Arkussinus und Arkuskosinus

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list-style-image: none; margin: 0;">arcsin (x)</li> <li style="list-style-type: none; list-style-image: none; margin: 0;">arccos (x)</li> </figcaption></figure><figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Datei:01_Umkehrfunktion.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d2/01_Umkehrfunktion.svg/370px-01_Umkehrfunktion.svg.png" decoding="async" width="370" height="217" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d2/01_Umkehrfunktion.svg/555px-01_Umkehrfunktion.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d2/01_Umkehrfunktion.svg/740px-01_Umkehrfunktion.svg.png 2x" data-file-width="436" data-file-height="256" /></a><figcaption>Beispiel: Umkehrung der Kosinus- und Sinusfunktion<a href="#cite_note-1">[1]</a></figcaption></figure> Der Arkussinus – geschrieben <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \arcsin }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arcsin</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \arcsin }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/627fcadc0495786c9167c5487a1a795bb1940edd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.962ex; height:2.176ex;" alt="{\displaystyle \arcsin }"> oder <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {asin} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>asin</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {asin} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5152fae0045d08c1e00b0df8df59bd869de9d07b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.018ex; height:2.176ex;" alt="{\displaystyle \operatorname {asin} }"> – und der Arkuskosinus (oder auch Arkuscosinus) – geschrieben <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \arccos }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arccos</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \arccos }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2f10121c1f582201ae32a56303e36bee4191336" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.218ex; height:1.676ex;" alt="{\displaystyle \arccos }"> oder <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {acos} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>acos</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {acos} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/37e6f7041331e63946df6ee619d432343086476e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.274ex; height:1.676ex;" alt="{\displaystyle \operatorname {acos} }"> – sind <a href="/wiki/Umkehrfunktion" title="Umkehrfunktion">Umkehrfunktionen</a> der (geeignet) <a href="/wiki/Einschr%C3%A4nkung" title="Einschränkung">eingeschränkten</a> <a href="/wiki/Sinus_und_Kosinus" title="Sinus und Kosinus">Sinus- bzw. Kosinusfunktion</a>. Sinus und Kosinus sind Funktionen, die einen Winkel auf einen Wert im <a href="/wiki/Intervall_(Mathematik)" title="Intervall (Mathematik)">Intervall</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [-1,1]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [-1,1]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51e3b7f14a6f70e614728c583409a0b9a8b9de01" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.461ex; height:2.843ex;" alt="{\displaystyle [-1,1]}"> abbilden; als deren Umkehrfunktionen bilden Arkussinus und Arkuskosinus einen Wert aus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [-1,1]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [-1,1]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51e3b7f14a6f70e614728c583409a0b9a8b9de01" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.461ex; height:2.843ex;" alt="{\displaystyle [-1,1]}"> wieder auf einen zugehörigen Winkel ab. Da Sinus und Kosinus periodische Funktionen sind, gibt es aber zu jedem Wert aus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [-1,1]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [-1,1]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51e3b7f14a6f70e614728c583409a0b9a8b9de01" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.461ex; height:2.843ex;" alt="{\displaystyle [-1,1]}"> unendlich viele zugehörige Winkel. Daher wird zur Umkehrung von Sinus und Kosinus deren <a href="/wiki/Definitionsmenge" title="Definitionsmenge">Definitionsmenge</a> auf das Intervall <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [-{\tfrac {\pi }{2}},{\tfrac {\pi }{2}}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [-{\tfrac {\pi }{2}},{\tfrac {\pi }{2}}]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c0215b3d39f1ccb8768a392d7ab3e9af48661c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:7.692ex; height:3.176ex;" alt="{\displaystyle [-{\tfrac {\pi }{2}},{\tfrac {\pi }{2}}]}"> für Sinus und auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [0,\pi ]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mi>π</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [0,\pi ]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e2a912eda6ef1afe46a81b518fe9da64a832751" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.822ex; height:2.843ex;" alt="{\displaystyle [0,\pi ]}"> für Kosinus eingeschränkt. Sinus und Kosinus sind auf diesen Intervallen streng monoton und daher umkehrbar. Zusammen mit dem <a href="/wiki/Arkustangens" class="mw-redirect" title="Arkustangens">Arkustangens</a> als Umkehrfunktion des (ebenfalls geeignet eingeschränkten) <a href="/wiki/Tangens" class="mw-redirect" title="Tangens">Tangens</a> bilden der Arkussinus und Arkuskosinus den Kern der Klasse der <a href="/wiki/Arkusfunktion" title="Arkusfunktion">Arkusfunktionen</a>. Aufgrund der in neuerer Zeit für Umkehrfunktionen gebräuchlichen Schreibweise <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f^{-1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e5cfa2f5c08d6fe7d046b73faa6e3f213acc802" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.653ex; height:3.009ex;" alt="{\displaystyle f^{-1}}"> beginnen die namentlich auf Taschenrechnern verbreiteten Schreibweisen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin ^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin ^{-1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/21bc4ca64ac415c9ae60fb4e60fe4bddee17b8ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.188ex; height:2.676ex;" alt="{\displaystyle \sin ^{-1}}"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos ^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos ^{-1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7eabf706b4642d521c6279a2f07ac9715c7679a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.444ex; height:2.676ex;" alt="{\displaystyle \cos ^{-1}}"> die klassische Schreibweise <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \arcsin }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arcsin</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \arcsin }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/627fcadc0495786c9167c5487a1a795bb1940edd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.962ex; height:2.176ex;" alt="{\displaystyle \arcsin }"> bzw. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \arccos }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arccos</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \arccos }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2f10121c1f582201ae32a56303e36bee4191336" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.218ex; height:1.676ex;" alt="{\displaystyle \arccos }"> zu verdrängen, was eventuell zu Verwechslungen mit den <a href="/wiki/Kehrwert" title="Kehrwert">Kehrwerten</a> des Sinus und Kosinus (<a href="/wiki/Sekans_und_Kosekans" title="Sekans und Kosekans">Kosekans und Sekans</a>) führen kann.<a href="#cite_note-2">[2]</a> <div id="toc" class="toc" role="navigation" aria-labelledby="mw-toc-heading"><input type="checkbox" role="button" id="toctogglecheckbox" class="toctogglecheckbox" style="display:none" /><div class="toctitle" lang="de" dir="ltr"><h2 id="mw-toc-heading">Inhaltsverzeichnis</h2><label class="toctogglelabel" for="toctogglecheckbox"></label></div> <ul> <li class="toclevel-1 tocsection-1"><a href="#Definitionen">1 Definitionen</a></li> <li class="toclevel-1 tocsection-2"><a href="#Eigenschaften">2 Eigenschaften</a></li> <li class="toclevel-1 tocsection-3"><a href="#Formeln_für_negative_Argumente">3 Formeln für negative Argumente</a></li> <li class="toclevel-1 tocsection-4"><a href="#Reihenentwicklungen">4 Reihenentwicklungen</a></li> <li class="toclevel-1 tocsection-5"><a href="#Verkettungen_mit_Sinus_und_Kosinus">5 Verkettungen mit Sinus und Kosinus</a></li> <li class="toclevel-1 tocsection-6"><a href="#Beziehung_zum_Arkustangens">6 Beziehung zum Arkustangens</a></li> <li class="toclevel-1 tocsection-7"><a href="#Additionstheoreme">7 Additionstheoreme</a></li> <li class="toclevel-1 tocsection-8"><a href="#Ableitungen">8 Ableitungen</a></li> <li class="toclevel-1 tocsection-9"><a href="#Integrale">9 Integrale</a> <ul> <li class="toclevel-2 tocsection-10"><a href="#Standardisierte_Integraldarstellungen">9.1 Standardisierte Integraldarstellungen</a></li> <li class="toclevel-2 tocsection-11"><a href="#Integralidentität_mit_dem_Logarithmus_Naturalis">9.2 Integralidentität mit dem Logarithmus Naturalis</a></li> <li class="toclevel-2 tocsection-12"><a href="#Integralidentität_mit_dem_Areatangens_Hyperbolicus">9.3 Integralidentität mit dem Areatangens Hyperbolicus</a></li> <li class="toclevel-2 tocsection-13"><a href="#Stammfunktionen_von_Arkussinus_und_Arkuskosinus">9.4 Stammfunktionen von Arkussinus und Arkuskosinus</a></li> <li class="toclevel-2 tocsection-14"><a href="#Stammfunktion_des_kardinalisierten_Arkussinus">9.5 Stammfunktion des kardinalisierten Arkussinus</a></li> </ul> </li> <li class="toclevel-1 tocsection-15"><a href="#Komplexe_Argumente">10 Komplexe Argumente</a></li> <li class="toclevel-1 tocsection-16"><a href="#Anmerkungen">11 Anmerkungen</a> <ul> <li class="toclevel-2 tocsection-17"><a href="#Wichtige_Funktionswerte">11.1 Wichtige Funktionswerte</a></li> <li class="toclevel-2 tocsection-18"><a href="#Kettenbruchdarstellung_des_Arkussinus">11.2 Kettenbruchdarstellung des Arkussinus</a></li> <li class="toclevel-2 tocsection-19"><a href="#Komplexe_Funktion">11.3 Komplexe Funktion</a></li> </ul> </li> <li class="toclevel-1 tocsection-20"><a href="#Siehe_auch">12 Siehe auch</a></li> <li class="toclevel-1 tocsection-21"><a href="#Literatur">13 Literatur</a></li> <li class="toclevel-1 tocsection-22"><a href="#Einzelnachweise">14 Einzelnachweise</a></li> </ul> </div> <div class="mw-heading mw-heading2"><h2 id="Definitionen">Definitionen</h2>[<a href="/w/index.php?title=Arkussinus_und_Arkuskosinus&veaction=edit&section=1" title="Abschnitt bearbeiten: Definitionen" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Arkussinus_und_Arkuskosinus&action=edit&section=1" title="Quellcode des Abschnitts bearbeiten: Definitionen">Quelltext bearbeiten</a>]</div> Die Sinusfunktion ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\pi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73efd1f6493490b058097060a572606d2c550a06" 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 }">-periodisch und innerhalb einer <a href="/wiki/Periodische_Funktion" title="Periodische Funktion">Periode</a> nicht <a href="/wiki/Injektivit%C3%A4t" class="mw-redirect" title="Injektivität">injektiv</a>. Daher muss ihr Definitionsbereich geeignet eingeschränkt werden, um eine umkehrbar-eindeutige Funktion zu erhalten. Da es für diese Einschränkung mehrere Möglichkeiten gibt, spricht man von Zweigen des Arkussinus. Meist wird der Hauptzweig (oder Hauptwert) <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \arcsin \colon [-1,1]\to \left[-{\frac {\pi }{2}},{\frac {\pi }{2}}\right],}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arcsin</mi> <mo>:</mo> <mo stretchy="false">[</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> <mo stretchy="false">→</mo> <mrow> <mo>[</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> </mrow> <mo>]</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \arcsin \colon [-1,1]\to \left[-{\frac {\pi }{2}},{\frac {\pi }{2}}\right],}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16005d8f13fe22748f868c3b42f9103c161298c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:27.478ex; height:4.843ex;" alt="{\displaystyle \arcsin \colon [-1,1]\to \left[-{\frac {\pi }{2}},{\frac {\pi }{2}}\right],}"></dd></dl> die Umkehrfunktion der <a href="/wiki/Einschr%C3%A4nkung_(Mathematik)" class="mw-redirect" title="Einschränkung (Mathematik)">Einschränkung</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin |_{\left[-{\frac {\pi }{2}},{\frac {\pi }{2}}\right]}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> </mrow> <mo>]</mo> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin |_{\left[-{\frac {\pi }{2}},{\frac {\pi }{2}}\right]}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61c44f77f1f62e69e6cfa4008dc70866a3be7206" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:10.611ex; height:4.343ex;" alt="{\displaystyle \sin |_{\left[-{\frac {\pi }{2}},{\frac {\pi }{2}}\right]}}"> der Sinusfunktion auf das Intervall <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left[-{\frac {\pi }{2}},{\frac {\pi }{2}}\right],}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>[</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> </mrow> <mo>]</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left[-{\frac {\pi }{2}},{\frac {\pi }{2}}\right],}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c80933c3f838c832f393b2701d11fbf961a433e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:10.407ex; height:4.843ex;" alt="{\displaystyle \left[-{\frac {\pi }{2}},{\frac {\pi }{2}}\right],}"> betrachtet. Analog zum Arkussinus wird der Hauptzweig des Arkuskosinus als die Umkehrfunktion von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos |_{[0,\pi ]}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mi>π</mi> <mo stretchy="false">]</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos |_{[0,\pi ]}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/21c2debc1e3cf07e8e689aa5b15c512aa843ca1e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:7.513ex; height:3.343ex;" alt="{\displaystyle \cos |_{[0,\pi ]}}"> definiert. Dies ergibt mit <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \arccos \colon [-1,1]\to [0,\pi ]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arccos</mi> <mo>:</mo> <mo stretchy="false">[</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> <mo stretchy="false">→</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mi>π</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \arccos \colon [-1,1]\to [0,\pi ]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea4c6551e2eb9568ea656354e031c39358fa5487" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.148ex; height:2.843ex;" alt="{\displaystyle \arccos \colon [-1,1]\to [0,\pi ]}"></dd></dl> ebenfalls eine <a href="/wiki/Bijektiv" class="mw-redirect" title="Bijektiv">bijektive</a> Funktion. Mittels <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \arccos(x)+\arcsin(x)={\frac {\pi }{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arccos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>arcsin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \arccos(x)+\arcsin(x)={\frac {\pi }{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ae68658ea9751a0e212492889748d24723d4837" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:26.565ex; height:4.676ex;" alt="{\displaystyle \arccos(x)+\arcsin(x)={\frac {\pi }{2}}}"></dd></dl> lassen sich diese beiden Funktionen ineinander umrechnen. <div class="mw-heading mw-heading2"><h2 id="Eigenschaften">Eigenschaften</h2>[<a href="/w/index.php?title=Arkussinus_und_Arkuskosinus&veaction=edit&section=2" title="Abschnitt bearbeiten: Eigenschaften" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Arkussinus_und_Arkuskosinus&action=edit&section=2" title="Quellcode des Abschnitts bearbeiten: Eigenschaften">Quelltext bearbeiten</a>]</div> <table class="wikitable"> <tbody><tr class="hintergrundfarbe6"> <th>  </th> <th>Arkussinus </th> <th>Arkuskosinus </th></tr> <tr> <th><a href="/wiki/Funktionsgraph" title="Funktionsgraph">Funktionsgraph</a> </th> <td><a href="/wiki/Datei:Mplwp_arcsin_piaxis.svg" class="mw-file-description" title="Arcsin"><img alt="Arcsin" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/01/Mplwp_arcsin_piaxis.svg/250px-Mplwp_arcsin_piaxis.svg.png" decoding="async" width="250" height="167" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/01/Mplwp_arcsin_piaxis.svg/375px-Mplwp_arcsin_piaxis.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/01/Mplwp_arcsin_piaxis.svg/500px-Mplwp_arcsin_piaxis.svg.png 2x" data-file-width="600" data-file-height="400" /></a> </td> <td><a href="/wiki/Datei:Mplwp_arccos_piaxis.svg" class="mw-file-description" title="Arccos"><img alt="Arccos" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/93/Mplwp_arccos_piaxis.svg/250px-Mplwp_arccos_piaxis.svg.png" decoding="async" width="250" height="167" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/93/Mplwp_arccos_piaxis.svg/375px-Mplwp_arccos_piaxis.