User:Lilyuserin/SVG/wifutab

• Trigonometric constants expressed in real radicals

Tabelle der algebraischen Werte für Winkelfunktionen

Alle Werte für Sinus und Cosinus von rationalen Vielfachen von 360° sind algebraische Zahlen. Alle Werte für Winkelfunktionen von Vielfachen von 3°-Winkeln können mit Hilfe von Rechenregeln für Sinus und Cosinus als Wurzeln der Werte von 0°, 30°, 36° und 45° ausgedrückt werden.

Es gibt nur wenige Werte, wo die Winkelfunktion eines rationalen Winkels (in Grad gemessen) eine rationale Zahl ergibt, nämlich 0°, 30°, 60° und 90°.

Für ganzzahlige Winkel, die kein Vielfaches von 3 sind, sind die Winkelfunktionen keine algebraischen Zahlen.

Allgemeine Abhängigkeiten

$\sin 0=0,\cos 0=1,\sin {\frac {\pi }{2}}=1,\cos {\frac {\pi }{2}}=0$

$\sin \pi =0,\cos \pi =-1,\sin \left(-{\frac {\pi }{2}}\right)=-1,\cos \left(-{\frac {\pi }{2}}\right)=0$

$\sin ^{2}\alpha +\cos ^{2}\alpha =1$

$\tan \alpha ={\frac {\sin \alpha }{\cos \alpha }}$

$\cot \alpha ={\frac {1}{\tan \alpha }}={\frac {\cos \alpha }{\sin \alpha }}$

$\sin(-\alpha )=-\sin \alpha$

$\cos(-\alpha )=\cos \alpha$

$\sin({\frac {\pi }{2}}-\alpha )=\cos \alpha$

$\sin(\alpha +{\frac {\pi }{2}})=\cos \alpha$

$\cos({\frac {\pi }{2}}-\alpha )=\sin \alpha$

$\cos(\alpha +{\frac {\pi }{2}})=-\sin \alpha$

$\sin(\alpha +\pi )=-\sin \alpha$

$\cos(\alpha +\pi )=-\cos \alpha$

$\sin(\pi -\alpha )=\sin \alpha$

$\cos(\pi -\alpha )=-\cos \alpha$

Allgemein gilt: Addition von ${\frac {\pi }{2}}$ zu einem Winkel vertauscht $\sin$ und $\cos$ zuzüglich etwaiger Vorzeichenwechsel, Addition von $\pi$ ändert evtl. das Vorzeichen.

Aufgrund dieser Summensätze genügt es, eine Winkelfunktion für einen Quadranten zu berechnen, der Rest lässt sich ableiten.

Winkelfunktionen von Winkelsummen

${\begin{aligned}\sin(\alpha \pm \beta )=\sin \alpha \cos \beta \pm \cos \alpha \sin \beta \\\cos(\alpha \pm \beta )=\cos \alpha \cos \beta \mp \sin \alpha \sin \beta \end{aligned}}$

$\sin 2\alpha =2\sin \alpha \cos \alpha ={\frac {2\tan \alpha }{1+\tan ^{2}\alpha }}$

$\cos 2\alpha =\cos ^{2}\alpha -\sin ^{2}\alpha$

$\sin 3\alpha =3\sin \alpha -4\sin ^{3}\alpha$

$\cos 3\alpha =4\cos ^{3}\alpha -3\cos \alpha$

$\sin ^{2}{\frac {\alpha }{2}}={\frac {1-\cos \alpha }{2}}$

${\begin{aligned}\sin(n\alpha )&=\sum _{k{\text{ odd}}}(-1)^{\frac {k-1}{2}}{n \choose k}\cos ^{n-k}\alpha \sin ^{k}\alpha ,\\\cos(n\alpha )&=\sum _{k{\text{ even}}}(-1)^{\frac {k}{2}}{n \choose k}\cos ^{n-k}\alpha \sin ^{k}\alpha \end{aligned}}$

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Algebraische Winkelfunktionswerte tabellarisch

