Formelsammlung Mathematik: Winkelfunktionen

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Symmetrien

Punktsymmetrie	Achsensymmetrie
${\begin{aligned}\sin(-x)&=-\sin(x),\\\tan(-x)&=-\tan(x),\\\cot(-x)&=-\cot(x),\\\csc(-x)&=-\csc(x)\end{aligned}}$	${\begin{aligned}\cos(-x)&=\cos(x),\\\sec(-x)&=\sec(x)\end{aligned}}$

Definition der Winkel- und Hyperbelfunktionen durch die e-Funktion

$\sin(z)={\frac {\mathrm {e} ^{\mathrm {i} z}-\mathrm {e} ^{-\mathrm {i} z}}{2\mathrm {i} }}$	$\sinh(z)={\frac {\mathrm {e} ^{z}-\mathrm {e} ^{-z}}{2}}$	$\sin(\mathrm {i} z)=\mathrm {i} \sinh(z)\,$	$\sinh(\mathrm {i} z)=\mathrm {i} \sin(z)\,$
$\cos(z)={\frac {\mathrm {e} ^{\mathrm {i} z}+\mathrm {e} ^{-\mathrm {i} z}}{2}}$	$\cosh(z)={\frac {\mathrm {e} ^{z}+\mathrm {e} ^{-z}}{2}}$	$\cos(\mathrm {i} z)=\cosh(z)\,$	$\cosh(\mathrm {i} z)=\cos(z)\,$
$\tan(z)={\frac {1}{\mathrm {i} }}\,{\frac {\mathrm {e} ^{\mathrm {i} z}-\mathrm {e} ^{-\mathrm {i} z}}{\mathrm {e} ^{\mathrm {i} z}+\mathrm {e} ^{-\mathrm {i} z}}}$	$\tanh(z)={\frac {\mathrm {e} ^{z}-\mathrm {e} ^{-z}}{\mathrm {e} ^{z}+\mathrm {e} ^{-z}}}$	$\tan(\mathrm {i} z)=\mathrm {i} \tanh(z)\,$	$\tanh(\mathrm {i} z)=\mathrm {i} \tan(z)\,$
$\cot(z)=\mathrm {i} \,{\frac {\mathrm {e} ^{\mathrm {i} z}+\mathrm {e} ^{-\mathrm {i} z}}{\mathrm {e} ^{\mathrm {i} z}-\mathrm {e} ^{-\mathrm {i} z}}}$	$\coth(z)={\frac {\mathrm {e} ^{z}+\mathrm {e} ^{-z}}{\mathrm {e} ^{z}-\mathrm {e} ^{-z}}}$	$\cot(\mathrm {i} z)={\frac {1}{\mathrm {i} }}\coth(z)\,$	$\coth(\mathrm {i} z)={\frac {1}{\mathrm {i} }}\cot(z)\,$
$\sec(z)={\frac {2}{\mathrm {e} ^{\mathrm {i} z}+\mathrm {e} ^{-\mathrm {i} z}}}$	$\operatorname {sech} (z)={\frac {2}{\mathrm {e} ^{z}+\mathrm {e} ^{-z}}}$	$\sec(\mathrm {i} z)=\operatorname {sech} (z)$	$\operatorname {sech} (\mathrm {i} z)=\sec(z)$
$\csc(z)={\frac {2\mathrm {i} }{\mathrm {e} ^{\mathrm {i} z}-\mathrm {e} ^{-\mathrm {i} z}}}$	$\operatorname {csch} (z)={\frac {2}{\mathrm {e} ^{z}-\mathrm {e} ^{-z}}}$	$\csc(\mathrm {i} z)={\frac {1}{\mathrm {i} }}\,\operatorname {csch} (z)$	$\operatorname {csch} (\mathrm {i} z)={\frac {1}{\mathrm {i} }}\,\csc(z)$

