899915c15cd07115ef4d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:26.655ex; height:7.509ex;" alt="{\displaystyle \csc(x)=\prod _{n=-\infty }^{\infty }\left({\frac {{\frac {\pi }{2}}+n\pi }{x+n\pi }}\right)}" loading="lazy"></dd>

</mrow>

</mrow>

</munderover>

<mrow>

<mfrac>

<mrow>

<mfrac>

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<mrow>

<mfrac>

</mfrac>

</mrow>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sec(x)=\prod _{n=-\infty }^{\infty }\left({\frac {{\frac {\pi }{2}}+n\pi }{x+n\pi +{\frac {\pi }{2}}}}\right)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/874ae90fbdbb1ae04d1f2e1e585ab41909d3e744.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:30.825ex; height:7.509ex;" alt="{\displaystyle \sec(x)=\prod _{n=-\infty }^{\infty }\left({\frac {{\frac {\pi }{2}}+n\pi }{x+n\pi +{\frac {\pi }{2}}}}\right)}" loading="lazy"></dd>

</mrow>

</mrow>

</munderover>

<mrow>

<mfrac>

<mrow>

<mfrac>

</mfrac>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \cot(x)=\prod _{n=-\infty }^{\infty }\left({\frac {x+n\pi +{\frac {\pi }{2}}}{x+n\pi }}\right)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/46ef5780d32ca536260b624f8446c98e4934fda6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:30.943ex; height:7.509ex;" alt="{\displaystyle \cot(x)=\prod _{n=-\infty }^{\infty }\left({\frac {x+n\pi +{\frac {\pi }{2}}}{x+n\pi }}\right)}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Zusammenhang_mit_der_komplexen_Exponentialfunktion">Zusammenhang mit der komplexen Exponentialfunktion</h3></div>

Ferner besteht zwischen den Funktionen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sin x}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/09b4b55580d6a821a07ad9fe35be88976917b10b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.572ex; height:2.176ex;" alt="{\displaystyle \sin x}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \cos x}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/184ba70c3a71df25a25c09f34cd7f8175a9b5280.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.828ex; height:1.676ex;" alt="{\displaystyle \cos x}" loading="lazy"> und der <a href="Exponentialfunktion#Exponentialfunktion_auf_den_komplexen_Zahlen" title="Exponentialfunktion">komplexen Exponentialfunktion</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exp(\mathrm {i} x)}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \exp(\mathrm {i} x)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4f645a50f11e7ee97b32b54095c2d52df2ed1a4f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.338ex; height:2.843ex;" alt="{\displaystyle \exp(\mathrm {i} x)}" loading="lazy"> folgender Zusammenhang:

</mrow>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \exp(\pm \mathrm {i} x)=\cos x\pm \mathrm {i} \sin x=e^{\pm \mathrm {i} x}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d67cb35b2557b25aebc2e8f2c5893442a195f1f2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.61ex; height:3.176ex;" alt="{\displaystyle \exp(\pm \mathrm {i} x)=\cos x\pm \mathrm {i} \sin x=e^{\pm \mathrm {i} x}}" loading="lazy"> (<a href="Eulersche_Formel" title="Eulersche Formel">Eulersche Formel</a>)</dd></dl>

Weiterhin wird <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos {x}+\mathrm {i} \sin {x}=:\operatorname {cis} (x)}">

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \cos {x}+\mathrm {i} \sin {x}=:\operatorname {cis} (x)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e28af42ed9007d7d96e8fd7d0bc88354d3841730.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.754ex; height:2.843ex;" alt="{\displaystyle \cos {x}+\mathrm {i} \sin {x}=:\operatorname {cis} (x)}" loading="lazy"> geschrieben.<a href="#cite_note-33">[33]</a>

Auf Grund der oben genannten Symmetrien gilt weiter:

<mfrac>

<mrow>

</mrow>

</mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \cos x={\frac {\exp(\mathrm {i} x)+\exp(-\mathrm {i} x)}{2}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b0313e645fccb1e14ea9e3d5bb148f9c24ea29cb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:28.087ex; height:5.676ex;" alt="{\displaystyle \cos x={\frac {\exp(\mathrm {i} x)+\exp(-\mathrm {i} x)}{2}}}" loading="lazy"></dd></dl>