Vielfache von $15^{0}={\frac {\pi }{12}}$ und $12^{0}={\frac {\pi }{15}}$

Winkel	Komplement	exakte Wurzel	dezimale Näherung	Kettenbruch
Vielfache von $15^{0}={\frac {\pi }{12}}$
$\sin 15^{0}=\sin {\frac {\pi }{12}}$	$\cos 75^{0}=\cos {\frac {5\pi }{12}}$	${\frac {1}{4}}\left({\sqrt {6}}-{\sqrt {2}}\right)$	0.258819045102520762348898837624048328349068901319930513814	[0;3,1,6,2,1,30,5,2,9,3,1,10,5,5,2,10,1,4,1,23,2,2,1,1,3,1,4,1,6]
$\cos 15^{0}=\cos {\frac {\pi }{12}}$	$\sin 75^{0}=\sin {\frac {5\pi }{12}}$	${\frac {1}{4}}\left({\sqrt {6}}+{\sqrt {2}}\right)$	0.965925826289068286749743199728897367633904839008404550402	[0;1,28,2,1,7,21,1,8,1,3,10,1,2,3,1,8,2,1,2,2,2,1,5,2,1,4,1,1,3,5]
$\sin 30^{0}=\sin {\frac {\pi }{6}}$	$\cos 60^{0}=\cos {\frac {\pi }{3}}$	${\frac {1}{2}}$	0.5	$[0;2]$
$\cos 30^{0}=\cos {\frac {\pi }{6}}$	$\sin 60^{0}=\sin {\frac {\pi }{3}}$	${\frac {\sqrt {3}}{2}}$	0.866025403784438646763723170752936183471402626905190314027	$[0;1,{\overline {6,2}}]$
$\sin 45^{0}=\sin {\frac {\pi }{4}}$	$\cos 45^{0}=\cos {\frac {\pi }{4}}$	${\frac {\sqrt {2}}{2}}$	0.707106781186547524400844362104849039284835937688474036588	$[0;1,{\overline {2}}]$
Vielfache von $12^{0}={\frac {\pi }{15}}$
$\sin 12^{0}=\sin {\frac {\pi }{15}}$	$\cos 78^{0}=\cos {\frac {13\pi }{30}}$	${\frac {1}{4}}{\sqrt {{\frac {1}{2}}\left(5+{\sqrt {5}}\right)}}-{\frac {1}{8}}{\sqrt {3}}\left({\sqrt {5}}-1\right)$	0.207911690817759337101742284405125166216584760627723836407	[0;4,1,4,3,1,10,116,12,2,12,2,2,32,4,1,2,2,1,4,1,1,1,1,1,1,1,6,7]
$\cos 12^{0}=\cos {\frac {\pi }{15}}$	$\sin 78^{0}=\sin {\frac {13\pi }{30}}$	${\frac {1}{8}}\left({\sqrt {6(5+{\sqrt {5}})}}+{\sqrt {5}}-1\right)$	0.978147600733805637928566747869599532459737808862677107885	[0;1,44,1,3,5,6,2,35,2,9,9,1,37,1,4,1,6,3,4,3,11,1,17,1,9,1,1,2,1]
$\sin 24^{0}=\sin {\frac {2\pi }{15}}$	$\cos 66^{0}=\cos {\frac {11\pi }{30}}$	${\frac {1}{8}}\left({\sqrt {15}}+{\sqrt {3}}-{\sqrt {2(5-{\sqrt {5}})}}\right)$	0.406736643075800207753985990341497612923139651066173436294	[0;2,2,5,1,1,6,7,2,1,2,8,2,1,61,1,3,2,1,2,1,2,1,1,1,2,7,1,1,1,3,1]