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/93/Mplwp_arccos_piaxis.svg/500px-Mplwp_arccos_piaxis.svg.png 2x" data-file-width="600" data-file-height="400" /></a> </td></tr> <tr> <th><a href="/wiki/Definitionsmenge" title="Definitionsmenge">Definitionsmenge</a> </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [-1,1]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [-1,1]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51e3b7f14a6f70e614728c583409a0b9a8b9de01" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.461ex; height:2.843ex;" alt="{\displaystyle [-1,1]}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [-1,1]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [-1,1]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51e3b7f14a6f70e614728c583409a0b9a8b9de01" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.461ex; height:2.843ex;" alt="{\displaystyle [-1,1]}"> </td></tr> <tr> <th><a href="/wiki/Bildmenge" class="mw-redirect" title="Bildmenge">Bildmenge</a> </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left[-{\frac {\pi }{2}},{\frac {\pi }{2}}\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>[</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left[-{\frac {\pi }{2}},{\frac {\pi }{2}}\right]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54381f086ac9ffe8306d413f813abcb616e95dee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:9.373ex; height:4.843ex;" alt="{\displaystyle \left[-{\frac {\pi }{2}},{\frac {\pi }{2}}\right]}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [0,\pi ]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mi>π</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [0,\pi ]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e2a912eda6ef1afe46a81b518fe9da64a832751" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.822ex; height:2.843ex;" alt="{\displaystyle [0,\pi ]}"> </td></tr> <tr> <th><a href="/wiki/Monotone_Funktion" class="mw-redirect" title="Monotone Funktion">Monotonie</a> </th> <td>streng monoton steigend </td> <td>streng monoton fallend </td></tr> <tr> <th><a href="/wiki/Symmetrie_(Geometrie)" title="Symmetrie (Geometrie)">Symmetrien</a> </th> <td><a href="/wiki/Ungerade_Funktion" class="mw-redirect" title="Ungerade Funktion">Ungerade Funktion</a> (<a href="/wiki/Punktsymmetrie" title="Punktsymmetrie">Punktsymmetrie</a> zu <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (0,0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (0,0)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d630d3e781a53b0a3559ae7e5b45f9479a3141a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.168ex; height:2.843ex;" alt="{\displaystyle (0,0)}">): <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \arcsin(-x)=-\arcsin(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arcsin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mi>arcsin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \arcsin(-x)=-\arcsin(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53647f9a40fbac38f7583597ad2d35882c69f687" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.304ex; height:2.843ex;" alt="{\displaystyle \arcsin(-x)=-\arcsin(x)}"> </td> <td>Punktsymmetrie zu <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(0,{\tfrac {\pi }{2}}\right)\colon }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <mn>0</mn> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mstyle> </mrow> </mrow> <mo>)</mo> </mrow> <mo>:</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(0,{\tfrac {\pi }{2}}\right)\colon }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/26766a9987cf752f64b4989336851fd085b7d26b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:7.138ex; height:3.343ex;" alt="{\displaystyle \left(0,{\tfrac {\pi }{2}}\right)\colon }"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \arccos(x)=\pi -\arccos(-x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arccos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>π</mi> <mo>−</mo> <mi>arccos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \arccos(x)=\pi -\arccos(-x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2bc22f3c351f26cfa6ad2e09ef346553d6aa0db9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.792ex; height:2.843ex;" alt="{\displaystyle \arccos(x)=\pi -\arccos(-x)}"> </td></tr> <tr> <th><a href="/wiki/Asymptote" title="Asymptote">Asymptoten</a> </th> <td>keine </td> <td>keine </td></tr> <tr> <th><a href="/wiki/Nullstelle" title="Nullstelle">Nullstellen</a> </th> <td>Eine Nullstelle bei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/953917eaf52f2e1baad54c8c9e3d6f9bb3710cdc" 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}"> </td> <td>Eine Nullstelle bei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee42176e76ae6b56d68c42ced807e08b962a2b54" 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=1}"> </td></tr> <tr> <th><a href="/wiki/Sprungstelle" class="mw-redirect" title="Sprungstelle">Sprungstellen</a> </th> <td>keine </td> <td>keine </td></tr> <tr> <th><a href="/wiki/Polstelle" title="Polstelle">Polstellen</a> </th> <td>keine </td> <td>keine </td></tr> <tr> <th><a href="/wiki/Extremum" class="mw-redirect" title="Extremum">Extrema</a> </th> <td>Globales Maximum <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {\pi }{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {\pi }{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4e31a202557dfbf326b44ebcc914ba3ab08fff1" 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}}}"> an der Stelle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}">, globales Minimum <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -{\tfrac {\pi }{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -{\tfrac {\pi }{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67a19acfb376df2a06f33d08ca28752fd43cb686" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:3.586ex; height:3.176ex;" alt="{\displaystyle -{\tfrac {\pi }{2}}}"> an der Stelle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/704fb0427140d054dd267925495e78164fee9aac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.971ex; height:2.343ex;" alt="{\displaystyle -1}"> </td> <td>Globales Maximum <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9be4ba0bb8df3af72e90a0535fabcc17431e540a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.332ex; height:1.676ex;" alt="{\displaystyle \pi }"> an der Stelle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/704fb0427140d054dd267925495e78164fee9aac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.971ex; height:2.343ex;" alt="{\displaystyle -1}">, globales Minimum <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}"> an der Stelle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}"> </td></tr> <tr> <th><a href="/wiki/Wendepunkt" title="Wendepunkt">Wendepunkte</a> </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (0,0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (0,0)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d630d3e781a53b0a3559ae7e5b45f9479a3141a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.168ex; height:2.843ex;" alt="{\displaystyle (0,0)}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(0,{\frac {\pi }{2}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <mn>0</mn> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(0,{\frac {\pi }{2}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef868c0486ec39cc42f223ebdfba02f0f59e1fb9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:7.14ex; height:4.843ex;" alt="{\displaystyle \left(0,{\frac {\pi }{2}}\right)}"> </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="Formeln_für_negative_Argumente">Formeln für negative Argumente</h2>[<a href="/w/index.php?title=Arkussinus_und_Arkuskosinus&veaction=edit&section=3" title="Abschnitt bearbeiten: Formeln für negative Argumente" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Arkussinus_und_Arkuskosinus&action=edit&section=3" title="Quellcode des Abschnitts bearbeiten: Formeln für negative Argumente">Quelltext bearbeiten</a>]</div> Aufgrund der Symmetrieeigenschaften gilt: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \arcsin(-x)=-\arcsin(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arcsin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mi>arcsin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \arcsin(-x)=-\arcsin(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53647f9a40fbac38f7583597ad2d35882c69f687" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.304ex; height:2.843ex;" alt="{\displaystyle \arcsin(-x)=-\arcsin(x)}"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \arccos(-x)=\pi -\arccos(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arccos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>π</mi> <mo>−</mo> <mi>arccos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \arccos(-x)=\pi -\arccos(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/66445bf7d606271f108bef813d1e415ccd4e8584" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.792ex; height:2.843ex;" alt="{\displaystyle \arccos(-x)=\pi -\arccos(x)}"></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Reihenentwicklungen">Reihenentwicklungen</h2>[<a href="/w/index.php?title=Arkussinus_und_Arkuskosinus&veaction=edit&section=4" title="Abschnitt bearbeiten: Reihenentwicklungen" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Arkussinus_und_Arkuskosinus&action=edit&section=4" title="Quellcode des Abschnitts bearbeiten: Reihenentwicklungen">Quelltext bearbeiten</a>]</div> Die <a href="/wiki/Taylorreihe" title="Taylorreihe">Taylorreihe</a> des Arkussinus erhält man durch Entwickeln der Ableitung in eine <a href="/wiki/Binomische_Reihe" title="Binomische Reihe">binomische Reihe</a> und anschließende Integration, sie ist gegeben durch: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\arcsin(x)&=\sum _{k=0}^{\infty }{\frac {(2k-1)!!}{(2k)!!}}{\frac {x^{2k+1}}{2k+1}}=\sum _{k=0}^{\infty }{\binom {2k}{k}}{\frac {x^{2k+1}}{4^{k}(2k+1)}}=\sum _{k=0}^{\infty }\,{\frac {\operatorname {CBC} (k)}{4^{k}(2k+1)}}\,x^{2k+1}=\\&={x+{\frac {1}{2}}\cdot {\frac {x^{3}}{3}}+{\frac {1\cdot 3}{2\cdot 4}}\cdot {\frac {x^{5}}{5}}+{\frac {1\cdot 3\cdot 5}{2\cdot 4\cdot 6}}\cdot {\frac {x^{7}}{7}}+\dotsb }\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>arcsin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>!</mo> <mo>!</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo stretchy="false">)</mo> <mo>!</mo> <mo>!</mo> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mrow> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mn>2</mn> <mi>k</mi> </mrow> <mi>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mrow> <msup> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>CBC</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <msup> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mo>=</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mn>3</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>⋅</mo> <mn>3</mn> </mrow> <mrow> <mn>2</mn> <mo>⋅</mo> <mn>4</mn> </mrow> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mn>5</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>⋅</mo> <mn>3</mn> <mo>⋅</mo> <mn>5</mn> </mrow> <mrow> <mn>2</mn> <mo>⋅</mo> <mn>4</mn> <mo>⋅</mo> <mn>6</mn> </mrow> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msup> <mn>7</mn> </mfrac> </mrow> <mo>+</mo> <mo>⋯</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\arcsin(x)&=\sum _{k=0}^{\infty }{\frac {(2k-1)!!}{(2k)!!}}{\frac {x^{2k+1}}{2k+1}}=\sum _{k=0}^{\infty }{\binom {2k}{k}}{\frac {x^{2k+1}}{4^{k}(2k+1)}}=\sum _{k=0}^{\infty }\,{\frac {\operatorname {CBC} (k)}{4^{k}(2k+1)}}\,x^{2k+1}=\\&={x+{\frac {1}{2}}\cdot {\frac {x^{3}}{3}}+{\frac {1\cdot 3}{2\cdot 4}}\cdot {\frac {x^{5}}{5}}+{\frac {1\cdot 3\cdot 5}{2\cdot 4\cdot 6}}\cdot {\frac {x^{7}}{7}}+\dotsb }\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/21cc460631af26044d4f70e1b7a6db8d7ff485d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.838ex; width:84.812ex; height:12.843ex;" alt="{\displaystyle {\begin{aligned}\arcsin(x)&=\sum _{k=0}^{\infty }{\frac {(2k-1)!!}{(2k)!!}}{\frac {x^{2k+1}}{2k+1}}=\sum _{k=0}^{\infty }{\binom {2k}{k}}{\frac {x^{2k+1}}{4^{k}(2k+1)}}=\sum _{k=0}^{\infty }\,{\frac {\operatorname {CBC} (k)}{4^{k}(2k+1)}}\,x^{2k+1}=\\&={x+{\frac {1}{2}}\cdot {\frac {x^{3}}{3}}+{\frac {1\cdot 3}{2\cdot 4}}\cdot {\frac {x^{5}}{5}}+{\frac {1\cdot 3\cdot 5}{2\cdot 4\cdot 6}}\cdot {\frac {x^{7}}{7}}+\dotsb }\end{aligned}}}"></dd></dl> Die <a href="/wiki/Taylorreihe" title="Taylorreihe">Taylorreihe</a> des Arkuskosinus ergibt sich aus der Beziehung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \arccos x={\tfrac {\pi }{2}}-\arcsin x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arccos</mi> <mo>⁡</mo> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>−</mo> <mi>arcsin</mi> <mo>⁡</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \arccos x={\tfrac {\pi }{2}}-\arcsin x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f68246e62a9bd048bcf1de95ffe72ab33a0266c2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:23.33ex; height:3.176ex;" alt="{\displaystyle \arccos x={\tfrac {\pi }{2}}-\arcsin x}">: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \arccos(x)={\frac {\pi }{2}}-\sum _{k=0}^{\infty }{\frac {(2k-1)!!}{(2k)!!}}{\frac {x^{2k+1}}{2k+1}}={\frac {\pi }{2}}-\sum _{k=0}^{\infty }{\binom {2k}{k}}{\frac {x^{2k+1}}{4^{k}(2k+1)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arccos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>!</mo> <mo>!</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo stretchy="false">)</mo> <mo>!</mo> <mo>!</mo> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mrow> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mn>2</mn> <mi>k</mi> </mrow> <mi>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mrow> <msup> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \arccos(x)={\frac {\pi }{2}}-\sum _{k=0}^{\infty }{\frac {(2k-1)!!}{(2k)!!}}{\frac {x^{2k+1}}{2k+1}}={\frac {\pi }{2}}-\sum _{k=0}^{\infty }{\binom {2k}{k}}{\frac {x^{2k+1}}{4^{k}(2k+1)}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b41a2ff6e31821e3ddb2e04a3e9113d8142d685" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:67.651ex; height:7.009ex;" alt="{\displaystyle \arccos(x)={\frac {\pi }{2}}-\sum _{k=0}^{\infty }{\frac {(2k-1)!!}{(2k)!!}}{\frac {x^{2k+1}}{2k+1}}={\frac {\pi }{2}}-\sum _{k=0}^{\infty }{\binom {2k}{k}}{\frac {x^{2k+1}}{4^{k}(2k+1)}}}"></dd></dl> Beide Reihen haben den <a href="/wiki/Konvergenzradius" title="Konvergenzradius">Konvergenzradius</a> 1. Der Ausdruck <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k!!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>!</mo> <mo>!</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k!!}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/62ad767fe1e8212343f3c2a88b17ecb554460423" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.505ex; height:2.176ex;" alt="{\displaystyle k!!}"> bezeichnet dabei die <a href="/wiki/Fakult%C3%A4t_(Mathematik)#Doppelfakultät" title="Fakultät (Mathematik)">Doppelfakultät</a> und mit dem Ausdruck CBC wird der <a href="/wiki/Mittlerer_Binomialkoeffizient" title="Mittlerer Binomialkoeffizient">Zentralbinomialkoeffizient</a> bezeichnet: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {CBC} (x)={2x \choose x}={\frac {(2x)!}{(x!)^{2}}}={\frac {\Pi (2x)}{\Pi (x)^{2}}}=\prod _{n=1}^{\infty }\left[\left(1+{\frac {x}{n}}\right)^{2}\left(1+{\frac {2x}{n}}\right)^{-1}\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>CBC</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mn>2</mn> <mi>x</mi> </mrow> <mi>x</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>x</mi> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>!</mo> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Π</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">Π</mi> <mo stretchy="false">(</mo> <mi>x</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <munderover> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow> <mo>[</mo> <mrow> <msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mi>n</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>x</mi> </mrow> <mi>n</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {CBC} (x)={2x \choose x}={\frac {(2x)!}{(x!)^{2}}}={\frac {\Pi (2x)}{\Pi (x)^{2}}}=\prod _{n=1}^{\infty }\left[\left(1+{\frac {x}{n}}\right)^{2}\left(1+{\frac {2x}{n}}\right)^{-1}\right]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35dde2db3e0a178b0c689321c3bfa85f34f75758" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:68.446ex; height:7.509ex;" alt="{\displaystyle \operatorname {CBC} (x)={2x \choose x}={\frac {(2x)!}{(x!)^{2}}}={\frac {\Pi (2x)}{\Pi (x)^{2}}}=\prod _{n=1}^{\infty }\left[\left(1+{\frac {x}{n}}\right)^{2}\left(1+{\frac {2x}{n}}\right)^{-1}\right]}"></dd></dl> So wird der Zentralbinomialkoeffizient mit Hilfe von der <a href="/wiki/Fakult%C3%A4t_(Mathematik)" title="Fakultät (Mathematik)">Fakultätsfunktion</a> beziehungsweise der Gaußschen Pifunktion definiert. Im Gegensatz zum Arkussinus selbst hat das Quadrat des Arkussinus in dessen <a href="/wiki/Maclaurinsche_Reihe" title="Maclaurinsche Reihe">MacLaurinschen Reihe</a> den Zentralbinomialkoeffizienten<a href="#cite_note-3">[3]</a> nicht im Zähler, sondern im Nenner: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\arcsin(x)^{2}&=\sum _{n=1}^{\infty }\,{\frac {2^{2n-1}}{n^{2}\operatorname {CBC} (n)}}\,x^{2n}=\\&={x^{2}+{\frac {1}{3}}\cdot x^{4}+{\frac {8}{45}}\cdot x^{6}+{\frac {4}{35}}\cdot x^{8}+\dotsb }\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>arcsin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mrow> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>CBC</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msup> <mo>=</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <mo>⋅</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>8</mn> <mn>45</mn> </mfrac> </mrow> <mo>⋅</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mn>35</mn> </mfrac> </mrow> <mo>⋅</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>8</mn> </mrow> </msup> <mo>+</mo> <mo>⋯</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\arcsin(x)^{2}&=\sum _{n=1}^{\infty }\,{\frac {2^{2n-1}}{n^{2}\operatorname {CBC} (n)}}\,x^{2n}=\\&={x^{2}+{\frac {1}{3}}\cdot x^{4}+{\frac {8}{45}}\cdot x^{6}+{\frac {4}{35}}\cdot x^{8}+\dotsb }\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44662830bd9213025d898b5f3b7bb6642bc3a32a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:50.984ex; height:12.509ex;" alt="{\displaystyle {\begin{aligned}\arcsin(x)^{2}&=\sum _{n=1}^{\infty }\,{\frac {2^{2n-1}}{n^{2}\operatorname {CBC} (n)}}\,x^{2n}=\\&={x^{2}+{\frac {1}{3}}\cdot x^{4}+{\frac {8}{45}}\cdot x^{6}+{\frac {4}{35}}\cdot x^{8}+\dotsb }\end{aligned}}}"></dd></dl> Das Gleiche gilt somit auch für den Quotienten aus Arkussinus und Pythagoräischer Gegenstückfunktion: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\frac {\arcsin(x)}{\sqrt {1-x^{2}}}}&=\sum _{n=1}^{\infty }\,{\frac {2^{2n-1}}{n\operatorname {CBC} (n)}}\,x^{2n-1}\\&={x+{\frac {2}{3}}\cdot x^{3}+{\frac {8}{15}}\cdot x^{5}+{\frac {16}{35}}\cdot x^{7}+\dotsb }\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>arcsin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mrow> <mi>n</mi> <mi>CBC</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mrow> <mo>⋅</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>8</mn> <mn>15</mn> </mfrac> </mrow> <mo>⋅</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>16</mn> <mn>35</mn> </mfrac> </mrow> <mo>⋅</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msup> <mo>+</mo> <mo>⋯</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\frac {\arcsin(x)}{\sqrt {1-x^{2}}}}&=\sum _{n=1}^{\infty }\,{\frac {2^{2n-1}}{n\operatorname {CBC} (n)}}\,x^{2n-1}\\&={x+{\frac {2}{3}}\cdot x^{3}+{\frac {8}{15}}\cdot x^{5}+{\frac {16}{35}}\cdot x^{7}+\dotsb }\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee7fcf244feae2243189faeca0c15cb78740c356" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:49.711ex; height:12.509ex;" alt="{\displaystyle {\begin{aligned}{\frac {\arcsin(x)}{\sqrt {1-x^{2}}}}&=\sum _{n=1}^{\infty }\,{\frac {2^{2n-1}}{n\operatorname {CBC} (n)}}\,x^{2n-1}\\&={x+{\frac {2}{3}}\cdot x^{3}+{\frac {8}{15}}\cdot x^{5}+{\frac {16}{35}}\cdot x^{7}+\dotsb }\end{aligned}}}"></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Verkettungen_mit_Sinus_und_Kosinus">Verkettungen mit Sinus und Kosinus</h2>[<a href="/w/index.php?title=Arkussinus_und_Arkuskosinus&veaction=edit&section=5" title="Abschnitt bearbeiten: Verkettungen mit Sinus und Kosinus" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Arkussinus_und_Arkuskosinus&action=edit&section=5" title="Quellcode des Abschnitts bearbeiten: Verkettungen mit Sinus und Kosinus">Quelltext bearbeiten</a>]</div> Für die Arkusfunktionen gelten unter anderem folgende Formeln: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin(\arccos(x))={\sqrt {1-x^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>arccos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin(\arccos(x))={\sqrt {1-x^{2}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/96715129befa92fe44fde97e001a8bed280dcb4a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.83ex; height:3.509ex;" alt="{\displaystyle \sin(\arccos(x))={\sqrt {1-x^{2}}}}">, denn für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=\arccos(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mi>arccos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=\arccos(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/180deffe95226787d18a945b5093e1a9f64d85b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.611ex; height:2.843ex;" alt="{\displaystyle y=\arccos(x)}"> gilt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y\in \left[0,{\pi }\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>∈</mo> <mrow> <mo>[</mo> <mrow> <mn>0</mn> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>π</mi> </mrow> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y\in \left[0,{\pi }\right]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6f788ef654230044a492dd40686fc6f393be8e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.818ex; height:2.843ex;" alt="{\displaystyle y\in \left[0,{\pi }\right]}"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin(y)={\sqrt {1-\cos ^{2}(y)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin(y)={\sqrt {1-\cos ^{2}(y)}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bdb4a040aab0fbe62b545cd2d422319fa8d2e256" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:22.376ex; height:4.843ex;" alt="{\displaystyle \sin(y)={\sqrt {1-\cos ^{2}(y)}}}">.</dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos(\arcsin(x))={\sqrt {1-x^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>arcsin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos(\arcsin(x))={\sqrt {1-x^{2}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/58b7e76b5f09150c7d53eefd98da99a7c6003b55" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.83ex; height:3.509ex;" alt="{\displaystyle \cos(\arcsin(x))={\sqrt {1-x^{2}}}}">, denn für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=\arcsin(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mi>arcsin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=\arcsin(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38e0923cd97e3e42f303d625dde6e5b468836824" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.355ex; height:2.843ex;" alt="{\displaystyle y=\arcsin(x)}"> gilt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y\in \left[-{\frac {\pi }{2}},{\frac {\pi }{2}}\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>∈</mo> <mrow> <mo>[</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y\in \left[-{\frac {\pi }{2}},{\frac {\pi }{2}}\right]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98937a3d0b8000c81a0eed570473405330b76605" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.369ex; height:4.843ex;" alt="{\displaystyle y\in \left[-{\frac {\pi }{2}},{\frac {\pi }{2}}\right]}"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos(y)={\sqrt {1-\sin ^{2}(y)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos(y)={\sqrt {1-\sin ^{2}(y)}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54bfb69c3316ba2f920feae2f6486b6d16811b98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:22.376ex; height:4.843ex;" alt="{\displaystyle \cos(y)={\sqrt {1-\sin ^{2}(y)}}}">.</dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin(\arctan(x))={\frac {x}{\sqrt {1+x^{2}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>arctan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <msqrt> <mn>1</mn> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin(\arctan(x))={\frac {x}{\sqrt {1+x^{2}}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45a32bc142eb5da5fe334aa7c496e99fc67ff33f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:26.915ex; height:6.009ex;" alt="{\displaystyle \sin(\arctan(x))={\frac {x}{\sqrt {1+x^{2}}}}}">, denn für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=\arctan(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mi>arctan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=\arctan(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47ab78600eb69cbc1af4ae58af2532a134b99275" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.859ex; height:2.843ex;" alt="{\displaystyle y=\arctan(x)}"> gilt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y\in \left]-{\frac {\pi }{2}},{\frac {\pi }{2}}\right[}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>∈</mo> <mrow> <mo>]</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> </mrow> <mo>[</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y\in \left]-{\frac {\pi }{2}},{\frac {\pi }{2}}\right[}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae3c69f5704cb758503bfa093c66044205b89ae9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.369ex; height:4.843ex;" alt="{\displaystyle y\in \left]-{\frac {\pi }{2}},{\frac {\pi }{2}}\right[}"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin(y)={\frac {\tan(y)}{\sqrt {1+\tan ^{2}(y)}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>tan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> <msqrt> <mn>1</mn> <mo>+</mo> <msup> <mi>tan</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> </msqrt> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin(y)={\frac {\tan(y)}{\sqrt {1+\tan ^{2}(y)}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9845e3ab141ce42970b01dd2720ad64f0b14cc12" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:23.46ex; height:8.509ex;" alt="{\displaystyle \sin(y)={\frac {\tan(y)}{\sqrt {1+\tan ^{2}(y)}}}}">.</dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos(\arctan(x))={\frac {1}{\sqrt {1+x^{2}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>arctan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>1</mn> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos(\arctan(x))={\frac {1}{\sqrt {1+x^{2}}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98a657dfc8a2af1dc1a57a7ea86536226babe99e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:27.171ex; height:6.509ex;" alt="{\displaystyle \cos(\arctan(x))={\frac {1}{\sqrt {1+x^{2}}}}}">, denn für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=\arctan(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mi>arctan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=\arctan(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47ab78600eb69cbc1af4ae58af2532a134b99275" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.859ex; height:2.843ex;" alt="{\displaystyle y=\arctan(x)}"> gilt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y\in \left]-{\frac {\pi }{2}},{\frac {\pi }{2}}\right[}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>∈</mo> <mrow> <mo>]</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> </mrow> <mo>[</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y\in \left]-{\frac {\pi }{2}},{\frac {\pi }{2}}\right[}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae3c69f5704cb758503bfa093c66044205b89ae9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.369ex; height:4.843ex;" alt="{\displaystyle y\in \left]-{\frac {\pi }{2}},{\frac {\pi }{2}}\right[}"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos(y)={\frac {1}{\sqrt {1+\tan ^{2}(y)}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>1</mn> <mo>+</mo> <msup> <mi>tan</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> </msqrt> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos(y)={\frac {1}{\sqrt {1+\tan ^{2}(y)}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06bf4ee1a5ae3d17fc277fd385ac6bbc6ad3d625" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:23.716ex; height:8.009ex;" alt="{\displaystyle \cos(y)={\frac {1}{\sqrt {1+\tan ^{2}(y)}}}}">.</dd></dl> <div class="mw-heading mw-heading2"><h2 id="Beziehung_zum_Arkustangens">Beziehung zum Arkustangens</h2>[<a href="/w/index.php?title=Arkussinus_und_Arkuskosinus&veaction=edit&section=6" title="Abschnitt bearbeiten: Beziehung zum Arkustangens" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Arkussinus_und_Arkuskosinus&action=edit&section=6" title="Quellcode des Abschnitts bearbeiten: Beziehung zum Arkustangens">Quelltext bearbeiten</a>]</div> Von besonderer Bedeutung in älteren Programmiersprachen ohne implementierte Arkussinus- und Arkuskosinusfunktion sind folgende Beziehungen, die es ermöglichen, den Arkussinus und Arkuskosinus aus dem vielleicht implementierten Arkustangens zu berechnen. Aufgrund obiger Formeln gilt <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \arcsin(x)=\arctan \left({\frac {x}{\sqrt {1-x^{2}}}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arcsin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>arctan</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \arcsin(x)=\arctan \left({\frac {x}{\sqrt {1-x^{2}}}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/356a4125c347df9d0ccee1926e93ef622486ddd9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:31.894ex; height:7.509ex;" alt="{\displaystyle \arcsin(x)=\arctan \left({\frac {x}{\sqrt {1-x^{2}}}}\right)}"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \arccos(x)={\frac {\pi }{2}}-\arctan \left({\frac {x}{\sqrt {1-x^{2}}}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arccos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <mi>arctan</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \arccos(x)={\frac {\pi }{2}}-\arctan \left({\frac {x}{\sqrt {1-x^{2}}}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/651f04689e8d4b51b7f5006e84e2af6226c8dc13" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:37.158ex; height:7.509ex;" alt="{\displaystyle \arccos(x)={\frac {\pi }{2}}-\arctan \left({\frac {x}{\sqrt {1-x^{2}}}}\right)}"></dd></dl> für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |x|<1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo><</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |x|<1.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c4e657241d23e0514c31745c2d302fffa61a77ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.531ex; height:2.843ex;" alt="{\displaystyle |x|<1.}"> Definiert man <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \arctan \left({\tfrac {1}{0}}\right):=\lim _{t\to \infty }\arctan(t)={\tfrac {\pi }{2}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arctan</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>0</mn> </mfrac> </mstyle> </mrow> <mo>)</mo> </mrow> <mo>:=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mi>arctan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \arctan \left({\tfrac {1}{0}}\right):=\lim _{t\to \infty }\arctan(t)={\tfrac {\pi }{2}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/82ece3b05cd99f24ad92e08a2b6e3bcdd3da4385" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:33.55ex; height:5.009ex;" alt="{\displaystyle \arctan \left({\tfrac {1}{0}}\right):=\lim _{t\to \infty }\arctan(t)={\tfrac {\pi }{2}},}"> so werden diese beiden Gleichungen auch für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=\pm 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mo>±</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=\pm 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae75026eda1ba1ac961f5003150288e9d2f3049b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.399ex; height:2.176ex;" alt="{\displaystyle x=\pm 1}"> richtig. Alternativ dazu kann man auch <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \arcsin(x)=2\arctan \left({\frac {x}{1+{\sqrt {1-x^{2}}}}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arcsin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>2</mn> <mi>arctan</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \arcsin(x)=2\arctan \left({\frac {x}{1+{\sqrt {1-x^{2}}}}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b5eca8ab240be45337dca5d00e20f4c938cf7d07" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:37.446ex; height:7.509ex;" alt="{\displaystyle \arcsin(x)=2\arctan \left({\frac {x}{1+{\sqrt {1-x^{2}}}}}\right)}"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \arccos(x)={\frac {\pi }{2}}-2\arctan \left({\frac {x}{1+{\sqrt {1-x^{2}}}}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arccos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <mn>2</mn> <mi>arctan</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \arccos(x)={\frac {\pi }{2}}-2\arctan \left({\frac {x}{1+{\sqrt {1-x^{2}}}}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bdf69569becd8d88c9a683a5a5a4a66d2696a677" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:42.71ex; height:7.509ex;" alt="{\displaystyle \arccos(x)={\frac {\pi }{2}}-2\arctan \left({\frac {x}{1+{\sqrt {1-x^{2}}}}}\right)}"></dd></dl> verwenden, was sich aus Obigem durch Anwenden der <a href="/wiki/Arkustangens_und_Arkuskotangens#Funktionalgleichungen" title="Arkustangens und Arkuskotangens">Funktionalgleichung des Arkustangens</a> ergibt und für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |x|\leq 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>≤</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |x|\leq 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ad95e9e3840f09ecffbe53c104ee1bbf639fd42" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.884ex; height:2.843ex;" alt="{\displaystyle |x|\leq 1}"> gilt. Für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -1<x\leq 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>1</mn> <mo><</mo> <mi>x</mi> <mo>≤</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -1<x\leq 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b3146f6393631821d264b39f7bf2a9fc1e60cb0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.66ex; height:2.343ex;" alt="{\displaystyle -1<x\leq 1}"> lässt sich Letzteres auch zu <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \arccos(x)=2\arctan \left({\sqrt {\frac {1-x}{1+x}}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arccos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>2</mn> <mi>arctan</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mi>x</mi> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <mi>x</mi> </mrow> </mfrac> </msqrt> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \arccos(x)=2\arctan \left({\sqrt {\frac {1-x}{1+x}}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d461bdd1b606d89e06e5def5ae232a70515ea5a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:32.384ex; height:6.343ex;" alt="{\displaystyle \arccos(x)=2\arctan \left({\sqrt {\frac {1-x}{1+x}}}\right)}"></dd></dl> vereinfachen. <div class="mw-heading mw-heading2"><h2 id="Additionstheoreme">Additionstheoreme</h2>[<a href="/w/index.php?title=Arkussinus_und_Arkuskosinus&veaction=edit&section=7" title="Abschnitt bearbeiten: Additionstheoreme" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Arkussinus_und_Arkuskosinus&action=edit&section=7" title="Quellcode des Abschnitts bearbeiten: Additionstheoreme">Quelltext bearbeiten</a>]</div> <div class="hauptartikel" role="navigation">→ Hauptartikel: <a href="/wiki/Formelsammlung_Trigonometrie#Additionstheoreme_für_Arkusfunktionen" title="Formelsammlung Trigonometrie">Additionstheoreme für Arkusfunktionen (Trigonometrie)</a></div> Die Additionstheoreme für Arkussinus und Arkuskosinus erhält man mit Hilfe der <a href="/wiki/Sinus_und_Kosinus#Additionstheoreme" title="Sinus und Kosinus">Additionstheoreme für Sinus und Kosinus</a>: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \arcsin x+\arcsin y=\left\{{\begin{array}{rcrl}\arcsin(\sin(\arcsin x+\arcsin y))&=&\arcsin \left(x{\sqrt {1-y^{2}}}+y{\sqrt {1-x^{2}}}\right)&{\text{wenn}}\quad xy\leq 0\quad {\text{oder}}\quad x^{2}+y^{2}\leq 1\\\pi -\arcsin(\sin(\arcsin x+\arcsin y))&=&\pi -\arcsin \left(x{\sqrt {1-y^{2}}}+y{\sqrt {1-x^{2}}}\right)&{\text{wenn}}\quad x>0\quad {\text{und}}\quad y>0\quad {\text{und}}\quad x^{2}+y^{2}>1\\-\pi -\arcsin(\sin(\arcsin x+\arcsin y))&=&-\pi -\arcsin \left(x{\sqrt {1-y^{2}}}+y{\sqrt {1-x^{2}}}\right)&{\text{wenn}}\quad x<0\quad {\text{und}}\quad y<0\quad {\text{und}}\quad x^{2}+y^{2}>1\\\end{array}}\right.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arcsin</mi> <mo>⁡</mo> <mi>x</mi> <mo>+</mo> <mi>arcsin</mi> <mo>⁡</mo> <mi>y</mi> <mo>=</mo> <mrow> <mo>{</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right center right left" rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>arcsin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>arcsin</mi> <mo>⁡</mo> <mi>x</mi> <mo>+</mo> <mi>arcsin</mi> <mo>⁡</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mtd> <mtd> <mo>=</mo> </mtd> <mtd> <mi>arcsin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>+</mo> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mrow> <mo>)</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>wenn</mtext> </mrow> <mspace width="1em" /> <mi>x</mi> <mi>y</mi> <mo>≤</mo> <mn>0</mn> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>oder</mtext> </mrow> <mspace width="1em" /> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>≤</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mi>π</mi> <mo>−</mo> <mi>arcsin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>arcsin</mi> <mo>⁡</mo> <mi>x</mi> <mo>+</mo> <mi>arcsin</mi> <mo>⁡</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mtd> <mtd> <mo>=</mo> </mtd> <mtd> <mi>π</mi> <mo>−</mo> <mi>arcsin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>+</mo> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mrow> <mo>)</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>wenn</mtext> </mrow> <mspace width="1em" /> <mi>x</mi> <mo>></mo> <mn>0</mn> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>und</mtext> </mrow> <mspace width="1em" /> <mi>y</mi> <mo>></mo> <mn>0</mn> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>und</mtext> </mrow> <mspace width="1em" /> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>></mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mi>π</mi> <mo>−</mo> <mi>arcsin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>arcsin</mi> <mo>⁡</mo> <mi>x</mi> <mo>+</mo> <mi>arcsin</mi> <mo>⁡</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mtd> <mtd> <mo>=</mo> </mtd> <mtd> <mo>−</mo> <mi>π</mi> <mo>−</mo> <mi>arcsin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>+</mo> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mrow> <mo>)</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>wenn</mtext> </mrow> <mspace width="1em" /> <mi>x</mi> <mo><</mo> <mn>0</mn> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>und</mtext> </mrow> <mspace width="1em" /> <mi>y</mi> <mo><</mo> <mn>0</mn> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>und</mtext> </mrow> <mspace width="1em" /> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>></mo> <mn>1</mn> </mtd> </mtr> </mtable> </mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \arcsin x+\arcsin y=\left\{{\begin{array}{rcrl}\arcsin(\sin(\arcsin x+\arcsin y))&=&\arcsin \left(x{\sqrt {1-y^{2}}}+y{\sqrt {1-x^{2}}}\right)&{\text{wenn}}\quad xy\leq 0\quad {\text{oder}}\quad x^{2}+y^{2}\leq 1\\\pi -\arcsin(\sin(\arcsin x+\arcsin y))&=&\pi -\arcsin \left(x{\sqrt {1-y^{2}}}+y{\sqrt {1-x^{2}}}\right)&{\text{wenn}}\quad x>0\quad {\text{und}}\quad y>0\quad {\text{und}}\quad x^{2}+y^{2}>1\\-\pi -\arcsin(\sin(\arcsin x+\arcsin y))&=&-\pi -\arcsin \left(x{\sqrt {1-y^{2}}}+y{\sqrt {1-x^{2}}}\right)&{\text{wenn}}\quad x<0\quad {\text{und}}\quad y<0\quad {\text{und}}\quad x^{2}+y^{2}>1\\\end{array}}\right.