Gegenseitige Darstellbarkeit von Winkelfunktionen

	sin	cos	tan	cot	sec	csc
sin2(x)	$\sin ^{2}(x)$	$1-\cos ^{2}(x)$	${\frac {\tan ^{2}(x)}{1+\tan ^{2}(x)}}$	${\frac {1}{\cot ^{2}(x)+1}}$	${\frac {\sec ^{2}(x)-1}{\sec ^{2}(x)}}$	${\frac {1}{\csc ^{2}(x)}}$
cos2(x)	$1-\sin ^{2}(x)$	$\cos ^{2}(x)$	${\frac {1}{1+\tan ^{2}(x)}}$	${\frac {\cot ^{2}(x)}{\cot ^{2}(x)+1}}$	${\frac {1}{\sec ^{2}(x)}}$	${\frac {\csc ^{2}(x)-1}{\csc ^{2}(x)}}$
tan2(x)	${\frac {\sin ^{2}(x)}{1-\sin ^{2}(x)}}$	${\frac {1-\cos ^{2}(x)}{\cos ^{2}(x)}}$	$\tan ^{2}(x)$	${\frac {1}{\cot ^{2}(x)}}$	$\sec ^{2}(x)-1$	${\frac {1}{\csc ^{2}(x)-1}}$
cot2(x)	${\frac {1-\sin ^{2}(x)}{\sin ^{2}(x)}}$	${\frac {\cos ^{2}(x)}{1-\cos ^{2}(x)}}$	${\frac {1}{\tan ^{2}(x)}}$	$\cot ^{2}(x)$	${\frac {1}{\sec ^{2}(x)-1}}$	$\csc ^{2}(x)-1$
sec2(x)	${\frac {1}{1-\sin ^{2}(x)}}$	${\frac {1}{\cos ^{2}(x)}}$	$1+\tan ^{2}(x)$	${\frac {\cot ^{2}(x)+1}{\cot ^{2}(x)}}$	$\sec ^{2}(x)$	${\frac {\csc ^{2}(x)}{\csc ^{2}(x)-1}}$
csc2(x)	${\frac {1}{\sin ^{2}(x)}}$	${\frac {1}{1-\cos ^{2}(x)}}$	${\frac {1+\tan ^{2}(x)}{\tan ^{2}(x)}}$	$\cot ^{2}(x)+1$	${\frac {\sec ^{2}(x)}{\sec ^{2}(x)-1}}$	$\csc ^{2}(x)$

Die Gleichungen gelten für alle $x\in \mathbb {R}$ mit Ausnahme der Polstellen. Stetig hebbare Definitionslücken können entsprechend ergänzt werden.

Man beachte, dass die Gleichungen nach dem Wurzelziehen nur betragsmäßig gültig sind, da beim Quadrieren die Vorzeichen verloren gehen.

Winkelfunktionen mit verschobenem Argument

	$\pm \varphi$	${\frac {\pi }{2}}\pm \varphi$	$\pi \pm \varphi$	${\frac {3\pi }{2}}\pm \varphi$
$\sin \,$	$\pm \sin$	$\cos \!$	$\mp \sin$	$-\cos \!$
$\cos \,$	$\cos \!$	$\mp \sin \!$	$-\cos \!$	$\pm \sin \!$
$\tan \,$	$\pm \tan \!$	$\mp \cot \!$	$\pm \tan \!$	$\mp \cot \!$
$\cot \,$	$\pm \cot \!$	$\mp \tan \!$	$\pm \cot \!$	$\mp \tan \!$
$\sec \,$	$\sec \!$	$\mp \csc \!$	$-\sec \!$	$\pm \csc \!$
$\csc \,$	$\pm \csc$	$\sec \!$	$\mp \csc$	$-\sec \!$

$1+\tan \left({\frac {\pi }{4}}-x\right)={\frac {2}{1+\tan x}}$

Additionstheoreme

$\sin(x\pm y)=\sin x\;\cos y\pm \sin y\;\cos x$

$\cos(x\pm y)=\cos x\;\cos y\mp \sin x\;\sin y$

$\tan(x\pm y)={\frac {\tan x\pm \tan y}{1\mp \tan x\;\tan y}}$

$\cot \left(x\pm y\right)={\frac {\pm \cot x\cot y\mp 1}{\cot x\pm \cot y}}$

$\sin(x+y)\,\sin(x-y)=\cos ^{2}y-\cos ^{2}x$

$\cos(x+y)\,\cos(x-y)=\cos ^{2}y-\sin ^{2}x$

Doppelwinkelfunktionen

$\sin(2x)=2\,\sin x\,\cos x={\frac {2\tan x}{1+\tan ^{2}x}}$

$\cos(2x)=\cos ^{2}x-\sin ^{2}x=1-2\sin ^{2}x=2\cos ^{2}x-1={\frac {1-\tan ^{2}x}{1+\tan ^{2}x}}$