<mfrac>

<mrow>

</mrow>

</mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sin x={\frac {\exp(\mathrm {i} x)-\exp(-\mathrm {i} x)}{2\mathrm {i} }}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1cd04620f6a14015c6e0b38b6c9e50a918d8364c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:27.832ex; height:5.676ex;" alt="{\displaystyle \sin x={\frac {\exp(\mathrm {i} x)-\exp(-\mathrm {i} x)}{2\mathrm {i} }}}" loading="lazy"></dd></dl>

Mit diesen Beziehungen können einige Additionstheoreme besonders einfach und elegant hergeleitet werden.

<div class="mw-heading mw-heading2"><h2 id="Sphärische_Trigonometrie">Sphärische Trigonometrie</h2></div>

Eine Formelsammlung für das <a href="Sph%C3%A4rische_Trigonometrie#Rechtwinkliges_Kugeldreieck" title="Sphärische Trigonometrie">rechtwinklige</a> und das <a href="Sph%C3%A4rische_Trigonometrie#Sätze_für_das_allgemeine_Kugeldreieck" title="Sphärische Trigonometrie">allgemeine Dreieck</a> auf der Kugeloberfläche findet sich in einem eigenen Kapitel.

<div class="mw-heading mw-heading2"><h2 id="Weblinks">Weblinks</h2></div>

<ul><li><a href="Abramowitz-Stegun" title="Abramowitz-Stegun">Abramowitz-Stegun</a>: <a rel="nofollow" class="external text" href="https://personal.math.ubc.ca/~cbm/aands/abramowitz_and_stegun.pdf">Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Einzelnachweise">Einzelnachweise</h2></div>

<li id="cite_note-1"><a href="#cite_ref-1">↑</a> <a href="Die_Wurzel" title="Die Wurzel">Die Wurzel</a> 2006/04+05, 104ff., ohne Beweis

</li>

<li id="cite_note-2"><a href="#cite_ref-2">↑</a> Joachim Mohr: <a rel="nofollow" class="external text" href="https://kilchb.de/cosinuswerte.php">Kosinus-, Sinus und Tangenswerte</a>, abgerufen am 1. Juni 2016

</li>

<li id="cite_note-3"><a href="#cite_ref-3">↑</a> Ausführliche Beweise in <a class="external text" href="https://de.wikibooks.org/wiki/Beweisarchiv:_Geometrie:_Trigonometrie:_Additionstheoreme">Wikibooks Beweisarchiv.</a>

</li>

<li id="cite_note-forster_anal1-4">↑ <a href="#cite_ref-forster_anal1_4-0">a</a> <a href="#cite_ref-forster_anal1_4-1">b</a> Otto Forster: Analysis 1. Differential- und Integralrechnung einer Veränderlichen. vieweg 1983, Seite 87.

</li>

<li id="cite_note-5"><a href="#cite_ref-5">↑</a> Roger B. Nelsen: Beweise ohne Worte, Deutschsprachige Ausgabe herausgegeben von Nicola Oswald, <a href="Springer_Spektrum" title="Springer Spektrum">Springer Spektrum</a>, Springer-Verlag <a href="Berlin" title="Berlin">Berlin</a> <a href="Heidelberg" title="Heidelberg">Heidelberg</a> 2016, ISBN 978-3-662-50330-0, Seite 44

</li>

<li id="cite_note-6"><a href="#cite_ref-6">↑</a> I. N. Bronstein, K. A. Semendjajew: Taschenbuch der Mathematik. 19. Auflage, 1979. B.G. Teubner Verlagsgesellschaft, Leipzig. S. 237.