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77e359a897570396cdb648287b8a155bc3f935ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.838ex; width:153.823ex; height:14.843ex;" alt="{\displaystyle \arcsin x+\arcsin y=\left\{{\begin{array}{rcrl}\arcsin(\sin(\arcsin x+\arcsin y))&=&\arcsin \left(x{\sqrt {1-y^{2}}}+y{\sqrt {1-x^{2}}}\right)&{\text{wenn}}\quad xy\leq 0\quad {\text{oder}}\quad x^{2}+y^{2}\leq 1\\\pi -\arcsin(\sin(\arcsin x+\arcsin y))&=&\pi -\arcsin \left(x{\sqrt {1-y^{2}}}+y{\sqrt {1-x^{2}}}\right)&{\text{wenn}}\quad x>0\quad {\text{und}}\quad y>0\quad {\text{und}}\quad x^{2}+y^{2}>1\\-\pi -\arcsin(\sin(\arcsin x+\arcsin y))&=&-\pi -\arcsin \left(x{\sqrt {1-y^{2}}}+y{\sqrt {1-x^{2}}}\right)&{\text{wenn}}\quad x<0\quad {\text{und}}\quad y<0\quad {\text{und}}\quad x^{2}+y^{2}>1\\\end{array}}\right.}"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \arccos x+\arccos y=\left\{{\begin{array}{rcrl}\arccos(\cos(\arccos x+\arccos y))&=&\arccos \left(xy-{\sqrt {1-x^{2}}}{\sqrt {1-y^{2}}}\right)&{\text{wenn}}\quad x+y\geq 0\\2\pi -\arccos(\cos(\arccos x+\arccos y))&=&2\pi -\arccos \left(xy-{\sqrt {1-x^{2}}}{\sqrt {1-y^{2}}}\right)&{\text{wenn}}\quad x+y<0\\\end{array}}\right.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arccos</mi> <mo>⁡</mo> <mi>x</mi> <mo>+</mo> <mi>arccos</mi> <mo>⁡</mo> <mi>y</mi> <mo>=</mo> <mrow> <mo>{</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right center right left" rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>arccos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>arccos</mi> <mo>⁡</mo> <mi>x</mi> <mo>+</mo> <mi>arccos</mi> <mo>⁡</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mtd> <mtd> <mo>=</mo> </mtd> <mtd> <mi>arccos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mi>y</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mrow> <mo>)</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>wenn</mtext> </mrow> <mspace width="1em" /> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo>≥</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> <mi>π</mi> <mo>−</mo> <mi>arccos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>arccos</mi> <mo>⁡</mo> <mi>x</mi> <mo>+</mo> <mi>arccos</mi> <mo>⁡</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mtd> <mtd> <mo>=</mo> </mtd> <mtd> <mn>2</mn> <mi>π</mi> <mo>−</mo> <mi>arccos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mi>y</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mrow> <mo>)</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>wenn</mtext> </mrow> <mspace width="1em" /> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo><</mo> <mn>0</mn> </mtd> </mtr> </mtable> </mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \arccos x+\arccos y=\left\{{\begin{array}{rcrl}\arccos(\cos(\arccos x+\arccos y))&=&\arccos \left(xy-{\sqrt {1-x^{2}}}{\sqrt {1-y^{2}}}\right)&{\text{wenn}}\quad x+y\geq 0\\2\pi -\arccos(\cos(\arccos x+\arccos y))&=&2\pi -\arccos \left(xy-{\sqrt {1-x^{2}}}{\sqrt {1-y^{2}}}\right)&{\text{wenn}}\quad x+y<0\\\end{array}}\right.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8aa02d61d15b7871ae249383df7c6e6cf18a4d2d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.338ex; width:124.154ex; height:9.843ex;" alt="{\displaystyle \arccos x+\arccos y=\left\{{\begin{array}{rcrl}\arccos(\cos(\arccos x+\arccos y))&=&\arccos \left(xy-{\sqrt {1-x^{2}}}{\sqrt {1-y^{2}}}\right)&{\text{wenn}}\quad x+y\geq 0\\2\pi -\arccos(\cos(\arccos x+\arccos y))&=&2\pi -\arccos \left(xy-{\sqrt {1-x^{2}}}{\sqrt {1-y^{2}}}\right)&{\text{wenn}}\quad x+y<0\\\end{array}}\right.}"></dd></dl> Daraus folgt insbesondere für doppelte Funktionswerte <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\arcsin x=\left\{{\begin{array}{rl}\arcsin \left(2x{\sqrt {1-x^{2}}}\right)&{\text{wenn}}\quad 2x^{2}\leq 1\\\pi -\arcsin \left(2x{\sqrt {1-x^{2}}}\right)&{\text{wenn}}\quad x>0\quad {\text{und}}\quad 2x^{2}>1\\-\pi -\arcsin \left(2x{\sqrt {1-x^{2}}}\right)&{\text{wenn}}\quad x<0\quad {\text{und}}\quad 2x^{2}>1\\\end{array}}\right.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>arcsin</mi> <mo>⁡</mo> <mi>x</mi> <mo>=</mo> <mrow> <mo>{</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left" rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>arcsin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mrow> <mo>)</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>wenn</mtext> </mrow> <mspace width="1em" /> <mn>2</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>≤</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mi>π</mi> <mo>−</mo> <mi>arcsin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mrow> <mo>)</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>wenn</mtext> </mrow> <mspace width="1em" /> <mi>x</mi> <mo>></mo> <mn>0</mn> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>und</mtext> </mrow> <mspace width="1em" /> <mn>2</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>></mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mi>π</mi> <mo>−</mo> <mi>arcsin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mrow> <mo>)</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>wenn</mtext> </mrow> <mspace width="1em" /> <mi>x</mi> <mo><</mo> <mn>0</mn> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>und</mtext> </mrow> <mspace width="1em" /> <mn>2</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>></mo> <mn>1</mn> </mtd> </mtr> </mtable> </mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\arcsin x=\left\{{\begin{array}{rl}\arcsin \left(2x{\sqrt {1-x^{2}}}\right)&{\text{wenn}}\quad 2x^{2}\leq 1\\\pi -\arcsin \left(2x{\sqrt {1-x^{2}}}\right)&{\text{wenn}}\quad x>0\quad {\text{und}}\quad 2x^{2}>1\\-\pi -\arcsin \left(2x{\sqrt {1-x^{2}}}\right)&{\text{wenn}}\quad x<0\quad {\text{und}}\quad 2x^{2}>1\\\end{array}}\right.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5bfbb6d41b9c7b3d0b0bef0021ffb669f2b9353" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.838ex; width:72.926ex; height:14.843ex;" alt="{\displaystyle 2\arcsin x=\left\{{\begin{array}{rl}\arcsin \left(2x{\sqrt {1-x^{2}}}\right)&{\text{wenn}}\quad 2x^{2}\leq 1\\\pi -\arcsin \left(2x{\sqrt {1-x^{2}}}\right)&{\text{wenn}}\quad x>0\quad {\text{und}}\quad 2x^{2}>1\\-\pi -\arcsin \left(2x{\sqrt {1-x^{2}}}\right)&{\text{wenn}}\quad x<0\quad {\text{und}}\quad 2x^{2}>1\\\end{array}}\right.}"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\arccos x=\left\{{\begin{array}{rl}\arccos \left(2x^{2}-1\right)&{\text{wenn}}\quad x\geq 0\\2\pi -\arccos \left(2x^{2}-1\right)&{\text{wenn}}\quad x<0\\\end{array}}\right.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>arccos</mi> <mo>⁡</mo> <mi>x</mi> <mo>=</mo> <mrow> <mo>{</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left" rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>arccos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>wenn</mtext> </mrow> <mspace width="1em" /> <mi>x</mi> <mo>≥</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> <mi>π</mi> <mo>−</mo> <mi>arccos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>wenn</mtext> </mrow> <mspace width="1em" /> <mi>x</mi> <mo><</mo> <mn>0</mn> </mtd> </mtr> </mtable> </mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\arccos x=\left\{{\begin{array}{rl}\arccos \left(2x^{2}-1\right)&{\text{wenn}}\quad x\geq 0\\2\pi -\arccos \left(2x^{2}-1\right)&{\text{wenn}}\quad x<0\\\end{array}}\right.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8da54c26d9df56facfb8ca979b90f1c82083f462" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:51.97ex; height:7.509ex;" alt="{\displaystyle 2\arccos x=\left\{{\begin{array}{rl}\arccos \left(2x^{2}-1\right)&{\text{wenn}}\quad x\geq 0\\2\pi -\arccos \left(2x^{2}-1\right)&{\text{wenn}}\quad x<0\\\end{array}}\right.}"></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Ableitungen">Ableitungen</h2>[<a href="/w/index.php?title=Arkussinus_und_Arkuskosinus&veaction=edit&section=8" title="Abschnitt bearbeiten: Ableitungen" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Arkussinus_und_Arkuskosinus&action=edit&section=8" title="Quellcode des Abschnitts bearbeiten: Ableitungen">Quelltext bearbeiten</a>]</div> <dl><dt>Arkussinus</dt> <dd></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\arcsin(x)={\frac {1}{\sqrt {1-x^{2}}}},\qquad -1<x<1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> </mrow> </mfrac> </mrow> <mi>arcsin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> <mo>,</mo> <mspace width="2em" /> <mo>−</mo> <mn>1</mn> <mo><</mo> <mi>x</mi> <mo><</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\arcsin(x)={\frac {1}{\sqrt {1-x^{2}}}},\qquad -1<x<1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b05627bc0da4103cd85fa94a6c760ff1dddde543" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:42.93ex; height:6.676ex;" alt="{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\arcsin(x)={\frac {1}{\sqrt {1-x^{2}}}},\qquad -1<x<1}"></dd></dl> <dl><dt>Arkuskosinus</dt> <dd></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\arccos(x)=-{\frac {1}{\sqrt {1-x^{2}}}},\qquad -1<x<1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> </mrow> </mfrac> </mrow> <mi>arccos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> <mo>,</mo> <mspace width="2em" /> <mo>−</mo> <mn>1</mn> <mo><</mo> <mi>x</mi> <mo><</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\arccos(x)=-{\frac {1}{\sqrt {1-x^{2}}}},\qquad -1<x<1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35e071a7b69da340e1fba2cd477337be85e1dbe2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:44.994ex; height:6.676ex;" alt="{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\arccos(x)=-{\frac {1}{\sqrt {1-x^{2}}}},\qquad -1<x<1}"></dd></dl> <dl><dt>Umrechnung</dt> <dd></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\arccos(x)=-{\frac {\mathrm {d} }{\mathrm {d} x}}\arcsin(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> </mrow> </mfrac> </mrow> <mi>arccos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> </mrow> </mfrac> </mrow> <mi>arcsin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\arccos(x)=-{\frac {\mathrm {d} }{\mathrm {d} x}}\arcsin(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76d07f383231cfa69f004c8e39512c78e2c537de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:31.055ex; height:5.509ex;" alt="{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\arccos(x)=-{\frac {\mathrm {d} }{\mathrm {d} x}}\arcsin(x)}"></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Integrale">Integrale</h2>[<a href="/w/index.php?title=Arkussinus_und_Arkuskosinus&veaction=edit&section=9" title="Abschnitt bearbeiten: Integrale" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Arkussinus_und_Arkuskosinus&action=edit&section=9" title="Quellcode des Abschnitts bearbeiten: Integrale">Quelltext bearbeiten</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Standardisierte_Integraldarstellungen">Standardisierte Integraldarstellungen</h3>[<a href="/w/index.php?title=Arkussinus_und_Arkuskosinus&veaction=edit&section=10" title="Abschnitt bearbeiten: Standardisierte Integraldarstellungen" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Arkussinus_und_Arkuskosinus&action=edit&section=10" title="Quellcode des Abschnitts bearbeiten: Standardisierte Integraldarstellungen">Quelltext bearbeiten</a>]</div> Die Integraldarstellungen des Arkussinus bzw. Arkuskosinus sind gegeben durch: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \arcsin(x)=\int \limits _{0}^{x}{\frac {\mathrm {d} t}{\sqrt {1-t^{2}}}}=\int \limits _{0}^{1}{\frac {x}{\sqrt {1-x^{2}y^{2}}}}\,\mathrm {d} y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arcsin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> <mo>=</mo> <munderover> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \arcsin(x)=\int \limits _{0}^{x}{\frac {\mathrm {d} t}{\sqrt {1-t^{2}}}}=\int \limits _{0}^{1}{\frac {x}{\sqrt {1-x^{2}y^{2}}}}\,\mathrm {d} y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c33ff6a94652ad3321bce58dacc8c8d74b0e5722" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:44.113ex; height:9.176ex;" alt="{\displaystyle \arcsin(x)=\int \limits _{0}^{x}{\frac {\mathrm {d} t}{\sqrt {1-t^{2}}}}=\int \limits _{0}^{1}{\frac {x}{\sqrt {1-x^{2}y^{2}}}}\,\mathrm {d} y}"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \arccos(x)=\int \limits _{x}^{1}{\frac {\mathrm {d} t}{\sqrt {1-t^{2}}}}={\frac {\pi }{2}}-\int \limits _{0}^{1}{\frac {x}{\sqrt {1-x^{2}y^{2}}}}\,\mathrm {d} y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arccos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <munderover> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \arccos(x)=\int \limits _{x}^{1}{\frac {\mathrm {d} t}{\sqrt {1-t^{2}}}}={\frac {\pi }{2}}-\int \limits _{0}^{1}{\frac {x}{\sqrt {1-x^{2}y^{2}}}}\,\mathrm {d} y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71e672ce460f7b00da69afb0714c58b1b269028b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:49.377ex; height:9.176ex;" alt="{\displaystyle \arccos(x)=\int \limits _{x}^{1}{\frac {\mathrm {d} t}{\sqrt {1-t^{2}}}}={\frac {\pi }{2}}-\int \limits _{0}^{1}{\frac {x}{\sqrt {1-x^{2}y^{2}}}}\,\mathrm {d} y}"></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Integralidentität_mit_dem_Logarithmus_Naturalis">Integralidentität mit dem Logarithmus Naturalis</h3>[<a href="/w/index.php?title=Arkussinus_und_Arkuskosinus&veaction=edit&section=11" title="Abschnitt bearbeiten: Integralidentität mit dem Logarithmus Naturalis" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Arkussinus_und_Arkuskosinus&action=edit&section=11" title="Quellcode des Abschnitts bearbeiten: Integralidentität mit dem Logarithmus Naturalis">Quelltext bearbeiten</a>]</div> Auch mit dem <a href="/wiki/Logarithmus" title="Logarithmus">Logarithmus Naturalis</a> kann für den Arkussinus eine Integralidentität aufgestellt werden: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{\sqrt {1-x^{2}}}}=\int \limits _{0}^{1}{\frac {4\,(y^{2}+1)}{\pi {\bigl [}(y^{2}+1)^{2}-4\,x^{2}y^{2}{\bigr ]}}}\,\mathrm {d} y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> <mo>=</mo> <munderover> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>4</mn> <mspace width="thinmathspace" /> <mo stretchy="false">(</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">[</mo> </mrow> </mrow> <mo stretchy="false">(</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>4</mn> <mspace width="thinmathspace" /> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">]</mo> </mrow> </mrow> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{\sqrt {1-x^{2}}}}=\int \limits _{0}^{1}{\frac {4\,(y^{2}+1)}{\pi {\bigl [}(y^{2}+1)^{2}-4\,x^{2}y^{2}{\bigr ]}}}\,\mathrm {d} y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/58c9c67b16f98b3f9fc4ae6ca54b24894251f243" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:40.238ex; height:9.176ex;" alt="{\displaystyle {\frac {1}{\sqrt {1-x^{2}}}}=\int \limits _{0}^{1}{\frac {4\,(y^{2}+1)}{\pi {\bigl [}(y^{2}+1)^{2}-4\,x^{2}y^{2}{\bigr ]}}}\,\mathrm {d} y}"></dd></dl> Durch Bildung der Ursprungsstammfunktion bezüglich <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" 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}"> entsteht folgende Formel: <dl><dd><table class="wikitable"> <tbody><tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \arcsin(x)=\int \limits _{0}^{1}{\frac {1}{\pi \,y}}\ln {\biggl (}{\frac {y^{2}+2xy+1}{y^{2}-2xy+1}}{\biggr )}\,\mathrm {d} y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arcsin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>π</mi> <mspace width="thinmathspace" /> <mi>y</mi> </mrow> </mfrac> </mrow> <mi>ln</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mi>x</mi> <mi>y</mi> <mo>+</mo> <mn>1</mn> </mrow> <mrow> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <mi>x</mi> <mi>y</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \arcsin(x)=\int \limits _{0}^{1}{\frac {1}{\pi \,y}}\ln {\biggl (}{\frac {y^{2}+2xy+1}{y^{2}-2xy+1}}{\biggr )}\,\mathrm {d} y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/62a70843066d00cbb569e4918ededa27bfd90231" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:41.003ex; height:9.176ex;" alt="{\displaystyle \arcsin(x)=\int \limits _{0}^{1}{\frac {1}{\pi \,y}}\ln {\biggl (}{\frac {y^{2}+2xy+1}{y^{2}-2xy+1}}{\biggr )}\,\mathrm {d} y}"> </td></tr></tbody></table></dd></dl> Die nun gezeigte Integralidentität wurde durch den Mathematiker James Harper entdeckt und in seinen Werken A simple proof of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1+1/2^{2}+1/3^{2}+\ldots =\pi ^{2}/6}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>+</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo>…</mo> <mo>=</mo> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>6</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1+1/2^{2}+1/3^{2}+\ldots =\pi ^{2}/6}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5c4c7870d1df228bdd6db634426c16d49059896" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.302ex; height:3.176ex;" alt="{\displaystyle 