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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Trigonometrischer Pythagoras</span></h1>
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<p>Als <b>trigonometrischer Pythagoras</b> wird die <a href="Identit%C3%A4tsgleichung" title="Identitätsgleichung">Identität</a>
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin ^{2}\alpha +\cos ^{2}\alpha =1}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>sin</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo><!-- --></mo>
<mi>α<!-- α --></mi>
<mo>+</mo>
<msup>
<mi>cos</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo><!-- --></mo>
<mi>α<!-- α --></mi>
<mo>=</mo>
<mn>1</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \sin ^{2}\alpha +\cos ^{2}\alpha =1}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c63e7175c35591236e8272cd18d264a0fbd5a7f5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:18.926ex; height:2.843ex;" alt="{\displaystyle \sin ^{2}\alpha +\cos ^{2}\alpha =1}" loading="lazy"></span></dd></dl>
<p>bezeichnet.<sup id="cite_ref-Papula_1-0" class="reference"><a href="#cite_note-Papula-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-KreulZiebarth2009_2-0" class="reference"><a href="#cite_note-KreulZiebarth2009-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>
Hierbei steht <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin ^{2}\alpha }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>sin</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo><!-- --></mo>
<mi>α<!-- α --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \sin ^{2}\alpha }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1a6fa3bb862aeb01860f7ca8ff344e23b3b83519.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.785ex; height:2.676ex;" alt="{\displaystyle \sin ^{2}\alpha }" loading="lazy"></span> für <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\sin \alpha )^{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">(</mo>
<mi>sin</mi>
<mo><!-- --></mo>
<mi>α<!-- α --></mi>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle (\sin \alpha )^{2}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/804ff271c79229388b9e56a889be12620dd3b473.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.594ex; height:3.176ex;" alt="{\displaystyle (\sin \alpha )^{2}}" loading="lazy"></span> und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos ^{2}\alpha }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>cos</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo><!-- --></mo>
<mi>α<!-- α --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \cos ^{2}\alpha }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/156b5e4a04bc91470828157e7c07307bf0493dcc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.04ex; height:2.676ex;" alt="{\displaystyle \cos ^{2}\alpha }" loading="lazy"></span> für <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\cos \alpha )^{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">(</mo>
<mi>cos</mi>
<mo><!-- --></mo>
<mi>α<!-- α --></mi>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle (\cos \alpha )^{2}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/716c85d295a03a0f628c8600bca90ea8059f9b2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.849ex; height:3.176ex;" alt="{\displaystyle (\cos \alpha )^{2}}" loading="lazy"></span>. Die Gültigkeit dieser Identität kann am <a href="Einheitskreis" title="Einheitskreis">Einheitskreis</a> gezeigt werden, mit Hilfe des <a href="Satz_des_Pythagoras" title="Satz des Pythagoras">Satzes von Pythagoras</a>, der auch namensgebend für diesen häufig benutzten Satz der <a href="Trigonometrie" title="Trigonometrie">Trigonometrie</a> ist.