</li>

<li id="cite_note-7"><a href="#cite_ref-7">↑</a> Roger B. Nelsen: Beweise ohne Worte, Deutschsprachige Ausgabe herausgegeben von Nicola Oswald, <a href="Springer_Spektrum" title="Springer Spektrum">Springer Spektrum</a>, Springer-Verlag <a href="Berlin" title="Berlin">Berlin</a> <a href="Heidelberg" title="Heidelberg">Heidelberg</a> 2016, ISBN 978-3-662-50330-0, Seite 46

</li>

<li id="cite_note-8"><a href="#cite_ref-8">↑</a> Milton Abramowitz and Irene A. Stegun, <a rel="nofollow" class="external text" href="https://personal.math.ubc.ca/~cbm/aands/abramowitz_and_stegun.pdf">22.3.15</a>, (s. a. oben „Weblinks“)

</li>

<li id="cite_note-9"><a href="#cite_ref-9">↑</a> Milton Abramowitz and Irene A. Stegun, <a rel="nofollow" class="external text" href="https://personal.math.ubc.ca/~cbm/aands/abramowitz_and_stegun.pdf">4.3.27</a>, (s. a. oben „Weblinks“)

</li>

<li id="cite_note-10"><a href="#cite_ref-10">↑</a> Milton Abramowitz and Irene A. Stegun, <a rel="nofollow" class="external text" href="https://personal.math.ubc.ca/~cbm/aands/abramowitz_and_stegun.pdf">4.3.29</a>, (s. a. oben „Weblinks“)

</li>

<li id="cite_note-11"><a href="#cite_ref-11">↑</a> <a href="Gradshteyn-Ryzhik" title="Gradshteyn-Ryzhik">I. S. Gradshteyn and I. M. Ryzhik</a>, Table of Integrals, Series, and Products, Academic Press, 5th edition (1994), ISBN 0-12-294755-X 1.333.4

</li>

<li id="cite_note-12"><a href="#cite_ref-12">↑</a> I. S. Gradshteyn and I. M. Ryzhik, ebenda 1.331.3 (Bei dieser Formel enthält Gradshteyn/Ryzhik allerdings einen Vorzeichenfehler)

</li>

<li id="cite_note-Bronstein2.5.2.1.3-13">↑ <a href="#cite_ref-Bronstein2.5.2.1.3_13-0">a</a> <a href="#cite_ref-Bronstein2.5.2.1.3_13-1">b</a> <a href="#cite_ref-Bronstein2.5.2.1.3_13-2">c</a> <a href="#cite_ref-Bronstein2.5.2.1.3_13-3">d</a> <a href="#cite_ref-Bronstein2.5.2.1.3_13-4">e</a> <a href="#cite_ref-Bronstein2.5.2.1.3_13-5">f</a> <a href="#cite_ref-Bronstein2.5.2.1.3_13-6">g</a> <a href="#cite_ref-Bronstein2.5.2.1.3_13-7">h</a> <a href="#cite_ref-Bronstein2.5.2.1.3_13-8">i</a> <a href="#cite_ref-Bronstein2.5.2.1.3_13-9">j</a> <a href="#cite_ref-Bronstein2.5.2.1.3_13-10">k</a> <a href="#cite_ref-Bronstein2.5.2.1.3_13-11">l</a> <a href="#cite_ref-Bronstein2.5.2.1.3_13-12">m</a> <a href="#cite_ref-Bronstein2.5.2.1.3_13-13">n</a> <a href="#cite_ref-Bronstein2.5.2.1.3_13-14">o</a> I. N. Bronstein, K. A. Semendjajew, <a href="Taschenbuch_der_Mathematik" title="Taschenbuch der Mathematik">Taschenbuch der Mathematik</a>, B. G. Teubner Verlagsgesellschaft Leipzig. 19. Auflage 1979. 2.5.2.1.3

</li>

<li id="cite_note-14"><a href="#cite_ref-14">↑</a> Milton Abramowitz and Irene A. Stegun, <a rel="nofollow" class="external text" href="https://personal.math.ubc.ca/~cbm/aands/abramowitz_and_stegun.pdf">4.3.28</a>, (s. a. oben „Weblinks“)

</li>

<li id="cite_note-15"><a href="#cite_ref-15">↑</a> Milton Abramowitz and Irene A. Stegun, <a rel="nofollow" class="external text" href="https://personal.math.ubc.ca/~cbm/aands/abramowitz_and_stegun.pdf">4.3.30</a>, (s. a. oben „Weblinks“)

</li>

<li id="cite_note-16"><a href="#cite_ref-16">↑</a> I. S. Gradshteyn and I. M. Ryzhik, ebenda 1.335.4