1+1/2^{2}+1/3^{2}+\ldots =\pi ^{2}/6}"> und Another simple proof of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1+1/2^{2}+1/3^{2}+\ldots =\pi ^{2}/6}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>+</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo>…</mo> <mo>=</mo> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>6</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1+1/2^{2}+1/3^{2}+\ldots =\pi ^{2}/6}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5c4c7870d1df228bdd6db634426c16d49059896" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.302ex; height:3.176ex;" alt="{\displaystyle 1+1/2^{2}+1/3^{2}+\ldots =\pi ^{2}/6}"><a href="#cite_note-4">[4]</a> aus dem Jahre 2003 behandelt. James Harper löste damit unter anderem das <a href="/wiki/Basler_Problem" title="Basler Problem">Basler Problem</a> und konnte einige weitere Integralidentitäten aufstellen, welche das Bindeglied zwischen den Arkusfunktionen und den Areafunktionen beziehungsweise Logarithmusfunktionen darstellen. Beispielsweise gilt folgendes Integral: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int \limits _{0}^{1}{\frac {1}{y}}\ln {\biggl (}{\frac {y^{2}+y+1}{y^{2}-y+1}}{\biggr )}\,\mathrm {d} y={\frac {\pi ^{2}}{6}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>y</mi> </mfrac> </mrow> <mi>ln</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>y</mi> <mo>+</mo> <mn>1</mn> </mrow> <mrow> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi>y</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>y</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mn>6</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int \limits _{0}^{1}{\frac {1}{y}}\ln {\biggl (}{\frac {y^{2}+y+1}{y^{2}-y+1}}{\biggr )}\,\mathrm {d} y={\frac {\pi ^{2}}{6}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6abe30fa42090fcca32242d20e1662bd3bab4ad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:30.922ex; height:9.176ex;" alt="{\displaystyle \int \limits _{0}^{1}{\frac {1}{y}}\ln {\biggl (}{\frac {y^{2}+y+1}{y^{2}-y+1}}{\biggr )}\,\mathrm {d} y={\frac {\pi ^{2}}{6}}}"></dd></dl> Eine analoge Integralidentität nach demselben Grundmuster kann für das Quadrat des Arkuskosinus hervorgebracht werden: <dl><dd><table class="wikitable"> <tbody><tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \arccos(x)^{2}={\frac {\pi ^{2}}{3}}-\int \limits _{0}^{1}{\frac {2}{y}}\ln(y^{2}+2xy+1)\,\mathrm {d} y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arccos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mn>3</mn> </mfrac> </mrow> <mo>−</mo> <munderover> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mi>y</mi> </mfrac> </mrow> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mi>x</mi> <mi>y</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \arccos(x)^{2}={\frac {\pi ^{2}}{3}}-\int \limits _{0}^{1}{\frac {2}{y}}\ln(y^{2}+2xy+1)\,\mathrm {d} y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c190a86430df1a6e6fe017f2f67e47a4409f61e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:43.83ex; height:9.176ex;" alt="{\displaystyle \arccos(x)^{2}={\frac {\pi ^{2}}{3}}-\int \limits _{0}^{1}{\frac {2}{y}}\ln(y^{2}+2xy+1)\,\mathrm {d} y}"> </td></tr></tbody></table></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Integralidentität_mit_dem_Areatangens_Hyperbolicus">Integralidentität mit dem Areatangens Hyperbolicus</h3>[<a href="/w/index.php?title=Arkussinus_und_Arkuskosinus&veaction=edit&section=12" title="Abschnitt bearbeiten: Integralidentität mit dem Areatangens Hyperbolicus" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Arkussinus_und_Arkuskosinus&action=edit&section=12" title="Quellcode des Abschnitts bearbeiten: Integralidentität mit dem Areatangens Hyperbolicus">Quelltext bearbeiten</a>]</div> Und mit dem <a href="/wiki/Areatangens_hyperbolicus_und_Areakotangens_hyperbolicus" title="Areatangens hyperbolicus und Areakotangens hyperbolicus">Areatangens Hyperbolicus</a> kann für den Arkussinus eine Integralidentität aufgestellt werden: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {2\arcsin(x)}{\sqrt {1-x^{2}}}}=\int \limits _{0}^{1}{\frac {2\,x}{\sqrt {(1-x^{2})(1-x^{2}y^{2})}}}\,\mathrm {d} y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>arcsin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> <mo>=</mo> <munderover> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mspace width="thinmathspace" /> <mi>x</mi> </mrow> <msqrt> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </msqrt> </mfrac> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {2\arcsin(x)}{\sqrt {1-x^{2}}}}=\int \limits _{0}^{1}{\frac {2\,x}{\sqrt {(1-x^{2})(1-x^{2}y^{2})}}}\,\mathrm {d} y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a4addd3bd317d1ef5b2c98b575f1eae0bee62b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:41.768ex; height:9.176ex;" alt="{\displaystyle {\frac {2\arcsin(x)}{\sqrt {1-x^{2}}}}=\int \limits _{0}^{1}{\frac {2\,x}{\sqrt {(1-x^{2})(1-x^{2}y^{2})}}}\,\mathrm {d} y}"></dd></dl> Durch Bildung der Ursprungsstammfunktion bezüglich <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" 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}"> entsteht folgende Formel: <dl><dd><table class="wikitable"> <tbody><tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \arcsin(x)^{2}=\int \limits _{0}^{1}{\frac {2}{y}}\left[\operatorname {artanh} {\bigl (}y{\bigr )}-\operatorname {artanh} \left({\frac {{\sqrt {1-x^{2}}}\,y}{\sqrt {1-x^{2}y^{2}}}}\right)\right]\,\mathrm {d} y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arcsin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <munderover> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mi>y</mi> </mfrac> </mrow> <mrow> <mo>[</mo> <mrow> <mi>artanh</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>−</mo> <mi>artanh</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mspace width="thinmathspace" /> <mi>y</mi> </mrow> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <mo>]</mo> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \arcsin(x)^{2}=\int \limits _{0}^{1}{\frac {2}{y}}\left[\operatorname {artanh} {\bigl (}y{\bigr )}-\operatorname {artanh} \left({\frac {{\sqrt {1-x^{2}}}\,y}{\sqrt {1-x^{2}y^{2}}}}\right)\right]\,\mathrm {d} y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2966539dd29cb3a2df95b97630e49981774f28c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:59.561ex; height:9.176ex;" alt="{\displaystyle \arcsin(x)^{2}=\int \limits _{0}^{1}{\frac {2}{y}}\left[\operatorname {artanh} {\bigl (}y{\bigr )}-\operatorname {artanh} \left({\frac {{\sqrt {1-x^{2}}}\,y}{\sqrt {1-x^{2}y^{2}}}}\right)\right]\,\mathrm {d} y}"> </td></tr></tbody></table></dd></dl> Wenn der Grenzwert von dieser Identität für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee42176e76ae6b56d68c42ced807e08b962a2b54" 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=1}"> berechnet wird, dann entsteht für dieses Integral über den <a href="/wiki/Areatangens_hyperbolicus_und_Areakotangens_hyperbolicus" title="Areatangens hyperbolicus und Areakotangens hyperbolicus">Areatangens Hyperbolicus</a> folgende Identität: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int \limits _{0}^{1}{\frac {1}{y}}\,\operatorname {artanh} (y)\,\mathrm {d} y={\frac {\pi ^{2}}{8}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>y</mi> </mfrac> </mrow> <mspace width="thinmathspace" /> <mi>artanh</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>y</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mn>8</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int \limits _{0}^{1}{\frac {1}{y}}\,\operatorname {artanh} (y)\,\mathrm {d} y={\frac {\pi ^{2}}{8}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fcba6cffa9e2d32ac780841c2d6c67a6a7186bac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:24.203ex; height:9.176ex;" alt="{\displaystyle \int \limits _{0}^{1}{\frac {1}{y}}\,\operatorname {artanh} (y)\,\mathrm {d} y={\frac {\pi ^{2}}{8}}}"></dd></dl> Und mit dieser Formel kann das <a href="/wiki/Basler_Problem" title="Basler Problem">Basler Problem</a> bewiesen werden: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int \limits _{0}^{1}{\frac {1}{y}}\,\operatorname {artanh} (y)\,\mathrm {d} y=\int \limits _{0}^{1}{\biggl (}{\frac {1}{y}}\sum _{n=1}^{\infty }{\frac {1}{2n-1}}\,y^{2n-1}{\biggr )}\,\mathrm {d} y=\sum _{n=1}^{\infty }{\frac {1}{(2n-1)^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>y</mi> </mfrac> </mrow> <mspace width="thinmathspace" /> <mi>artanh</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>y</mi> <mo>=</mo> <munderover> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>y</mi> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>y</mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int \limits _{0}^{1}{\frac {1}{y}}\,\operatorname {artanh} (y)\,\mathrm {d} y=\int \limits _{0}^{1}{\biggl (}{\frac {1}{y}}\sum _{n=1}^{\infty }{\frac {1}{2n-1}}\,y^{2n-1}{\biggr )}\,\mathrm {d} y=\sum _{n=1}^{\infty }{\frac {1}{(2n-1)^{2}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/217de34e3ff70ac91e93dba8cee8abebf63ba48b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:66.128ex; height:9.176ex;" alt="{\displaystyle \int \limits _{0}^{1}{\frac {1}{y}}\,\operatorname {artanh} (y)\,\mathrm {d} y=\int \limits _{0}^{1}{\biggl (}{\frac {1}{y}}\sum _{n=1}^{\infty }{\frac {1}{2n-1}}\,y^{2n-1}{\biggr )}\,\mathrm {d} y=\sum _{n=1}^{\infty }{\frac {1}{(2n-1)^{2}}}}"></dd></dl> Daraus folgt: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{n=1}^{\infty }{\frac {1}{(2n-1)^{2}}}={\frac {\pi ^{2}}{8}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mn>8</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{n=1}^{\infty }{\frac {1}{(2n-1)^{2}}}={\frac {\pi ^{2}}{8}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/031afbb5062fa1b519c8b4ea7a6113a82cb974eb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:20.325ex; height:6.843ex;" alt="{\displaystyle \sum _{n=1}^{\infty }{\frac {1}{(2n-1)^{2}}}={\frac {\pi ^{2}}{8}}}"></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Stammfunktionen_von_Arkussinus_und_Arkuskosinus">Stammfunktionen von Arkussinus und Arkuskosinus</h3>[<a href="/w/index.php?title=Arkussinus_und_Arkuskosinus&veaction=edit&section=13" title="Abschnitt bearbeiten: Stammfunktionen von Arkussinus und Arkuskosinus" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Arkussinus_und_Arkuskosinus&action=edit&section=13" title="Quellcode des Abschnitts bearbeiten: Stammfunktionen von Arkussinus und Arkuskosinus">Quelltext bearbeiten</a>]</div> <dl><dt>Arkussinus</dt> <dd></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int \arcsin \left({\frac {x}{a}}\right)\,\mathrm {d} x=x\,\arcsin \left({\frac {x}{a}}\right)+{\sqrt {a^{2}-x^{2}}}+C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∫</mo> <mi>arcsin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mi>a</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> <mo>=</mo> <mi>x</mi> <mspace width="thinmathspace" /> <mi>arcsin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mi>a</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>+</mo> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int \arcsin \left({\frac {x}{a}}\right)\,\mathrm {d} x=x\,\arcsin \left({\frac {x}{a}}\right)+{\sqrt {a^{2}-x^{2}}}+C}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c85556ffb7a3de9ebfbbb32b6d70465562deedf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:49.878ex; height:5.676ex;" alt="{\displaystyle \int \arcsin \left({\frac {x}{a}}\right)\,\mathrm {d} x=x\,\arcsin \left({\frac {x}{a}}\right)+{\sqrt {a^{2}-x^{2}}}+C}"></dd></dl> <dl><dt>Arkuskosinus</dt> <dd></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int \arccos \left({\frac {x}{a}}\right)\,\mathrm {d} x=x\,\arccos \left({\frac {x}{a}}\right)-{\sqrt {a^{2}-x^{2}}}+C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∫</mo> <mi>arccos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mi>a</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> <mo>=</mo> <mi>x</mi> <mspace width="thinmathspace" /> <mi>arccos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mi>a</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>+</mo> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int \arccos \left({\frac {x}{a}}\right)\,\mathrm {d} x=x\,\arccos \left({\frac {x}{a}}\right)-{\sqrt {a^{2}-x^{2}}}+C}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/49065807d215ee4af1a4a10cb441446af27dee3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:50.389ex; height:5.676ex;" alt="{\displaystyle \int \arccos \left({\frac {x}{a}}\right)\,\mathrm {d} x=x\,\arccos \left({\frac {x}{a}}\right)-{\sqrt {a^{2}-x^{2}}}+C}"></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Stammfunktion_des_kardinalisierten_Arkussinus">Stammfunktion des kardinalisierten Arkussinus</h3>[<a href="/w/index.php?title=Arkussinus_und_Arkuskosinus&veaction=edit&section=14" title="Abschnitt bearbeiten: Stammfunktion des kardinalisierten Arkussinus" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Arkussinus_und_Arkuskosinus&action=edit&section=14" title="Quellcode des Abschnitts bearbeiten: Stammfunktion des kardinalisierten Arkussinus">Quelltext bearbeiten</a>]</div> Wenn der Arkussinus durch die identische Abbildungsfunktion geteilt wird, dann stellt diese Funktion den kardinalisierten Arkussinus dar. Die ursprüngliche Stammfunktion des kardinalisierten Arkussinus ist das sogenannte <a href="/wiki/Arkustangensintegral#Arkussinusintegral" title="Arkustangensintegral">Arkussinusintegral</a> und dies ist eine nicht elemenare Funktion: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int \limits _{0}^{x}{\frac {1}{y}}\arcsin(y)\,\mathrm {d} y=\int _{0}^{1}{\frac {1}{z}}\arcsin(xz)\,\mathrm {d} z=\operatorname {Si} _{2}(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>y</mi> </mfrac> </mrow> <mi>arcsin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>y</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>z</mi> </mfrac> </mrow> <mi>arcsin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mi>z</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>z</mi> <mo>=</mo> <msub> <mi>Si</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int \limits _{0}^{x}{\frac {1}{y}}\arcsin(y)\,\mathrm {d} y=\int _{0}^{1}{\frac {1}{z}}\arcsin(xz)\,\mathrm {d} z=\operatorname {Si} _{2}(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4480024237aa115196ad81f70fe8c9224c3597a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:48.306ex; height:8.843ex;" alt="{\displaystyle \int \limits _{0}^{x}{\frac {1}{y}}\arcsin(y)\,\mathrm {d} y=\int _{0}^{1}{\frac {1}{z}}\arcsin(xz)\,\mathrm {d} z=\operatorname {Si} _{2}(x)}"></dd></dl> Nach dem <a href="/wiki/Fundamentalsatz_der_Analysis" title="Fundamentalsatz der Analysis">Fundamentalsatz der Infinitesimalrechnung</a> gilt somit: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\operatorname {Si} _{2}(x)={\frac {1}{x}}\arcsin(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> </mrow> </mfrac> </mrow> <msub> <mi>Si</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> </mrow> <mi>arcsin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\operatorname {Si} _{2}(x)={\frac {1}{x}}\arcsin(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6af48d2436b40fedf84aa0b77e68498f3cb392e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:24.73ex; height:5.509ex;" alt="{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\operatorname {Si} _{2}(x)={\frac {1}{x}}\arcsin(x)}">das bekannteste Beispiel für einen Wert dieser Stammfunktion:</dd></dl> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int \limits _{0}^{1}{\frac {1}{x}}\arcsin(x)\,\mathrm {d} x=\operatorname {Si} _{2}(1)={\frac {\pi }{2}}\ln(2)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> </mrow> <mi>arcsin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> <mo>=</mo> <msub> <mi>Si</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int \limits _{0}^{1}{\frac {1}{x}}\arcsin(x)\,\mathrm {d} x=\operatorname {Si} _{2}(1)={\frac {\pi }{2}}\ln(2)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f800d083b87b69053afd548054b315348748508a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:36.873ex; height:9.176ex;" alt="{\displaystyle \int \limits _{0}^{1}{\frac {1}{x}}\arcsin(x)\,\mathrm {d} x=\operatorname {Si} _{2}(1)={\frac {\pi }{2}}\ln(2)}"></dd></dl> Mit dem <a href="/wiki/Satz_von_Fubini" title="Satz von Fubini">Satz von Fubini</a> kann der nun genannte Wert des Integrals bewiesen werden: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Si} _{2}(1)=\int \limits _{0}^{1}{\frac {1}{x}}\arcsin(x)\,\mathrm {d} x=\int \limits _{0}^{1}\int \limits _{0}^{1}{\frac {{\sqrt {1-x^{2}}}\,y}{(1-x^{2}y^{2}){\sqrt {1-y^{2}}}}}\,\mathrm {d} y\,\mathrm {d} x=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Si</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> </mrow> <mi>arcsin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> <mo>=</mo> <munderover> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </munderover> <munderover> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mspace width="thinmathspace" /> <mi>y</mi> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>y</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Si} _{2}(1)=\int \limits _{0}^{1}{\frac {1}{x}}\arcsin(x)\,\mathrm {d} x=\int \limits _{0}^{1}\int \limits _{0}^{1}{\frac {{\sqrt {1-x^{2}}}\,y}{(1-x^{2}y^{2}){\sqrt {1-y^{2}}}}}\,\mathrm {d} y\,\mathrm {d} x=}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/49e0a316ab17a3de2c6563e5d81be4544d126405" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:62.654ex; height:9.176ex;" alt="{\displaystyle \operatorname {Si} _{2}(1)=\int \limits _{0}^{1}{\frac {1}{x}}\arcsin(x)\,\mathrm {d} x=\int \limits _{0}^{1}\int \limits _{0}^{1}{\frac {{\sqrt {1-x^{2}}}\,y}{(1-x^{2}y^{2}){\sqrt {1-y^{2}}}}}\,\mathrm {d} y\,\mathrm {d} x=}"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle =\int \limits _{0}^{1}\int \limits _{0}^{1}{\frac {{\sqrt {1-x^{2}}}\,y}{(1-x^{2}y^{2}){\sqrt {1-y^{2}}}}}\,\mathrm {d} x\,\mathrm {d} y=\int \limits _{0}^{1}{\frac {\pi \,y}{2{\sqrt {1-y^{2}}}(1+{\sqrt {1-y^{2}}}\,)}}\,\mathrm {d} y={\frac {\pi }{2}}\ln(2)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <munderover> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </munderover> <munderover> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mspace width="thinmathspace" /> <mi>y</mi> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>y</mi> <mo>=</mo> <munderover> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>π</mi> <mspace width="thinmathspace" /> <mi>y</mi> </mrow> <mrow> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mspace