</p>
<div class="mw-heading mw-heading2"><h2 id="Geometrische_Herleitung">Geometrische Herleitung</h2></div>
<p>Als Grundlage dient der <a href="Satz_des_Pythagoras" title="Satz des Pythagoras">Satz des Pythagoras</a>. Er besagt, dass in einem rechtwinkligen Dreieck mit der <a href="Hypotenuse" class="mw-redirect" title="Hypotenuse">Hypotenuse</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>c</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle c}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/86a67b81c2de995bd608d5b2df50cd8cd7d92455.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}" loading="lazy"></span> und den <a href="Kathete" class="mw-redirect" title="Kathete">Katheten</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>a</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle a}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"></span> und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>b</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle b}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f11423fbb2e967f986e36804a8ae4271734917c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}" loading="lazy"></span>
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a^{2}+b^{2}=c^{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>a</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo>+</mo>
<msup>
<mi>b</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo>=</mo>
<msup>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle a^{2}+b^{2}=c^{2}}</annotation>
</semantics>
</math></span><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"></span></dd></dl>
<p>gilt. Wird der Winkel <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>α<!-- α --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation>
</semantics>
</math></span><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"></span> im besagten rechtwinkligen Dreieck so gewählt, dass <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>a</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle a}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"></span> seine Gegenkathete und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>b</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle b}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f11423fbb2e967f986e36804a8ae4271734917c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}" loading="lazy"></span> seine Ankathete ist, so gilt allgemein
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=\sin \alpha \cdot c}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>a</mi>
<mo>=</mo>
<mi>sin</mi>
<mo><!-- --></mo>
<mi>α<!-- α --></mi>
<mo>⋅<!-- ⋅ --></mo>
<mi>c</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle a=\sin \alpha \cdot c}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e6ee967641f7d2f3cc1c1d72dd792f32ce51f037.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.745ex; height:2.176ex;" alt="{\displaystyle a=\sin \alpha \cdot c}" loading="lazy"></span>,</dd>
<dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b=\cos \alpha \cdot c}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>b</mi>
<mo>=</mo>
<mi>cos</mi>
<mo><!-- --></mo>
<mi>α<!-- α --></mi>
<mo>⋅<!-- ⋅ --></mo>
<mi>c</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle b=\cos \alpha \cdot c}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/916db8c22fb15a34bf9dd53f7ccc74b872840fff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.768ex; height:2.176ex;" alt="{\displaystyle b=\cos \alpha \cdot c}" loading="lazy"></span>.</dd></dl>
<p>Einsetzen beider Gleichungen in den Satz des Pythagoras ergibt dann
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\sin ^{2}\alpha +\cos ^{2}\alpha )\cdot c^{2}=c^{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">(</mo>
<msup>
<mi>sin</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo><!-- --></mo>
<mi>α<!-- α --></mi>
<mo>+</mo>
<msup>
<mi>cos</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo><!-- --></mo>
<mi>α<!-- α --></mi>
<mo stretchy="false">)</mo>
<mo>⋅<!-- ⋅ --></mo>
<msup>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo>=</mo>
<msup>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle (\sin ^{2}\alpha +\cos ^{2}\alpha )\cdot c^{2}=c^{2}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fb1442f05282e04573c917d133b5bda7cacf759f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.374ex; height:3.176ex;" alt="{\displaystyle (\sin ^{2}\alpha +\cos ^{2}\alpha )\cdot c^{2}=c^{2}}" loading="lazy"></span>,</dd>
<dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin ^{2}\alpha +\cos ^{2}\alpha =1}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>sin</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo><!-- --></mo>
<mi>α<!-- α --></mi>
<mo>+</mo>
<msup>
<mi>cos</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo><!-- --></mo>
<mi>α<!-- α --></mi>
<mo>=</mo>
<mn>1</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \sin ^{2}\alpha +\cos ^{2}\alpha =1}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c63e7175c35591236e8272cd18d264a0fbd5a7f5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:18.926ex; height:2.843ex;" alt="{\displaystyle \sin ^{2}\alpha +\cos ^{2}\alpha =1}" loading="lazy"></span>.</dd></dl>
<div class="mw-heading mw-heading3"><h3 id="Geometrische_Veranschaulichung">Geometrische Veranschaulichung</h3></div>
<p>In der nebenstehenden Skizze sind der <a href="Einheitskreis" title="Einheitskreis">Einheitskreis</a>, das heißt ein Kreis mit Radius 1, und ein rechtwinkliges Dreieck mit Hypotenusenlänge 1 im Einheitskreis dargestellt. Der Satz des Pythagoras gilt hier für einen beliebigen Wert des Winkels <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>α<!-- α --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation>
</semantics>
</math></span><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"></span> im Einheitskreis und zeigt sofort die Gültigkeit des „trigonometrischen Pythagoras“.