</li>

<li id="cite_note-17"><a href="#cite_ref-17">↑</a> I. S. Gradshteyn and I. M. Ryzhik, ebenda 1.335.5

</li>

<li id="cite_note-18"><a href="#cite_ref-18">↑</a> I. S. Gradshteyn and I. M. Ryzhik, ebenda 1.331.3

</li>

<li id="cite_note-19"><a href="#cite_ref-19">↑</a> Roger B. Nelsen: Beweise ohne Worte, Deutschsprachige Ausgabe herausgegeben von Nicola Oswald, <a href="Springer_Spektrum" title="Springer Spektrum">Springer Spektrum</a>, Springer-Verlag <a href="Berlin" title="Berlin">Berlin</a> <a href="Heidelberg" title="Heidelberg">Heidelberg</a> 2016, ISBN 978-3-662-50330-0, Seite 49

</li>

<li id="cite_note-20"><a href="#cite_ref-20">↑</a> I. S. Gradshteyn and I. M. Ryzhik, ebenda 1.321.1

</li>

<li id="cite_note-21"><a href="#cite_ref-21">↑</a> I. S. Gradshteyn and I. M. Ryzhik, ebenda 1.321.2

</li>

<li id="cite_note-22"><a href="#cite_ref-22">↑</a> I. S. Gradshteyn and I. M. Ryzhik, ebenda 1.321.3

</li>

<li id="cite_note-23"><a href="#cite_ref-23">↑</a> I. S. Gradshteyn and I. M. Ryzhik, ebenda 1.321.4

</li>

<li id="cite_note-24"><a href="#cite_ref-24">↑</a> I. S. Gradshteyn and I. M. Ryzhik, ebenda 1.321.5

</li>

<li id="cite_note-25"><a href="#cite_ref-25">↑</a> I. S. Gradshteyn and I. M. Ryzhik, ebenda 1.323.1

</li>

<li id="cite_note-26"><a href="#cite_ref-26">↑</a> I. S. Gradshteyn and I. M. Ryzhik, ebenda 1.323.2

</li>

<li id="cite_note-27"><a href="#cite_ref-27">↑</a> I. S. Gradshteyn and I. M. Ryzhik, ebenda 1.323.3

</li>

<li id="cite_note-28"><a href="#cite_ref-28">↑</a> I. S. Gradshteyn and I. M. Ryzhik, ebenda 1.323.4

</li>

<li id="cite_note-29"><a href="#cite_ref-29">↑</a> I. S. Gradshteyn and I. M. Ryzhik, ebenda 1.323.5

</li>

<li id="cite_note-30"><a href="#cite_ref-30">↑</a> Weisstein, Eric W.: <a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/HarmonicAdditionTheorem.html">Harmonic Addition Theorem.</a> Abgerufen am 20. Januar 2018 (englisch).<span style="display: none;" class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Adc&rfr_id=info%3Asid%2Fde.wikipedia.org%3AFormelsammlung+Trigonometrie&rft.title=Harmonic+Addition+Theorem&rft.description=Harmonic+Addition+Theorem&rft.identifier=http%3A%2F%2Fmathworld.wolfram.com%2FHarmonicAdditionTheorem.html&rft.creator=Weisstein%2C+Eric+W.&rft.language=en">

</li>

<li id="cite_note-31"><a href="#cite_ref-31">↑</a> Milton Abramowitz and Irene A. Stegun, <a rel="nofollow" class="external text" href="https://personal.math.ubc.ca/~cbm/aands/abramowitz_and_stegun.pdf">4.3.67</a>, (s. a. oben „Weblinks“)

</li>

<li id="cite_note-32"><a href="#cite_ref-32">↑</a> Milton Abramowitz and Irene A. Stegun, <a rel="nofollow" class="external text" href="https://personal.math.ubc.ca/~cbm/aands/abramowitz_and_stegun.pdf">4.3.70</a>, (s. a. oben „Weblinks“)

</li>

<li id="cite_note-33"><a href="#cite_ref-33">↑</a> Herbert Amann, Joachim Escher: Analysis I, Birkhäuser Verlag, Basel 2006, 3. Auflage, S. 292 und 298

</li>

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