width="thinmathspace" /> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>y</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =\int \limits _{0}^{1}\int \limits _{0}^{1}{\frac {{\sqrt {1-x^{2}}}\,y}{(1-x^{2}y^{2}){\sqrt {1-y^{2}}}}}\,\mathrm {d} x\,\mathrm {d} y=\int \limits _{0}^{1}{\frac {\pi \,y}{2{\sqrt {1-y^{2}}}(1+{\sqrt {1-y^{2}}}\,)}}\,\mathrm {d} y={\frac {\pi }{2}}\ln(2)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b191b8f6ff22001d56edbb3d90c9f5b8c720a6a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:77.607ex; height:9.176ex;" alt="{\displaystyle =\int \limits _{0}^{1}\int \limits _{0}^{1}{\frac {{\sqrt {1-x^{2}}}\,y}{(1-x^{2}y^{2}){\sqrt {1-y^{2}}}}}\,\mathrm {d} x\,\mathrm {d} y=\int \limits _{0}^{1}{\frac {\pi \,y}{2{\sqrt {1-y^{2}}}(1+{\sqrt {1-y^{2}}}\,)}}\,\mathrm {d} y={\frac {\pi }{2}}\ln(2)}"></dd></dl> Mit diesem Arkussinusintegral kann ebenso das sogenannte Arkustangensintegral direkt erzeugt werden: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\,\mathrm {Ti} _{2}{\bigl [}x(1+{\sqrt {1-x^{2}}})^{-1}{\bigr ]}=4\,\mathrm {Si} _{2}{\bigl (}{\tfrac {1}{2}}{\sqrt {1+x}}-{\tfrac {1}{2}}{\sqrt {1-x}}\,{\bigr )}-\mathrm {Si} _{2}(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mspace width="thinmathspace" /> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> <mi mathvariant="normal">i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">[</mo> </mrow> </mrow> <mi>x</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">]</mo> </mrow> </mrow> <mo>=</mo> <mn>4</mn> <mspace width="thinmathspace" /> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">S</mi> <mi mathvariant="normal">i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>+</mo> <mi>x</mi> </msqrt> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <mi>x</mi> </msqrt> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">S</mi> <mi mathvariant="normal">i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\,\mathrm {Ti} _{2}{\bigl [}x(1+{\sqrt {1-x^{2}}})^{-1}{\bigr ]}=4\,\mathrm {Si} _{2}{\bigl (}{\tfrac {1}{2}}{\sqrt {1+x}}-{\tfrac {1}{2}}{\sqrt {1-x}}\,{\bigr )}-\mathrm {Si} _{2}(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/293a660717b808825cd8677f29743e2f53f272de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:64.878ex; height:3.843ex;" alt="{\displaystyle 2\,\mathrm {Ti} _{2}{\bigl [}x(1+{\sqrt {1-x^{2}}})^{-1}{\bigr ]}=4\,\mathrm {Si} _{2}{\bigl (}{\tfrac {1}{2}}{\sqrt {1+x}}-{\tfrac {1}{2}}{\sqrt {1-x}}\,{\bigr )}-\mathrm {Si} _{2}(x)}"></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Komplexe_Argumente">Komplexe Argumente</h2>[<a href="/w/index.php?title=Arkussinus_und_Arkuskosinus&veaction=edit&section=15" title="Abschnitt bearbeiten: Komplexe Argumente" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Arkussinus_und_Arkuskosinus&action=edit&section=15" title="Quellcode des Abschnitts bearbeiten: Komplexe Argumente">Quelltext bearbeiten</a>]</div> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\arcsin(a+b\,\mathrm {i} )=\quad {\frac {\operatorname {sgn^{+}} {a}}{2}}\cdot \arccos &\left({\sqrt {(a^{2}+b^{2}-1)^{2}+4b^{2}}}-(a^{2}+b^{2})\right)\\+\;\mathrm {i} \cdot {\frac {\operatorname {sgn^{+}} {b}}{2}}\cdot \operatorname {arcosh} &\left({\sqrt {(a^{2}+b^{2}-1)^{2}+4b^{2}}}+(a^{2}+b^{2})\right)\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>arcsin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">g</mi> <msup> <mi mathvariant="normal">n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> </mrow> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>⋅</mo> <mi>arccos</mi> </mtd> <mtd> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo stretchy="false">(</mo> <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> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>4</mn> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>−</mo> <mo stretchy="false">(</mo> <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 stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>+</mo> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">g</mi> <msup> <mi mathvariant="normal">n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> </mrow> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>⋅</mo> <mi>arcosh</mi> </mtd> <mtd> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo stretchy="false">(</mo> <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> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>4</mn> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>+</mo> <mo stretchy="false">(</mo> <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 stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\arcsin(a+b\,\mathrm {i} )=\quad {\frac {\operatorname {sgn^{+}} {a}}{2}}\cdot \arccos &\left({\sqrt {(a^{2}+b^{2}-1)^{2}+4b^{2}}}-(a^{2}+b^{2})\right)\\+\;\mathrm {i} \cdot {\frac {\operatorname {sgn^{+}} {b}}{2}}\cdot \operatorname {arcosh} &\left({\sqrt {(a^{2}+b^{2}-1)^{2}+4b^{2}}}+(a^{2}+b^{2})\right)\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4bb41cacf2c83b1c43ee1ddcf83ee73158247a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.171ex; width:72.3ex; height:11.509ex;" alt="{\displaystyle {\begin{aligned}\arcsin(a+b\,\mathrm {i} )=\quad {\frac {\operatorname {sgn^{+}} {a}}{2}}\cdot \arccos &\left({\sqrt {(a^{2}+b^{2}-1)^{2}+4b^{2}}}-(a^{2}+b^{2})\right)\\+\;\mathrm {i} \cdot {\frac {\operatorname {sgn^{+}} {b}}{2}}\cdot \operatorname {arcosh} &\left({\sqrt {(a^{2}+b^{2}-1)^{2}+4b^{2}}}+(a^{2}+b^{2})\right)\end{aligned}}}">   mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a,b\in \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a,b\in \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe0bf0e2324bac843fab916c46d1b3349017a616" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.78ex; height:2.509ex;" alt="{\displaystyle a,b\in \mathbb {R} }"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \arccos(a+b\,\mathrm {i} )={\frac {\pi }{2}}-\arcsin(a+b\,\mathrm {i} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arccos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <mi>arcsin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \arccos(a+b\,\mathrm {i} )={\frac {\pi }{2}}-\arcsin(a+b\,\mathrm {i} )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ea1456dfd6b3c1dec429550e1f4ec1b56a03712" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:36.109ex; height:4.676ex;" alt="{\displaystyle \arccos(a+b\,\mathrm {i} )={\frac {\pi }{2}}-\arcsin(a+b\,\mathrm {i} )}"></dd></dl> Zur Funktion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {arcosh} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arcosh</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {arcosh} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af6b5863b04ea834ec696ad51de830ba60c9f268" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.478ex; height:2.176ex;" alt="{\displaystyle \operatorname {arcosh} }"> siehe <a href="/wiki/Areakosinus_hyperbolicus" class="mw-redirect" title="Areakosinus hyperbolicus">Areakosinus hyperbolicus</a>, und für die Funktion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {sgn^{+}} \colon \mathbb {R} \to \{-1,1\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">g</mi> <msup> <mi mathvariant="normal">n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> </mrow> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">→</mo> <mo fence="false" stretchy="false">{</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {sgn^{+}} \colon \mathbb {R} \to \{-1,1\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b39fd6120ca34621021705fce2c5b006bd9a571b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.7ex; height:3.009ex;" alt="{\displaystyle \operatorname {sgn^{+}} \colon \mathbb {R} \to \{-1,1\}}"> gilt <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {sgn^{+}} (x):=2\cdot \Theta (x)-1={\begin{cases}+1&{\text{für }}x\geq 0\\-1&{\text{für }}x<0\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">g</mi> <msup> <mi mathvariant="normal">n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> </mrow> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>:=</mo> <mn>2</mn> <mo>⋅</mo> <mi mathvariant="normal">Θ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mn>1</mn> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mo>+</mo> <mn>1</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>für </mtext> </mrow> <mi>x</mi> <mo>≥</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>für </mtext> </mrow> <mi>x</mi> <mo><</mo> <mn>0</mn> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {sgn^{+}} (x):=2\cdot \Theta (x)-1={\begin{cases}+1&{\text{für }}x\geq 0\\-1&{\text{für }}x<0\end{cases}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d99bd3ec29ed3579d0a7c611967a48fd3a8e0a3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:43.786ex; height:7.509ex;" alt="{\displaystyle \operatorname {sgn^{+}} (x):=2\cdot \Theta (x)-1={\begin{cases}+1&{\text{für }}x\geq 0\\-1&{\text{für }}x<0\end{cases}}}"></dd></dl> mit der <a href="/wiki/Heaviside-Funktion" title="Heaviside-Funktion">Heaviside-Funktion</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc927b19f46d005b4720db7a0f96cd5b6f1a0d9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \Theta }">. <div class="mw-heading mw-heading2"><h2 id="Anmerkungen">Anmerkungen</h2>[<a href="/w/index.php?title=Arkussinus_und_Arkuskosinus&veaction=edit&section=16" title="Abschnitt bearbeiten: Anmerkungen" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Arkussinus_und_Arkuskosinus&action=edit&section=16" title="Quellcode des Abschnitts bearbeiten: Anmerkungen">Quelltext bearbeiten</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Wichtige_Funktionswerte">Wichtige Funktionswerte</h3>[<a href="/w/index.php?title=Arkussinus_und_Arkuskosinus&veaction=edit&section=17" title="Abschnitt bearbeiten: Wichtige Funktionswerte" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Arkussinus_und_Arkuskosinus&action=edit&section=17" title="Quellcode des Abschnitts bearbeiten: Wichtige Funktionswerte">Quelltext bearbeiten</a>]</div> Siehe auch: <a href="/wiki/Sinus_und_Kosinus#Wichtigste_Funktionswerte" title="Sinus und Kosinus">Sinus und Kosinus: Wichtige Funktionswerte</a> Die folgende Tabelle listet die wichtigen Funktionswerte der beiden <a href="/wiki/Arkusfunktion" title="Arkusfunktion">Arkusfunktionen</a> auf.<a href="#cite_note-5">[5]</a> <table class="wikitable"> <tbody><tr> <th><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" 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}"> </th> <th colspan="2"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \arcsin(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arcsin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \arcsin(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f111f8c1e5de2f92ee17eeabd64c4ea1bcd55196" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.101ex; height:2.843ex;" alt="{\displaystyle \arcsin(x)}"> </th> <th colspan="2"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \arccos(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arccos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \arccos(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3baa15507445b7175a6143e9d16ba98a9849374c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.357ex; height:2.843ex;" alt="{\displaystyle \arccos(x)}"> </th></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0^{\circ }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0^{\circ }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd0e1e92cf5770c2bfbb1de8b4b7bf904c9deef" 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 }}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 90^{\circ }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>90</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 90^{\circ }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c326d317eddef3ad3e6625e018a708e290a039f6" 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 }}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\pi }{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\pi }{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98f98bef5d4981ff6e2aa827d4699e347fb30db2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:2.168ex; height:4.676ex;" alt="{\displaystyle {\frac {\pi }{2}}}"> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a11cfb2fdb143693b1daf78fcb5c11a023cb1c55" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:1.999ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{2}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 30^{\circ }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>30</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 30^{\circ }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/42f29df9c101d2a8dae3f1552342cfe4c3adb76c" 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 }}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\pi }{6}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>6</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\pi }{6}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da430867901fd359c000b52f2bd70b36cf5e2182" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:2.168ex; height:4.676ex;" alt="{\displaystyle {\frac {\pi }{6}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 60^{\circ }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>60</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 60^{\circ }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08c42292485b447b7f627a7accd90d5b439c11d0" 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 }}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\pi }{3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>3</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\pi }{3}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c83c684a603005cda4feb8eea0254143ffb0e16" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:2.168ex; height:4.676ex;" alt="{\displaystyle {\frac {\pi }{3}}}"> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{2}}{\sqrt {2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{2}}{\sqrt {2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ff1b44c4decd955dc0ab5b32dd2c83fbe33583a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:5.097ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{2}}{\sqrt {2}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 45^{\circ }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>45</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 45^{\circ }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c28223ddedeb94a84bb15474cc64b5ce436cbe50" 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 }}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\pi }{4}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\pi }{4}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f89d7c88c1c93dce69a46052a8e276e231063de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:2.168ex; height:4.676ex;" alt="{\displaystyle {\frac {\pi }{4}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 45^{\circ }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>45</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 45^{\circ }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c28223ddedeb94a84bb15474cc64b5ce436cbe50" 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 }}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\pi }{4}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\pi }{4}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f89d7c88c1c93dce69a46052a8e276e231063de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:2.168ex; height:4.676ex;" alt="{\displaystyle {\frac {\pi }{4}}}"> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{2}}{\sqrt {3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{2}}{\sqrt {3}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f98f16503bd3aba49a297c4057511ace0ae2c95f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:5.097ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{2}}{\sqrt {3}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 60^{\circ }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>60</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 60^{\circ }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08c42292485b447b7f627a7accd90d5b439c11d0" 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 }}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\pi }{3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>3</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\pi }{3}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c83c684a603005cda4feb8eea0254143ffb0e16" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:2.168ex; height:4.676ex;" alt="{\displaystyle {\frac {\pi }{3}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 30^{\circ }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>30</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 30^{\circ }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/42f29df9c101d2a8dae3f1552342cfe4c3adb76c" 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 }}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\pi }{6}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>6</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\pi }{6}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da430867901fd359c000b52f2bd70b36cf5e2182" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:2.168ex; height:4.676ex;" alt="{\displaystyle {\frac {\pi }{6}}}"> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 90^{\circ }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>90</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 90^{\circ }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c326d317eddef3ad3e6625e018a708e290a039f6" 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 }}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\pi }{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\pi }{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98f98bef5d4981ff6e2aa827d4699e347fb30db2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:2.168ex; height:4.676ex;" alt="{\displaystyle {\frac {\pi }{2}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0^{\circ }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0^{\circ }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd0e1e92cf5770c2bfbb1de8b4b7bf904c9deef" 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 }}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}"> </td></tr> </tbody></table> Weitere wichtige Werte sind: <table class="wikitable"> <tbody><tr> <th><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" 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}"> </th> <th