$\cos 24^{0}=\cos {\frac {2\pi }{15}}$	$\sin 66^{0}=\sin {\frac {11\pi }{30}}$	${\frac {1}{8}}\left({\sqrt {6(5-{\sqrt {5}})}}+{\sqrt {5}}+1\right)$	0.913545457642600895502127571985317177940810459377474545060	[0;1,10,1,1,3,4,10,3,2,2,1,8,2,1,1,8,1,10,5,4,5,2,51,1,1,1,1,1,24]
$\sin 36^{0}=\sin {\frac {\pi }{5}}$	$\cos 54^{0}=\cos {\frac {3\pi }{10}}$	${\frac {\sqrt {2}}{4}}{\sqrt {5-{\sqrt {5}}}}={\sqrt {{\frac {5}{8}}-{\frac {\sqrt {5}}{8}}}}$	0.587785252292473129168705954639072768597652437643145991072	[0;1,1,2,2,1,6,1,56,1,54,1,1,1,10,1,16,1,4,4,1,2,18,1,1,9,2,1,3,1]
$\cos 36^{0}=\cos {\frac {\pi }{5}}$	$\sin 54^{0}=\sin {\frac {3\pi }{10}}$	${\frac {1}{4}}(1+{\sqrt {5}})={\frac {\varphi }{2}}$	0.809016994374947424102293417182819058860154589902881431067	$[0;1,{\overline {4}}]$
$\cos 48^{0}=\cos {\frac {4\pi }{15}}$	$\sin 42^{0}=\sin {\frac {7\pi }{30}}$	${\frac {1}{8}}\left({\sqrt {30+{\sqrt {180}}}}-{\sqrt {5}}+1\right)$	0.669130606358858213826273330686780473599583218959795676817	[0;1,2,44,1,3,5,6,2,35,2,9,9,1,37,1,4,1,6,3,4,3,11,1,17,1,9,1,1,2]
$\sin 48^{0}=\sin {\frac {4\pi }{15}}$	$\cos 42^{0}=\cos {\frac {7\pi }{30}}$	${\frac {1}{8}}\left({\sqrt {1}}5-{\sqrt {3}}+{\sqrt {10+{\sqrt {2}}0}}\right)$	0.743144825477394235014697048974256977189113873498026386040	[0;1,2,1,8,2,1,2,1,1,1,1,2,4,2,1,10,1,1,3,2,7,2,1,2,1,6,2,3,1,32]
$\sin 72^{0}=\sin {\frac {2\pi }{5}}$	$\cos 18^{0}=\cos {\frac {\pi }{10}}$	${\sqrt {{\frac {5}{8}}+{\frac {\sqrt {5}}{8}}}}$	0.951056516295153572116439333379382143405698634125750222447	[0;1,19,2,3,6,5,1,1,1,3,2,1,25,2,2,1,1,11,1,2,1,7,1,1,10,1,7,23,1]
$\cos 72^{0}=\cos {\frac {2\pi }{5}}$	$\sin 18^{0}=\sin {\frac {\pi }{10}}$	${\frac {1}{2}}({\sqrt {5}}-1)={\frac {1}{2\varphi }}$	0.309016994374947424102293417182819058860154589902881431067	$[0;3,{\overline {4}}]$