</p>
<div class="mw-heading mw-heading2"><h2 id="Analytische_Herleitung">Analytische Herleitung</h2></div>
<p>Für stumpfe und überstumpfe Winkel <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>α<!-- α --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation>
</semantics>
</math></span><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"></span> ist die Beweiskraft der Anschauung problematisch, da für solche (mindestens) eine Winkelfunktion negative Werte hat; was sind „negative Seiten“ eines rechtwinkligen Dreiecks? Ein analytischer Beweis zeigt, dass der trigonometrische Pythagoras für beliebige <a href="Reelle_Zahl" title="Reelle Zahl">reelle</a> und <a href="Komplexe_Zahl" title="Komplexe Zahl">komplexe</a> Argumente <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>α<!-- α --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation>
</semantics>
</math></span><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"></span> der verwendeten Winkelfunktionen gilt.
</p><p>Mit der <a href="Komplexe_Zahl" title="Komplexe Zahl">imaginären Einheit</a> und der <a href="Binomische_Formeln#Die_Formeln" title="Binomische Formeln">dritten binomischen Formel</a> lässt sich faktorisieren:
</p><p><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos ^{2}(\alpha )+\sin ^{2}(\alpha )=\cos ^{2}(\alpha )-\mathrm {i} ^{2}\cdot \sin ^{2}(\alpha )=(\cos(\alpha )+\mathrm {i} \cdot \sin(\alpha ))\cdot (\cos(\alpha )-\mathrm {i} \cdot \sin(\alpha ));}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>cos</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo><!-- --></mo>
<mo stretchy="false">(</mo>
<mi>α<!-- α --></mi>
<mo stretchy="false">)</mo>
<mo>+</mo>
<msup>
<mi>sin</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
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</msup>
<mo><!-- --></mo>
<mo stretchy="false">(</mo>
<mi>α<!-- α --></mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<msup>
<mi>cos</mi>
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<mn>2</mn>
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<mo><!-- --></mo>
<mo stretchy="false">(</mo>
<mi>α<!-- α --></mi>
<mo stretchy="false">)</mo>
<mo>−<!-- − --></mo>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">i</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo>⋅<!-- ⋅ --></mo>
<msup>
<mi>sin</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo><!-- --></mo>
<mo stretchy="false">(</mo>
<mi>α<!-- α --></mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mo stretchy="false">(</mo>
<mi>cos</mi>
<mo><!-- --></mo>
<mo stretchy="false">(</mo>
<mi>α<!-- α --></mi>
<mo stretchy="false">)</mo>
<mo>+</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">i</mi>
</mrow>
<mo>⋅<!-- ⋅ --></mo>
<mi>sin</mi>
<mo><!-- --></mo>
<mo stretchy="false">(</mo>
<mi>α<!-- α --></mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">)</mo>
<mo>⋅<!-- ⋅ --></mo>
<mo stretchy="false">(</mo>
<mi>cos</mi>
<mo><!-- --></mo>
<mo stretchy="false">(</mo>
<mi>α<!-- α --></mi>
<mo stretchy="false">)</mo>
<mo>−<!-- − --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">i</mi>
</mrow>
<mo>⋅<!-- ⋅ --></mo>
<mi>sin</mi>
<mo><!-- --></mo>
<mo stretchy="false">(</mo>
<mi>α<!-- α --></mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">)</mo>