colspan="2"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \arcsin(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arcsin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \arcsin(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f111f8c1e5de2f92ee17eeabd64c4ea1bcd55196" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.101ex; height:2.843ex;" alt="{\displaystyle \arcsin(x)}"> </th> <th colspan="2"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \arccos(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arccos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \arccos(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3baa15507445b7175a6143e9d16ba98a9849374c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.357ex; height:2.843ex;" alt="{\displaystyle \arccos(x)}"> </th></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {1}{4}}({\sqrt {6}}-{\sqrt {2}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mstyle> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>6</mn> </msqrt> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{4}}({\sqrt {6}}-{\sqrt {2}})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/94170142bb0318c3f62a3de4c838ab5e6117e1d7" 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}})}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 15^{\circ }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>15</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 15^{\circ }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6b2ba390041accdbd09de910f720bd6e873d137" 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 }}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {\pi }{12}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>π</mi> <mn>12</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {\pi }{12}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c4003cb335c111eff28d24f990d71ce744bdbce2" 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}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 75^{\circ }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>75</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 75^{\circ }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/565c6cb6b28eeab72c86cdb73940a1ad0d2d1d26" 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 }}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {5\pi }{12}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>5</mn> <mi>π</mi> </mrow> <mn>12</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {5\pi }{12}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/400cec5f3cfa4c977c82ec1002e208ea770c244b" 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}}}"> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {1}{4}}\left({\sqrt {5}}-1\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mstyle> </mrow> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{4}}\left({\sqrt {5}}-1\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac14cb52b40374b7bdc630767d704a3cb65e7e69" 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)}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 18^{\circ }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>18</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 18^{\circ }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16f8ae67f2d56e4095fdf0d10ebc72183313de82" 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 }}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {\pi }{10}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>π</mi> <mn>10</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {\pi }{10}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d4412f525ba517b7c31c7056cfa691f4162e09" 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}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 72^{\circ }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>72</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 72^{\circ }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0cfa1907f003a92fb16d6a32aa7a4f5da99f112" 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 }}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {2\pi }{5}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> </mrow> <mn>5</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {2\pi }{5}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5508e6906bf83215949906383498fa5b3402b730" 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}}}"> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {1}{4}}{\sqrt {10-2{\sqrt {5}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>10</mn> <mo>−</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </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="https://wikimedia.org/api/rest_v1/media/math/render/svg/34eaf04b764379d9da2dd47604f6d40b1e220cef" 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}}}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 36^{\circ }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>36</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 36^{\circ }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34fb8091def5cc9caece92b514b07fa743cf2fc3" 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 }}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {\pi }{5}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>π</mi> <mn>5</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {\pi }{5}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3dc8ac67580c4e2439889672d4b14d9b5ad67720" 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}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 54^{\circ }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>54</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 54^{\circ }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47c792d186c1239c4e574044a944723cb58dba48" 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 }}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {3\pi }{10}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>3</mn> <mi>π</mi> </mrow> <mn>10</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {3\pi }{10}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32545c8405bd937079e45f70e79fcc5eebff2c6e" 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}}}"> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {1}{4}}\left(1+{\sqrt {5}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mstyle> </mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{4}}\left(1+{\sqrt {5}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a55716e6401a3f1e5d5b60c5e224246659a869b" 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)}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 54^{\circ }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>54</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 54^{\circ }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47c792d186c1239c4e574044a944723cb58dba48" 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 }}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {3\pi }{10}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>3</mn> <mi>π</mi> </mrow> <mn>10</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {3\pi }{10}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32545c8405bd937079e45f70e79fcc5eebff2c6e" 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}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 36^{\circ }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>36</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 36^{\circ }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34fb8091def5cc9caece92b514b07fa743cf2fc3" 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 }}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {\pi }{5}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>π</mi> <mn>5</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {\pi }{5}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3dc8ac67580c4e2439889672d4b14d9b5ad67720" 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}}}"> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {1}{4}}{\sqrt {10+2{\sqrt {5}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>10</mn> <mo>+</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </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="https://wikimedia.org/api/rest_v1/media/math/render/svg/25396f9653504beff444cf52b5b217ff3a1a5ae2" 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}}}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 72^{\circ }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>72</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 72^{\circ }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0cfa1907f003a92fb16d6a32aa7a4f5da99f112" 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 }}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {2\pi }{5}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> </mrow> <mn>5</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {2\pi }{5}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5508e6906bf83215949906383498fa5b3402b730" 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}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 18^{\circ }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>18</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 18^{\circ }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16f8ae67f2d56e4095fdf0d10ebc72183313de82" 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 }}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {\pi }{10}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>π</mi> <mn>10</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {\pi }{10}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d4412f525ba517b7c31c7056cfa691f4162e09" 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}}}"> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {1}{4}}({\sqrt {6}}+{\sqrt {2}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mstyle> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>6</mn> </msqrt> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{4}}({\sqrt {6}}+{\sqrt {2}})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a456da8059e2a0e125bb309b52724ec105c921c" 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}})}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 75^{\circ }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>75</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 75^{\circ }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/565c6cb6b28eeab72c86cdb73940a1ad0d2d1d26" 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 }}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {5\pi }{12}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>5</mn> <mi>π</mi> </mrow> <mn>12</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {5\pi }{12}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/400cec5f3cfa4c977c82ec1002e208ea770c244b" 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}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 15^{\circ }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>15</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 15^{\circ }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6b2ba390041accdbd09de910f720bd6e873d137" 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 }}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {\pi }{12}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>π</mi> <mn>12</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {\pi }{12}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c4003cb335c111eff28d24f990d71ce744bdbce2" 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}}}"> </td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="Kettenbruchdarstellung_des_Arkussinus">Kettenbruchdarstellung des Arkussinus</h3>[<a href="/w/index.php?title=Arkussinus_und_Arkuskosinus&veaction=edit&section=18" title="Abschnitt bearbeiten: Kettenbruchdarstellung des Arkussinus" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Arkussinus_und_Arkuskosinus&action=edit&section=18" title="Quellcode des Abschnitts bearbeiten: Kettenbruchdarstellung des Arkussinus">Quelltext bearbeiten</a>]</div> H. S. Wall fand 1948 für den Arkussinus folgende Darstellung als <a href="/wiki/Kettenbruch" title="Kettenbruch">Kettenbruch</a>: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \arcsin(x)={\frac {x{\sqrt {1-x^{2}}}}{1-{\cfrac {1\cdot 2x^{2}}{3-{\cfrac {1\cdot 2x^{2}}{5-{\cfrac {3\cdot 4x^{2}}{7-{\cfrac {3\cdot 4x^{2}}{9-{\cfrac {5\cdot 6x^{2}}{11-\ldots }}}}}}}}}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arcsin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>⋅</mo> <mn>2</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>⋅</mo> <mn>2</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mo>⋅</mo> <mn>4</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mo>⋅</mo> <mn>4</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>9</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> <mo>⋅</mo> <mn>6</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> <mo>−</mo> <mo>…</mo> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \arcsin(x)={\frac {x{\sqrt {1-x^{2}}}}{1-{\cfrac {1\cdot 2x^{2}}{3-{\cfrac {1\cdot 2x^{2}}{5-{\cfrac {3\cdot 4x^{2}}{7-{\cfrac {3\cdot 4x^{2}}{9-{\cfrac {5\cdot 6x^{2}}{11-\ldots }}}}}}}}}}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fff634a6e41f4ad4c7d95e35e7413da9ca71513" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -21.005ex; width:45.119ex; height:25.343ex;" alt="{\displaystyle \arcsin(x)={\frac {x{\sqrt {1-x^{2}}}}{1-{\cfrac {1\cdot 2x^{2}}{3-{\cfrac {1\cdot 2x^{2}}{5-{\cfrac {3\cdot 4x^{2}}{7-{\cfrac {3\cdot 4x^{2}}{9-{\cfrac {5\cdot 6x^{2}}{11-\ldots }}}}}}}}}}}}}"></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Komplexe_Funktion">Komplexe Funktion</h3>[<a href="/w/index.php?title=Arkussinus_und_Arkuskosinus&veaction=edit&section=19" title="Abschnitt bearbeiten: Komplexe Funktion" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Arkussinus_und_Arkuskosinus&action=edit&section=19" title="Quellcode des Abschnitts bearbeiten: Komplexe Funktion">Quelltext bearbeiten</a>]</div> Man kann Arkussinus und Arkuskosinus auch durch den Hauptzweig des komplexen <a href="/wiki/Logarithmus" title="Logarithmus">Logarithmus</a> ausdrücken: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \arcsin z=-\mathrm {i} \,\ln \left(\mathrm {i} z+{\sqrt {1-z^{2}}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arcsin</mi> <mo>⁡</mo> <mi>z</mi> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mspace width="thinmathspace" /> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mi>z</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \arcsin z=-\mathrm {i} \,\ln \left(\mathrm {i} z+{\sqrt {1-z^{2}}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b01406aebcac59aa303257e99636bb1dd7844df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:31.526ex; height:4.843ex;" alt="{\displaystyle \arcsin z=-\mathrm {i} \,\ln \left(\mathrm {i} z+{\sqrt {1-z^{2}}}\right)}"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \arccos z=-\mathrm {i} \,\ln \left(z+\mathrm {i} {\sqrt {1-z^{2}}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arccos</mi> <mo>⁡</mo> <mi>z</mi> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mspace width="thinmathspace" /> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>z</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \arccos z=-\mathrm {i} \,\ln \left(z+\mathrm {i} {\sqrt {1-z^{2}}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/760d0b082a320c8f4c98c0e40e65f25a7f27aff8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:31.782ex; height:4.843ex;" alt="{\displaystyle \arccos z=-\mathrm {i} \,\ln \left(z+\mathrm {i} {\sqrt {1-z^{2}}}\right)}"></dd></dl> Diese beiden Formeln kann man wie folgt herleiten: Für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \arcsin z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arcsin</mi> <mo>⁡</mo> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \arcsin z}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/401a712ab3ed004dd2e2e9152634f0ac515c9c89" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.437ex; height:2.176ex;" alt="{\displaystyle \arcsin z}">: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\sin(x)&={\frac {\mathrm {e} ^{\mathrm {i} x}-\mathrm {e} ^{-\mathrm {i} x}}{2\mathrm {i} }}\\{\frac {\mathrm {e} ^{\mathrm {i} x}-\mathrm {e} ^{-\mathrm {i} x}}{2\mathrm {i} }}&=z\\\mathrm {e} ^{\mathrm {i} x}-{\frac {1}{\mathrm {e} ^{\mathrm {i} x}}}&=2z\mathrm {i} \\(\mathrm {e} ^{\mathrm {i} x})^{2}-1&=2z\mathrm {i} \mathrm {e} ^{\mathrm {i} x}\\(\mathrm {e} ^{\mathrm {i} x})^{2}-2z\mathrm {i} \mathrm {e} ^{\mathrm {i} x}-1&=0\\\mathrm {e} ^{\mathrm {i} x}&=-{\frac {-2z\mathrm {i} }{2}}\pm {\sqrt {\left({\frac {-2z\mathrm {i} }{2}}\right)^{2}-(-1)}}\\\mathrm {e} ^{\mathrm {i} x}&=z\mathrm {i} \pm {\sqrt {1-z^{2}}}\\\mathrm {i} x&=\ln(z\mathrm {i} \pm {\sqrt {1-z^{2}}})\\x&={\frac {\ln(z\mathrm {i} \pm {\sqrt {1-z^{2}}})}{\mathrm {i} }}\\x&={\frac {\ln(z\mathrm {i} \pm {\sqrt {1-z^{2}}})\,\mathrm {i} }{\mathrm {i} ^{2}}}\\x&={\frac {\ln(z\mathrm {i} \pm {\sqrt {1-z^{2}}})\,\mathrm {i} }{-1}}\\x&=-\mathrm {i} \,\ln(z\mathrm {i} \pm {\sqrt {1-z^{2}}})\\\arcsin z&=-\mathrm {i} \,\ln(z\mathrm {i} \pm {\sqrt {1-z^{2}}})\\\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <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>x</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>x</mi> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <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>x</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>x</mi> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>z</mi> </mtd> </mtr> <mtr> <mtd> <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>x</mi> </mrow> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <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>x</mi> </mrow> </msup> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>2</mn> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mo stretchy="false">(</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>x</mi> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>2</mn> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <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>x</mi> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <mo