Innenwinkel von regulären Polygonen mit einem Vielfachen von 6 Ecken

Winkel	Komplement	exakte Wurzel	dezimale Näherung	Kettenbruch
Regelmäßiges 120-Eck (Hexatonicosagon)
$\sin 1.5^{0}=\sin {\frac {\pi }{120}}$	$\cos 88.5^{0}=\cos {\frac {59\pi }{120}}$	${\frac {1}{16}}\left({\sqrt {2+{\sqrt {2}}}}\right)\left({\sqrt {15}}+{\sqrt {3}}-{\sqrt {10-2{\sqrt {5}}}}\right)-\left({\sqrt {2-{\sqrt {2}}}}\right)\left({\sqrt {30-6{\sqrt {5}}}}+{\sqrt {5}}+1\right)$	0.026176948307873152610611685554112663793391027680108638218	[0;38,4,1,25,160,1,6,74,2,2,2,1,11,1,195,9,5,1,6,2,1,2,8,5,1,1,1]
$\sin 1.5^{0}=\cos {\frac {\pi }{120}}$	$\cos 88.5^{0}=\cos {\frac {59\pi }{120}}$	${\frac {1}{16}}\left({\sqrt {2+{\sqrt {2}}}}\right)\left({\sqrt {30-6{\sqrt {5}}}}+{\sqrt {5}}+1\right)+\left({\sqrt {2-{\sqrt {2}}}}\right)\left({\sqrt {15}}+{\sqrt {3}}-{\sqrt {10-2{\sqrt {5}}}}\right)$	0.999657324975557280036760888367679875949875971241078461264	[0;1,2917,4,1,1,1,1,2,2,1,1,4,4,21,2,2,2,10,2,1,3,10,2,1,1,1,11,1]
Regelmäßiges 60-Eck (Hexacontagon)
$\sin 3^{0}=\sin {\frac {\pi }{60}}$	$\cos 87^{0}=\cos {\frac {29\pi }{60}}$	${\frac {1}{16}}\left(2(1-{\sqrt {3}}){\sqrt {5+{\sqrt {5}}}}+({\sqrt {10}}-{\sqrt {2}})({\sqrt {3}}+1)\right)$	0.052335956242943832722118629609078418731018253940164920483	[0;19,9,3,6,1,3,2,3,3,2341,1,5,2,8,3,1,1,9,1,1,1,1,7,5,1,2,1,1,3]
$\sin 3^{0}=\cos {\frac {\pi }{60}}$	$\cos 87^{0}=\cos {\frac {29\pi }{60}}$	${\frac {1}{16}}\left(2(1+{\sqrt {3}}){\sqrt {5+{\sqrt {5}}}}+({\sqrt {10}}-{\sqrt {2}})({\sqrt {3}}-1)\right)$	0.998629534754573873784492058439436580590952290767785532441	[0;1,728,1,2,8,1,1,10,11,1,3,3,2,5,4,2,5,1,1,1,2,2,1,3,3,5,7,164]
Regelmäßiges 30-Eck (Hexacontagon)
$\sin 6^{0}=\sin {\frac {\pi }{30}}$	$\cos 84^{0}=\cos {\frac {7\pi }{15}}$	${\frac {1}{8}}\left({\sqrt {30-{\sqrt {180}}}}-{\sqrt {5}}-1\right)$	0.104528463267653471399834154802498119080655869474593113993	[0;9,1,1,3,4,10,3,2,2,1,8,2,1,1,8,1,10,5,4,5,2,51,1,1,1,1,1,24,1]
$\sin 6^{0}=\cos {\frac {\pi }{30}}$	$\cos 84^{0}=\cos {\frac {7\pi }{14}}$	${\frac {1}{8}}\left({\sqrt {10-{\sqrt {20}}}}+{\sqrt {3}}+{\sqrt {15}}\right)$	0.994521895368273336922691944980570381520792088709319427366	[0;1,181,1,1,5,14,2,2,1,2,1,1,3,3,5,5,2,37,1,6,46,2,1,11,1,5,2,10]