<mo>;</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \cos ^{2}(\alpha )+\sin ^{2}(\alpha )=\cos ^{2}(\alpha )-\mathrm {i} ^{2}\cdot \sin ^{2}(\alpha )=(\cos(\alpha )+\mathrm {i} \cdot \sin(\alpha ))\cdot (\cos(\alpha )-\mathrm {i} \cdot \sin(\alpha ));}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3f153b0bed0a341e908db2b4418b3563fba963dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:85.994ex; height:3.176ex;" alt="{\displaystyle \cos ^{2}(\alpha )+\sin ^{2}(\alpha )=\cos ^{2}(\alpha )-\mathrm {i} ^{2}\cdot \sin ^{2}(\alpha )=(\cos(\alpha )+\mathrm {i} \cdot \sin(\alpha ))\cdot (\cos(\alpha )-\mathrm {i} \cdot \sin(\alpha ));}" loading="lazy"></span>
</p><p>da der Cosinus eine <a href="Gerade_und_ungerade_Funktionen#Definition" title="Gerade und ungerade Funktionen">gerade Funktion</a> und der Sinus eine <a href="Gerade_und_ungerade_Funktionen#Definition" title="Gerade und ungerade Funktionen">ungerade Funktion</a> ist, folgt mit der <a href="Eulersche_Formel" title="Eulersche Formel">Eulerschen Formel</a> weiter:
</p><p><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\cos(\alpha )+\mathrm {i} \cdot \sin(\alpha ))\cdot (\cos(\alpha )-\mathrm {i} \cdot \sin(\alpha ))=(\cos(\alpha )+\mathrm {i} \cdot \sin(\alpha ))\cdot (\cos(-\alpha )+\mathrm {i} \cdot \sin(-\alpha ))=\mathrm {e} ^{\mathrm {i} \alpha }\cdot \mathrm {e} ^{-\mathrm {i} \alpha }=\mathrm {e} ^{0}=1,\quad }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">(</mo>
<mi>cos</mi>
<mo><!-- --></mo>
<mo stretchy="false">(</mo>
<mi>α<!-- α --></mi>
<mo stretchy="false">)</mo>
<mo>+</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">i</mi>
</mrow>
<mo>⋅<!-- ⋅ --></mo>
<mi>sin</mi>
<mo><!-- --></mo>
<mo stretchy="false">(</mo>
<mi>α<!-- α --></mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">)</mo>
<mo>⋅<!-- ⋅ --></mo>
<mo stretchy="false">(</mo>
<mi>cos</mi>
<mo><!-- --></mo>
<mo stretchy="false">(</mo>
<mi>α<!-- α --></mi>
<mo stretchy="false">)</mo>
<mo>−<!-- − --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">i</mi>
</mrow>
<mo>⋅<!-- ⋅ --></mo>
<mi>sin</mi>
<mo><!-- --></mo>
<mo stretchy="false">(</mo>
<mi>α<!-- α --></mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mo stretchy="false">(</mo>
<mi>cos</mi>
<mo><!-- --></mo>
<mo stretchy="false">(</mo>
<mi>α<!-- α --></mi>
<mo stretchy="false">)</mo>
<mo>+</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">i</mi>
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<mo>⋅<!-- ⋅ --></mo>
<mi>sin</mi>
<mo><!-- --></mo>
<mo stretchy="false">(</mo>
<mi>α<!-- α --></mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">)</mo>
<mo>⋅<!-- ⋅ --></mo>
<mo stretchy="false">(</mo>
<mi>cos</mi>
<mo><!-- --></mo>
<mo stretchy="false">(</mo>
<mo>−<!-- − --></mo>
<mi>α<!-- α --></mi>
<mo stretchy="false">)</mo>
<mo>+</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">i</mi>
</mrow>
<mo>⋅<!-- ⋅ --></mo>
<mi>sin</mi>
<mo><!-- --></mo>
<mo stretchy="false">(</mo>
<mo>−<!-- − --></mo>
<mi>α<!-- α --></mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">)</mo>
<mo>=</mo>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">e</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">i</mi>
</mrow>
<mi>α<!-- α --></mi>
</mrow>
</msup>
<mo>⋅<!-- ⋅ --></mo>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">e</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">i</mi>
</mrow>
<mi>α<!-- α --></mi>
</mrow>
</msup>
<mo>=</mo>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">e</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msup>
<mo>=</mo>
<mn>1</mn>
<mo>,</mo>