stretchy="false">(</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>x</mi> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <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>x</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <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>x</mi> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <mn>2</mn> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <mn>2</mn> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </msqrt> </mrow> </mtd> </mtr> <mtr> <mtd> <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>x</mi> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mi>x</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mi>x</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>x</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>x</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>x</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mspace width="thinmathspace" /> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mi>arcsin</mi> <mo>⁡</mo> <mi>z</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mspace width="thinmathspace" /> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\sin(x)&={\frac {\mathrm {e} ^{\mathrm {i} x}-\mathrm {e} ^{-\mathrm {i} x}}{2\mathrm {i} }}\\{\frac {\mathrm {e} ^{\mathrm {i} x}-\mathrm {e} ^{-\mathrm {i} x}}{2\mathrm {i} }}&=z\\\mathrm {e} ^{\mathrm {i} x}-{\frac {1}{\mathrm {e} ^{\mathrm {i} x}}}&=2z\mathrm {i} \\(\mathrm {e} ^{\mathrm {i} x})^{2}-1&=2z\mathrm {i} \mathrm {e} ^{\mathrm {i} x}\\(\mathrm {e} ^{\mathrm {i} x})^{2}-2z\mathrm {i} \mathrm {e} ^{\mathrm {i} x}-1&=0\\\mathrm {e} ^{\mathrm {i} x}&=-{\frac {-2z\mathrm {i} }{2}}\pm {\sqrt {\left({\frac {-2z\mathrm {i} }{2}}\right)^{2}-(-1)}}\\\mathrm {e} ^{\mathrm {i} x}&=z\mathrm {i} \pm {\sqrt {1-z^{2}}}\\\mathrm {i} x&=\ln(z\mathrm {i} \pm {\sqrt {1-z^{2}}})\\x&={\frac {\ln(z\mathrm {i} \pm {\sqrt {1-z^{2}}})}{\mathrm {i} }}\\x&={\frac {\ln(z\mathrm {i} \pm {\sqrt {1-z^{2}}})\,\mathrm {i} }{\mathrm {i} ^{2}}}\\x&={\frac {\ln(z\mathrm {i} \pm {\sqrt {1-z^{2}}})\,\mathrm {i} }{-1}}\\x&=-\mathrm {i} \,\ln(z\mathrm {i} \pm {\sqrt {1-z^{2}}})\\\arcsin z&=-\mathrm {i} \,\ln(z\mathrm {i} \pm {\sqrt {1-z^{2}}})\\\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40fba4efc0ed23d7a4681957bc076160d91c52a8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -32.505ex; width:51.93ex; height:66.176ex;" alt="{\displaystyle {\begin{aligned}\sin(x)&={\frac {\mathrm {e} ^{\mathrm {i} x}-\mathrm {e} ^{-\mathrm {i} x}}{2\mathrm {i} }}\\{\frac {\mathrm {e} ^{\mathrm {i} x}-\mathrm {e} ^{-\mathrm {i} x}}{2\mathrm {i} }}&=z\\\mathrm {e} ^{\mathrm {i} x}-{\frac {1}{\mathrm {e} ^{\mathrm {i} x}}}&=2z\mathrm {i} \\(\mathrm {e} ^{\mathrm {i} x})^{2}-1&=2z\mathrm {i} \mathrm {e} ^{\mathrm {i} x}\\(\mathrm {e} ^{\mathrm {i} x})^{2}-2z\mathrm {i} \mathrm {e} ^{\mathrm {i} x}-1&=0\\\mathrm {e} ^{\mathrm {i} x}&=-{\frac {-2z\mathrm {i} }{2}}\pm {\sqrt {\left({\frac {-2z\mathrm {i} }{2}}\right)^{2}-(-1)}}\\\mathrm {e} ^{\mathrm {i} x}&=z\mathrm {i} \pm {\sqrt {1-z^{2}}}\\\mathrm {i} x&=\ln(z\mathrm {i} \pm {\sqrt {1-z^{2}}})\\x&={\frac {\ln(z\mathrm {i} \pm {\sqrt {1-z^{2}}})}{\mathrm {i} }}\\x&={\frac {\ln(z\mathrm {i} \pm {\sqrt {1-z^{2}}})\,\mathrm {i} }{\mathrm {i} ^{2}}}\\x&={\frac {\ln(z\mathrm {i} \pm {\sqrt {1-z^{2}}})\,\mathrm {i} }{-1}}\\x&=-\mathrm {i} \,\ln(z\mathrm {i} \pm {\sqrt {1-z^{2}}})\\\arcsin z&=-\mathrm {i} \,\ln(z\mathrm {i} \pm {\sqrt {1-z^{2}}})\\\end{aligned}}}"></dd></dl> Für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \arccos z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arccos</mi> <mo>⁡</mo> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \arccos z}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b9ce68d984228896f64b0e2c4c12022a6cc87a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.693ex; height:1.676ex;" alt="{\displaystyle \arccos z}">: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\cos(x)&={\frac {\mathrm {e} ^{\mathrm {i} x}+\mathrm {e} ^{-\mathrm {i} x}}{2}}\\{\frac {\mathrm {e} ^{\mathrm {i} x}+\mathrm {e} ^{-\mathrm {i} x}}{2}}&=z\\\mathrm {e} ^{\mathrm {i} x}+{\frac {1}{\mathrm {e} ^{\mathrm {i} x}}}&=2z\\(\mathrm {e} ^{\mathrm {i} x})^{2}+1&=2z\mathrm {e} ^{\mathrm {i} x}\\(\mathrm {e} ^{\mathrm {i} x})^{2}-2z\mathrm {e} ^{\mathrm {i} x}+1&=0\\\mathrm {e} ^{\mathrm {i} x}&=-{\frac {-2z}{2}}\pm {\sqrt {\left({\frac {-2z}{2}}\right)^{2}-1)}}\\\mathrm {e} ^{\mathrm {i} x}&=z\pm {\sqrt {z^{2}-1}}\\\mathrm {i} x&=\ln(z\pm \mathrm {i} {\sqrt {1-z^{2}}})\\x&={\frac {\ln(z\pm \mathrm {i} {\sqrt {1-z^{2}}})}{\mathrm {i} }}\\x&={\frac {\ln(z\pm \mathrm {i} {\sqrt {1-z^{2}}})\,\mathrm {i} }{\mathrm {i} ^{2}}}\\x&={\frac {\ln(z\pm \mathrm {i} {\sqrt {1-z^{2}}})\,\mathrm {i} }{-1}}\\x&=-\mathrm {i} \,\ln(z\pm \mathrm {i} {\sqrt {1-z^{2}}})\\\arccos z&=-\mathrm {i} \,\ln(z\pm \mathrm {i} {\sqrt {1-z^{2}}})\\\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <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>x</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>x</mi> </mrow> </msup> </mrow> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <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>x</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>x</mi> </mrow> </msup> </mrow> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>z</mi> </mtd> </mtr> <mtr> <mtd> <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>x</mi> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <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>x</mi> </mrow> </msup> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>2</mn> <mi>z</mi> </mtd> </mtr> <mtr> <mtd> <mo stretchy="false">(</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>x</mi> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>2</mn> <mi>z</mi> <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>x</mi> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <mo stretchy="false">(</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>x</mi> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <mi>z</mi> <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>x</mi> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <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>x</mi> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <mn>2</mn> <mi>z</mi> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <mn>2</mn> <mi>z</mi> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </msqrt> </mrow> </mtd> </mtr> <mtr> <mtd> <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>x</mi> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>z</mi> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> </msqrt> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mi>x</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mi>x</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>x</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>x</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>x</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mspace width="thinmathspace" /> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mi>arccos</mi> <mo>⁡</mo> <mi>z</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mspace width="thinmathspace" /> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\cos(x)&={\frac {\mathrm {e} ^{\mathrm {i} x}+\mathrm {e} ^{-\mathrm {i} x}}{2}}\\{\frac {\mathrm {e} ^{\mathrm {i} x}+\mathrm {e} ^{-\mathrm {i} x}}{2}}&=z\\\mathrm {e} ^{\mathrm {i} x}+{\frac {1}{\mathrm {e} ^{\mathrm {i} x}}}&=2z\\(\mathrm {e} ^{\mathrm {i} x})^{2}+1&=2z\mathrm {e} ^{\mathrm {i} x}\\(\mathrm {e} ^{\mathrm {i} x})^{2}-2z\mathrm {e} ^{\mathrm {i} x}+1&=0\\\mathrm {e} ^{\mathrm {i} x}&=-{\frac {-2z}{2}}\pm {\sqrt {\left({\frac {-2z}{2}}\right)^{2}-1)}}\\\mathrm {e} ^{\mathrm {i} x}&=z\pm {\sqrt {z^{2}-1}}\\\mathrm {i} x&=\ln(z\pm \mathrm {i} {\sqrt {1-z^{2}}})\\x&={\frac {\ln(z\pm \mathrm {i} {\sqrt {1-z^{2}}})}{\mathrm {i} }}\\x&={\frac {\ln(z\pm \mathrm {i} {\sqrt {1-z^{2}}})\,\mathrm {i} }{\mathrm {i} ^{2}}}\\x&={\frac {\ln(z\pm \mathrm {i} {\sqrt {1-z^{2}}})\,\mathrm {i} }{-1}}\\x&=-\mathrm {i} \,\ln(z\pm \mathrm {i} {\sqrt {1-z^{2}}})\\\arccos z&=-\mathrm {i} \,\ln(z\pm \mathrm {i} {\sqrt {1-z^{2}}})\\\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/80003f6e0d7dc87ae3ad5207d54515ef12ce917d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -32.505ex; width:47.277ex; height:66.176ex;" alt="{\displaystyle {\begin{aligned}\cos(x)&={\frac {\mathrm {e} ^{\mathrm {i} x}+\mathrm {e} ^{-\mathrm {i} x}}{2}}\\{\frac {\mathrm {e} ^{\mathrm {i} x}+\mathrm {e} ^{-\mathrm {i} x}}{2}}&=z\\\mathrm {e} ^{\mathrm {i} x}+{\frac {1}{\mathrm {e} ^{\mathrm {i} x}}}&=2z\\(\mathrm {e} ^{\mathrm {i} x})^{2}+1&=2z\mathrm {e} ^{\mathrm {i} x}\\(\mathrm {e} ^{\mathrm {i} x})^{2}-2z\mathrm {e} ^{\mathrm {i} x}+1&=0\\\mathrm {e} ^{\mathrm {i} x}&=-{\frac {-2z}{2}}\pm {\sqrt {\left({\frac {-2z}{2}}\right)^{2}-1)}}\\\mathrm {e} ^{\mathrm {i} x}&=z\pm {\sqrt {z^{2}-1}}\\\mathrm {i} x&=\ln(z\pm \mathrm {i} {\sqrt {1-z^{2}}})\\x&={\frac {\ln(z\pm \mathrm {i} {\sqrt {1-z^{2}}})}{\mathrm {i} }}\\x&={\frac {\ln(z\pm \mathrm {i} {\sqrt {1-z^{2}}})\,\mathrm {i} }{\mathrm {i} ^{2}}}\\x&={\frac {\ln(z\pm \mathrm {i} {\sqrt {1-z^{2}}})\,\mathrm {i} }{-1}}\\x&=-\mathrm {i} \,\ln(z\pm \mathrm {i} {\sqrt {1-z^{2}}})\\\arccos z&=-\mathrm {i} \,\ln(z\pm \mathrm {i} {\sqrt {1-z^{2}}})\\\end{aligned}}}"></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Siehe_auch">Siehe auch</h2>[<a href="/w/index.php?title=Arkussinus_und_Arkuskosinus&veaction=edit&section=20" title="Abschnitt bearbeiten: Siehe auch" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Arkussinus_und_Arkuskosinus&action=edit&section=20" title="Quellcode des Abschnitts bearbeiten: Siehe auch">Quelltext bearbeiten</a>]</div> <ul><li><a href="/wiki/Formelsammlung_Trigonometrie" title="Formelsammlung Trigonometrie">Formelsammlung Trigonometrie</a></li> <li><a href="/wiki/Trigonometrische_Funktion" title="Trigonometrische Funktion">Trigonometrische Funktionen</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2>[<a href="/w/index.php?title=Arkussinus_und_Arkuskosinus&veaction=edit&section=21" title="Abschnitt bearbeiten: Literatur" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Arkussinus_und_Arkuskosinus&action=edit&section=21" title="Quellcode des Abschnitts bearbeiten: Literatur">Quelltext bearbeiten</a>]</div> <ul><li>Ilja Bronstein, Konstantin Semendjajew: <a href="/wiki/Taschenbuch_der_Mathematik" title="Taschenbuch der Mathematik">Taschenbuch der Mathematik</a>. B.G. Teubner, Stuttgart 1991. <a href="/wiki/Spezial:ISBN-Suche/3871444928" class="internal mw-magiclink-isbn">ISBN 3-87144-492-8</a>.</li></ul> <ul><li><a href="/wiki/Ilja_Nikolajewitsch_Bronstein" class="mw-redirect" title="Ilja Nikolajewitsch Bronstein">I. N. Bronstein</a>, <a href="/wiki/K._A._Semendjajev" class="mw-redirect" title="K. A. Semendjajev">K. A. Semendjajev</a>, G. Musiol, H. Mühlig (Hrsg.): <cite style="font-style:italic">Taschenbuch der Mathematik</cite>. 7., vollständig überarbeitete und ergänzte Auflage. <a href="/wiki/Verlag_Harri_Deutsch" title="Verlag Harri Deutsch">Verlag Harri Deutsch</a>, Frankfurt am Main 2008, <a href="/wiki/Spezial:ISBN-Suche/9783817120079" class="internal mw-magiclink-isbn">ISBN 978-3-8171-2007-9</a>, S. 85–88.<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:Arkussinus+und+Arkuskosinus&rft.btitle=Taschenbuch+der+Mathematik&rft.date=2008&rft.edition=7.%2C+vollst%C3%A4ndig+%C3%BCberarbeitete+und+erg%C3%A4nzte&rft.genre=book&rft.isbn=9783817120079&rft.pages=85-88&rft.place=Frankfurt+am+Main&rft.pub=Verlag+Harri+Deutsch" style="display:none"> </li> <li>G.Huvent: Autour de la primitive de tp coth (αt/2). 3. Februar 2002. Seite 5</li></ul> <ul><li>James D. Harper: A simple proof of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1+1/2^{2}+1/3^{2}+\ldots =\pi ^{2}/6}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>+</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo>…</mo> <mo>=</mo> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>6</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1+1/2^{2}+1/3^{2}+\ldots =\pi ^{2}/6}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5c4c7870d1df228bdd6db634426c16d49059896" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.302ex; height:3.176ex;" alt="{\displaystyle 1+1/2^{2}+1/3^{2}+\ldots =\pi ^{2}/6}"> The American Mathematical Monthly 109(6) (Jun. – Jul., 2003) 540–541.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Einzelnachweise">Einzelnachweise</h2>[<a href="/w/index.php?title=Arkussinus_und_Arkuskosinus&veaction=edit&section=22" title="Abschnitt bearbeiten: Einzelnachweise" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Arkussinus_und_Arkuskosinus&action=edit&section=22" title="Quellcode des Abschnitts bearbeiten: Einzelnachweise">Quelltext bearbeiten</a>]</div> <ol class="references"> <li id="cite_note-1"><a href="#cite_ref-1">↑</a> Wolfgang Zeuge: <cite style="font-style:italic">Nützliche und schöne Geometrie</cite>. 3.3 Die Umkehrfunktionen. Springer Spektrum, Berlin 2021, <a href="/wiki/Spezial:ISBN-Suche/9783662638316" class="internal mw-magiclink-isbn">ISBN 978-3-662-63831-6</a>, S. 46.<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:Arkussinus+und+Arkuskosinus&rft.au=Wolfgang+Zeuge&rft.btitle=N%C3%BCtzliche+und+sch%C3%B6ne+Geometrie&rft.date=2021&rft.genre=book&rft.isbn=9783662638316&rft.pages=46&rft.place=Berlin&rft.pub=Springer+Spektrum" style="display:none">  </li> <li id="cite_note-2"><a href="#cite_ref-2">↑</a> <a href="/wiki/Eric_Weisstein" title="Eric Weisstein">Eric W. Weisstein</a>: <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/InverseTrigonometricFunctions.html">Inverse Trigonometric Functions.</a> In: <a href="/wiki/MathWorld" title="MathWorld">MathWorld</a> (englisch). </li> <li id="cite_note-3"><a href="#cite_ref-3">↑</a> Derrick Henry Lehmer: Interesting Series Involving the Central Binomial Coefficient. Volume 92, 1985. Seite 452 </li> <li id="cite_note-4"><a href="#cite_ref-4">↑</a> James D.Harper, Another simple proof of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1+{\frac {1}{2^{2}}}+{\frac {1}{3^{2}}}+\cdots ={\frac {\pi ^{2}}{6}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>+</mo> <mo>⋯</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mn>6</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1+{\frac {1}{2^{2}}}+{\frac {1}{3^{2}}}+\cdots ={\frac {\pi ^{2}}{6}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bcda50500fb8c8b5a42469fbd14845e6d3354351" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:24.836ex; height:6.176ex;" alt="{\displaystyle 1+{\frac {1}{2^{2}}}+{\frac {1}{3^{2}}}+\cdots ={\frac {\pi ^{2}}{6}}}">, American Mathematical Monthly, Band 110, Nr. 6, 2003, S. 540–541 </li> <li id="cite_note-5"><a href="#cite_ref-5">↑</a> Georg Hoever: <cite style="font-style:italic">Höhere Mathematik kompakt</cite>. Springer Spektrum, Berlin Heidelberg 2014, <a href="/wiki/Spezial:ISBN-Suche/9783662439944" class="internal mw-magiclink-isbn">ISBN 978-3-662-43994-4</a> (<a rel="nofollow" class="external text" href="https://books.google.de/books?id=jQnvAwAAQBAJ&pg=PA25#v=onepage">eingeschränkte Vorschau</a> in der Google-Buchsuche).<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:Arkussinus+und+Arkuskosinus&rft.au=Georg+Hoever&rft.btitle=H%C3%B6here+Mathematik+kompakt&rft.date=2014&rft.genre=book&rft.isbn=9783662439944&rft.place=Berlin+Heidelberg&rft.pub=Springer+Spektrum" style="display:none">  </li> </ol> <style data-mw-deduplicate="TemplateStyles:r248673343">.mw-parser-output div.NavFrame{border-width:1px;border-style:solid;border-left-color:var(--dewiki-rahmenfarbe1);border-right-color:var(--dewiki-rahmenfarbe1);border-top-color:var(--dewiki-rahmenfarbe1);border-bottom-color:var(--dewiki-rahmenfarbe1);clear:both;font-size:95%;margin-top:1.5em;min-height:0;padding:2px;text-align:center}.mw-parser-output div.NavPic{float:left;padding:2px}.mw-parser-output div.NavHead{background-color:var(--dewiki-hintergrundfarbe5);font-weight:bold}.mw-parser-output div.NavFrame:after{clear:both;content:"";display:block}.mw-parser-output div.NavFrame+div.NavFrame,.mw-parser-output div.NavFrame+link+div.NavFrame,.mw-parser-output div.NavFrame+style+div.NavFrame{margin-top:-1px}.mw-parser-output .NavToggle{float:right;font-size:x-small}@media screen{html.skin-theme-clientpref-night .mw-parser-output .NavPic span[typeof="mw:File"] img{background-color:#c8ccd1}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .NavPic span[typeof="mw:File"] img{background-color:#c8ccd1}}</style><div class="NavFrame navigation-not-searchable" role="navigation"> <div class="NavHead"><a href="/wiki/Trigonometrische_Funktion" title="Trigonometrische Funktion">Trigonometrische Funktion</a></div> <div class="NavContent"> <a href="/wiki/Trigonometrische_Funktion" title="Trigonometrische Funktion">Primäre trigonometrische Funktionen</a> <a href="/wiki/Sinus_und_Kosinus" title="Sinus und Kosinus">Sinus und Kosinus</a> | <a href="/wiki/Tangens_und_Kotangens" title="Tangens und Kotangens">Tangens und Kotangens</a> | <a href="/wiki/Sekans_und_Kosekans" title="Sekans und Kosekans">Sekans und Kosekans</a>  <a href="/wiki/Arkusfunktion" title="Arkusfunktion">Umkehrfunktionen (Arkusfunktionen)</a> <a class="mw-selflink selflink">Arkussinus und Arkuskosinus</a> | <a href="/wiki/Arkustangens_und_Arkuskotangens" title="Arkustangens und Arkuskotangens">Arkustangens und Arkuskotangens</a> | <a href="/wiki/Arkussekans_und_Arkuskosekans" title="Arkussekans und Arkuskosekans">Arkussekans und Arkuskosekans</a>  <a href="/wiki/Hyperbelfunktion" title="Hyperbelfunktion">Hyperbelfunktionen</a> <a href="/wiki/Sinus_hyperbolicus_und_Kosinus_hyperbolicus" title="Sinus hyperbolicus und Kosinus hyperbolicus">Sinus hyperbolicus und Kosinus hyperbolicus</a> | <a href="/wiki/Tangens_hyperbolicus_und_Kotangens_hyperbolicus" title="Tangens hyperbolicus und Kotangens hyperbolicus">Tangens hyperbolicus und Kotangens hyperbolicus</a> | <a href="/wiki/Sekans_hyperbolicus_und_Kosekans_hyperbolicus" title="Sekans hyperbolicus und Kosekans hyperbolicus">Sekans hyperbolicus und Kosekans hyperbolicus</a>  <a href="/wiki/Areafunktion" title="Areafunktion">Areafunktionen</a> <a href="/wiki/Areasinus_hyperbolicus_und_Areakosinus_hyperbolicus" title="Areasinus hyperbolicus und Areakosinus hyperbolicus">Areasinus hyperbolicus und Areakosinus hyperbolicus</a> | <a href="/wiki/Areatangens_hyperbolicus_und_Areakotangens_hyperbolicus" title="Areatangens hyperbolicus und Areakotangens hyperbolicus">Areatangens hyperbolicus und Areakotangens hyperbolicus</a> | <a href="/wiki/Areasekans_hyperbolicus_und_Areakosekans_hyperbolicus" title="Areasekans hyperbolicus und Areakosekans hyperbolicus">Areasekans hyperbolicus und Areakosekans hyperbolicus</a>  </div> </div></div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1&useformat=desktop" alt="" width="1" height="1" style="border: none; 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Arkussinus und Arkuskosinus – Wikipedia