Innenwinkel von regulären Polygonen mit einem Vielfachen von 8 Ecken

Winkel	Komplement	exakte Wurzel	dezimale Näherung	Kettenbruch
Regelmäßiges 96-Eck (Enneacontahexagon)
$\sin 1.875^{0}=\sin {\frac {\pi }{96}}$	$\cos 88.125^{0}=\cos {\frac {47\pi }{48}}$	${\frac {1}{2}}{\sqrt {2-{\sqrt {2+{\sqrt {2+{\sqrt {2+{\sqrt {3}}}}}}}}}}$	0.032719082821776142063659926317288126114962627251696196878	[0;30,1,1,3,2,5,8,1,15,2,3,1,1,5,1,1,1,4,1,7,8,1,1,8,2,96,1,10,1]
$\cos 1.875^{0}=\cos {\frac {\pi }{96}}$	$\sin 88.125^{0}=\sin {\frac {47\pi }{48}}$	${\frac {1}{2}}{\sqrt {2+{\sqrt {2+{\sqrt {2+{\sqrt {2+{\sqrt {3}}}}}}}}}}$	0.999464587476365644429836446242859945883640953153130896033	[0;1,1866,1,2,1,1,4,55,1,1,5,3,1,1,3,43,1,5,1,23,1,1,8,1,4,2,2,2]
Regelmäßiges 80-Eck (Octacontagon)
$\sin 2.25^{0}=\sin {\frac {\pi }{80}}$	$\cos 87.75^{0}=\cos {\frac {39\pi }{40}}$	${\frac {1}{2}}{\sqrt {2-{\sqrt {2+{\sqrt {2+{\sqrt {\frac {5+{\sqrt {5}}}{2}}}}}}}}}$	0.039259815759068609020803363798333589680182017977252297768	[0;25,2,8,4,1,1,77,10,1,3,1,43,1,2,1,4,2,3,1,12,4,2,3,1,11,23,1,1]
$\cos 2.25^{0}=\cos {\frac {\pi }{80}}$	$\sin 87.75^{0}=\sin {\frac {39\pi }{40}}$	${\frac {1}{2}}{\sqrt {2+{\sqrt {2+{\sqrt {2+{\sqrt {\frac {5+{\sqrt {5}}}{2}}}}}}}}}$	0.999229036240722934737126260341461625270674854841439422506	[0;1,1296,12,1,5,1,1,1,1,114,1,1,1,138,50,1,1,1,10,1,1,2,14,4,11]
Regelmäßiges 64-Eck (Hexacontatetragon)
$\sin 2.8125^{0}=\sin {\frac {\pi }{64}}$	$\cos 87.1875^{0}=\cos {\frac {31\pi }{32}}$	${\frac {1}{2}}{\sqrt {2-{\sqrt {2+{\sqrt {2+{\sqrt {2+{\sqrt {2}}}}}}}}}}$	0.049067674327418014254954976942682658314745363025752920210	[0;20,2,1,1,1,2,2,24,5,185,6,1,4,1,1,1,31,3,4,7,2,25,3,1,5,6,1,1]
$\cos 2.8125^{0}=\cos {\frac {\pi }{64}}$	$\sin 87.1875^{0}=\sin {\frac {31\pi }{32}}$	${\frac {1}{2}}{\sqrt {2+{\sqrt {2+{\sqrt {2+{\sqrt {2+{\sqrt {2}}}}}}}}}}$	0.998795456205172392714771604759100694443203614704611794342	[0;1,829,5,3,1,2,1,2,2,4,1,5,15,1,2,1,1,1,1,2,1,1,6,1,1,1,15,2,1]
Regelmäßiges 48-Eck (Tetracontaoctagon)
$\sin 3.75^{0}=\sin {\frac {\pi }{48}}$	$\cos 86.25^{0}=\cos {\frac {23\pi }{24}}$	${\frac {1}{2}}{\sqrt {2-{\sqrt {2+{\sqrt {2+{\sqrt {3}}}}}}}}$	0.065403129230143066815315558775175441440632303945054017831	[0;15,3,2,4,1,1,2,1,1,2,7,3,9,4,1,10,1,12,1,2,5,2,2,1,5,1,1,2,10]
$\cos 3.75^{0}=\cos {\frac {\pi }{48}}$	$\sin 86.25^{0}=\sin {\frac {23\pi }{24}}$	${\frac {1}{2}}{\sqrt {2+{\sqrt {2+{\sqrt {2+{\sqrt {3}}}}}}}}$	0.997858923238603506738069791272777604531866396325966418074	[0;1,466,18,3,1,1,1,1,1,8,1,1,2,1,5,1,2,2,1,4,6,1,7,1,7,37,3,3,1]
Regelmäßiges 40-Eck (Tetracontagon)
$\sin 4.5^{0}=\sin {\frac {\pi }{40}}$	$\cos 84.5^{0}=\cos {\frac {19\pi }{20}}$	${\frac {1}{2}}{\sqrt {2-{\sqrt {2+{\sqrt {\frac {5+{\sqrt {5}}}{2}}}}}}}$	0.078459095727844945032960245993459698681955877106031561346	[0;12,1,2,1,13,8,7,1,1,1,7,5,2,2,1,1,2,2,3,2,1,3,2,3,1,2,107,1,1]
$\cos 4.5^{0}=\cos {\frac {\pi }{40}}$	$\sin 84.5^{0}=\sin {\frac {19\pi }{20}}$	${\frac {1}{2}}{\sqrt {2+{\sqrt {2+{\sqrt {\frac {5+{\sqrt {5}}}{2}}}}}}}$	0.996917333733127976197773408742044420158926881594843892194	[0;1,323,2,1,1,6,1,2,7,1,4,16,1,1,5,4,6,2,3,4,4,2,5,1,6,3,1,186,8]
Regelmäßiges 32-Eck (Triacontadigon)
$\sin 5.625^{0}=\sin {\frac {\pi }{32}}$	$\cos 84.375^{0}=\cos {\frac {15\pi }{16}}$	${\frac {1}{2}}{\sqrt {2-{\sqrt {2+{\sqrt {2+{\sqrt {2}}}}}}}}$	0.098017140329560601994195563888641845861136673167500567257	[0;10,4,1,16,1,1,1,1,2,1,2,15,1,1,1,2,7,1,1,11,1,2,12,4,2,31,10,1]
$\cos 5.625^{0}=\cos {\frac {\pi }{32}}$	$\sin 84.375^{0}=\sin {\frac {15\pi }{16}}$	${\frac {1}{2}}{\sqrt {2+{\sqrt {2+{\sqrt {2+{\sqrt {2}}}}}}}}$	0.995184726672196886244836953109479921575474868729857061833	[0;1,206,1,2,18,1,1,1,1,2,2,2,1,5,1,1,2,1,1,1,1,1,1,3,49,4,6,5,1]