<mspace width="1em"></mspace>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle (\cos(\alpha )+\mathrm {i} \cdot \sin(\alpha ))\cdot (\cos(\alpha )-\mathrm {i} \cdot \sin(\alpha ))=(\cos(\alpha )+\mathrm {i} \cdot \sin(\alpha ))\cdot (\cos(-\alpha )+\mathrm {i} \cdot \sin(-\alpha ))=\mathrm {e} ^{\mathrm {i} \alpha }\cdot \mathrm {e} ^{-\mathrm {i} \alpha }=\mathrm {e} ^{0}=1,\quad }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e99ad6d76cb66ac70b3fa80a47a6b7c953684d16.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:112.237ex; height:3.176ex;" alt="{\displaystyle (\cos(\alpha )+\mathrm {i} \cdot \sin(\alpha ))\cdot (\cos(\alpha )-\mathrm {i} \cdot \sin(\alpha ))=(\cos(\alpha )+\mathrm {i} \cdot \sin(\alpha ))\cdot (\cos(-\alpha )+\mathrm {i} \cdot \sin(-\alpha ))=\mathrm {e} ^{\mathrm {i} \alpha }\cdot \mathrm {e} ^{-\mathrm {i} \alpha }=\mathrm {e} ^{0}=1,\quad }" loading="lazy"></span>q.e.d.
</p>
<div class="mw-heading mw-heading2"><h2 id="Anwendungsbeispiele">Anwendungsbeispiele</h2></div>
<p>Der trigonometrische Pythagoras wird häufig für Umformungen trigonometrischer Terme verwendet, etwa beim Zusammenhang zwischen <a href="Tangens" class="mw-redirect" title="Tangens">Tangens</a> und <a href="Kosinus" class="mw-redirect" title="Kosinus">Kosinus</a> oder bei den verschiedenen Formeln für den Kosinus eines verdoppelten Winkels.
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tan ^{2}\alpha +1={\frac {\sin ^{2}\alpha }{\cos ^{2}\alpha }}+{\frac {\cos ^{2}\alpha }{\cos ^{2}\alpha }}={\frac {\sin ^{2}\alpha +\cos ^{2}\alpha }{\cos ^{2}\alpha }}={\frac {1}{\cos ^{2}\alpha }}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>tan</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo><!-- --></mo>
<mi>α<!-- α --></mi>
<mo>+</mo>
<mn>1</mn>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<msup>
<mi>sin</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo><!-- --></mo>
<mi>α<!-- α --></mi>
</mrow>
<mrow>
<msup>
<mi>cos</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo><!-- --></mo>
<mi>α<!-- α --></mi>
</mrow>
</mfrac>
</mrow>
<mo>+</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<msup>
<mi>cos</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo><!-- --></mo>
<mi>α<!-- α --></mi>
</mrow>
<mrow>
<msup>
<mi>cos</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo><!-- --></mo>
<mi>α<!-- α --></mi>
</mrow>
</mfrac>
</mrow>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<msup>
<mi>sin</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo><!-- --></mo>
<mi>α<!-- α --></mi>
<mo>+</mo>
<msup>
<mi>cos</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo><!-- --></mo>
<mi>α<!-- α --></mi>
</mrow>
<mrow>
<msup>
<mi>cos</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo><!-- --></mo>
<mi>α<!-- α --></mi>
</mrow>
</mfrac>
</mrow>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mrow>
<msup>
<mi>cos</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo><!-- --></mo>
<mi>α<!-- α --></mi>
</mrow>
</mfrac>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \tan ^{2}\alpha +1={\frac {\sin ^{2}\alpha }{\cos ^{2}\alpha }}+{\frac {\cos ^{2}\alpha }{\cos ^{2}\alpha }}={\frac {\sin ^{2}\alpha +\cos ^{2}\alpha }{\cos ^{2}\alpha }}={\frac {1}{\cos ^{2}\alpha }}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/73d28f5f9be00c0895d3e7b04dad4ace05af11de.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:58.557ex; height:6.009ex;" alt="{\displaystyle \tan ^{2}\alpha +1={\frac {\sin ^{2}\alpha }{\cos ^{2}\alpha }}+{\frac {\cos ^{2}\alpha }{\cos ^{2}\alpha }}={\frac {\sin ^{2}\alpha +\cos ^{2}\alpha }{\cos ^{2}\alpha }}={\frac {1}{\cos ^{2}\alpha }}}" loading="lazy"></span></dd>
<dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos(2\alpha )=\cos ^{2}\alpha -\sin ^{2}\alpha =2\cos ^{2}\alpha -1=1-2\sin ^{2}\alpha }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>cos</mi>
<mo><!-- --></mo>
<mo stretchy="false">(</mo>
<mn>2</mn>
<mi>α<!-- α --></mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<msup>
<mi>cos</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
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<msup>
<mi>sin</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
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<mn>2</mn>
<msup>
<mi>cos</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
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</msup>
<mo><!-- --></mo>
<mi>α<!-- α --></mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
<mo>=</mo>
<mn>1</mn>
<mo>−<!-- − --></mo>
<mn>2</mn>
<msup>
<mi>sin</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
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</msup>
<mo><!-- --></mo>
<mi>α<!-- α --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \cos(2\alpha )=\cos ^{2}\alpha -\sin ^{2}\alpha =2\cos ^{2}\alpha -1=1-2\sin ^{2}\alpha }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b6f71dd7104584423a4e47c8dfbb698240767c63.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:54.46ex; height:3.176ex;" alt="{\displaystyle \cos(2\alpha )=\cos ^{2}\alpha -\sin ^{2}\alpha =2\cos ^{2}\alpha -1=1-2\sin ^{2}\alpha }" loading="lazy"></span></dd></dl>
<div class="mw-heading mw-heading2"><h2 id="Einzelnachweise">Einzelnachweise</h2></div>
<ol class="references">
<li id="cite_note-Papula-1"><span class="mw-cite-backlink"><a href="#cite_ref-Papula_1-0">↑</a></span> <span class="reference-text"><span class="book">Lothar Papula: <cite style="font-style:italic">Mathematik für Ingenieure und Naturwissenschaftler Band 1</cite>. Springer, ISBN 978-3-8348-1749-5, <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em"> </span>251</span> (<a rel="nofollow" class="external text" href="http://books.google.com/books?id=vTuCSeEal6AC&pg=PA251">google.com</a>).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Trigonometrischer+Pythagoras&rft.au=Lothar+Papula&rft.btitle=Mathematik+f%C3%BCr+Ingenieure+und+Naturwissenschaftler+Band+1&rft.genre=book&rft.isbn=9783834817495&rft.pages=251&rft.pub=Springer" style="display:none"> </span></span></span>
</li>
<li id="cite_note-KreulZiebarth2009-2"><span class="mw-cite-backlink"><a href="#cite_ref-KreulZiebarth2009_2-0">↑</a></span> <span class="reference-text"><span class="book">Hans Kreul, Harald Ziebarth: <cite style="font-style:italic">Mathematik leicht gemacht</cite>. Harri Deutsch Verlag, 2009, ISBN 978-3-8171-1836-6, <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em"> </span>94</span> (<a rel="nofollow" class="external text" href="http://books.google.com/books?id=p7ZQlJG6ClcC&pg=PA596">google.com</a>).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Trigonometrischer+Pythagoras&rft.au=Hans+Kreul%2C+Harald+Ziebarth&rft.btitle=Mathematik+leicht+gemacht&rft.date=2009-08&rft.genre=book&rft.isbn=9783817118366&rft.pages=94&rft.pub=Harri+Deutsch+Verlag" style="display:none"> </span></span></span>
</li>
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