Innenwinkel von regulären Polygonen mit 24 Ecken und weniger

Winkel	Komplement	exakte Wurzel	dezimale Näherung	Kettenbruch
Regelmäßiges 24-Eck ()
$\sin 7.5^{0}=\sin {\frac {\pi }{24}}$	$\cos 82.5^{0}=\cos {\frac {11\pi }{12}}$	${\frac {1}{2}}{\sqrt {2-{\sqrt {2+{\sqrt {3}}}}}}$	0.130526192220051591548406227895489010193740704811732251890	[0;7,1,1,1,20,35,1,4,1,1,8,7,8,12,3,1,9,2,88,2,14,1,1,5,14,1,1,2]
$\cos 7.5^{0}=\cos {\frac {\pi }{24}}$	$\sin 82.5^{0}=\sin {\frac {11\pi }{12}}$	${\frac {1}{2}}{\sqrt {2+{\sqrt {2+{\sqrt {3}}}}}}$	0.991444861373810411144557526928562871277738274448102271458	[0;1,115,1,7,1,162,37,1,2,3,3,1,2,5,2,11,4,1,1,1,1,1,7,3,1,2,1,3]
Regelmäßiges 20-Eck ()
$\sin 9^{0}=\sin {\frac {\pi }{20}}$	$\cos 81^{0}=\cos {\frac {9\pi }{10}}$	${\frac {1}{2}}{\sqrt {2-{\sqrt {\frac {5+{\sqrt {5}}}{2}}}}}$	0.312868930080461738020210638934333784627799784171321580169	[0;3,5,10,2,2,72,2,1,1,1,7,2,130,3,1,3,35,8,1,9,1,4,11,3,1,2,3,2]
$\cos 9^{0}=\cos {\frac {\pi }{20}}$	$\sin 81^{0}=\sin {\frac {9\pi }{10}}$	${\frac {1}{2}}{\sqrt {2+{\sqrt {\frac {5+{\sqrt {5}}}{2}}}}}$	1.975376681190275452380080495386874521516813723179760869847	[1;1,39,1,1,1,1,2,1,3,7,86,1,5,1,1,8,1,1,7,1,3,1,4,3,1,1,1,1,3,4]
Regelmäßiges 16-Eck ()
$\sin 11.25^{0}=\sin {\frac {\pi }{16}}$	$\cos 78.75^{0}=\cos {\frac {7\pi }{16}}$	${\frac {1}{2}}{\sqrt {2-{\sqrt {2+{\sqrt {2}}}}}}$	0.195090322016128267848284868477022240927691617751954807754	[0;5,7,1,17,1,13,3,2,7,1,1,8,1,1,1,40,1,2,69,271,70,1,1,1,3,1,42]
$\cos 11.25^{0}=\cos {\frac {\pi }{16}}$	$\sin 78.75^{0}=\sin {\frac {7\pi }{16}}$	${\frac {1}{2}}{\sqrt {2+{\sqrt {2+{\sqrt {2}}}}}}$	0.980785280403230449126182236134239036973933730893336095002	[0;1,51,23,43,8,1,2,1,3,1,15,6,2,3,1,1,2,7,2,1,44,1,1,2,2,2,11,3]
Regelmäßiges 15-Eck ()
$\sin 12^{0}=\sin {\frac {\pi }{15}}$	$\cos 78^{0}=\cos {\frac {13\pi }{30}}$	${\frac {1}{8}}\left({\sqrt {2(5+{\sqrt {5}})}}+{\sqrt {5}}-{\sqrt {15}}\right)$	siehe Vielfache von 12° und 15°	siehe Vielfache von 12° und 15°
$\cos 12^{0}=\cos {\frac {\pi }{15}}$	$\sin 78^{0}=\sin {\frac {13\pi }{30}}$	${\frac {1}{8}}\left({\sqrt {6(5+{\sqrt {5}})}}+{\sqrt {5}}-1\right)$	siehe Vielfache von 12° und 15°	siehe Vielfache von 12° und 15°
Regelmäßiges 12-Eck ()
$\sin 15^{0}=\sin {\frac {\pi }{12}}$	$\cos 75^{0}=\cos {\frac {11\pi }{12}}$	${\frac {1}{4}}\left({\sqrt {6}}-{\sqrt {2}}\right)$	siehe Vielfache von 12° und 15°	siehe Vielfache von 12° und 15°
$\cos 15^{0}=\cos {\frac {\pi }{12}}$	$\sin 75^{0}=\sin {\frac {11\pi }{12}}$	${\frac {1}{4}}\left({\sqrt {6}}+{\sqrt {2}}\right)$	siehe Vielfache von 12° und 15°	siehe Vielfache von 12° und 15°