$\tan(2x)={\frac {2\tan x}{1-\tan ^{2}x}}={\frac {2}{\cot x-\tan x}}$

$\cot(2x)={\frac {\cot ^{2}x-1}{2\cot x}}={\frac {\cot x-\tan x}{2}}$

Winkelfunktionen für weitere Vielfache

$\sin(3x)=3\sin x-4\sin ^{3}x\!$

$\sin(4x)=8\sin x\;\cos ^{3}x-4\sin x\;\cos x$

$\sin(5x)=16\sin x\;\cos ^{4}x-12\sin x\;\cos ^{2}x+\sin x$

$\sin(nx)=n\;\sin x\;\cos ^{n-1}x-{n \choose 3}\sin ^{3}x\;\cos ^{n-3}x+{n \choose 5}\sin ^{5}x\;\cos ^{n-5}x\;-\;+\;\dots$

$\cos(3x)=4\cos ^{3}x-3\cos x\!$

$\cos(4x)=8\cos ^{4}x-8\cos ^{2}x+1\!$

$\cos(5x)=16\cos ^{5}x-20\cos ^{3}x+5\cos x\!$

$\cos(nx)=\cos ^{n}x-{n \choose 2}\sin ^{2}x\;\cos ^{n-2}x+{n \choose 4}\sin ^{4}x\;\cos ^{n-4}x\;-\;+\;\dots$

$\tan(nx)={\frac {\sum _{k=0}^{\infty }(-1)^{k}\,{n \choose 2k+1}\,\tan ^{2k+1}}{\sum _{k=0}^{\infty }(-1)^{k}\,{n \choose 2k}\,\tan ^{2k}}}$

$\cot(nx)=(-1)^{n-1}\,\left({\frac {\sum _{k=0}^{\infty }(-1)^{k}\,{n \choose 2k+1}\,\cot ^{2k+1}}{\sum _{k=0}^{\infty }(-1)^{k}\,{n \choose 2k}\,\cot ^{2k}}}\right)^{(-1)^{n-1}}$

Rekursionsformeln mit $n,x\in \mathbb {C}$:

${\begin{aligned}\cos(nx)&=2\cos(x)\cos((n-1)x)+\cos((n-2)x),\\\sin(nx)&=2\cos(x)\sin((n-1)x)+\sin((n-2)x).\end{aligned}}$

Halbwinkelformeln

$\sin {\frac {x}{2}}={\sqrt {\frac {1-\cos x}{2}}}$ für $x\in [0,2\pi ]$

$\cos {\frac {x}{2}}={\sqrt {\frac {1+\cos x}{2}}}$ für $x\in [-\pi ,\pi ]$

$\tan {\frac {x}{2}}={\sqrt {\frac {1-\cos x}{1+\cos x}}}={\frac {1-\cos x}{\sin(x)}}={\frac {\sin x}{1+\cos x}}$ für $x\in ]-\pi ,\pi [$

$\cot {\frac {x}{2}}={\sqrt {\frac {1+\cos x}{1-\cos x}}}={\frac {1+\cos x}{\sin x}}={\frac {\sin x}{1-\cos x}}$ für $x\in ]-\pi ,\pi [$

Identitäten

Aus den Additionstheoremen lassen sich Identitäten ableiten:

$\sin x\pm \sin y=2\sin {\frac {x\pm y}{2}}\,\cos {\frac {x\mp y}{2}}$

$\cos x\pm \cos y=\pm 2\;{\begin{matrix}\cos \\\sin \end{matrix}}\left({\frac {x+y}{2}}\right){\begin{matrix}\cos \\\sin \end{matrix}}\left({\frac {x-y}{2}}\right)$

$\tan x\pm \tan y={\frac {\sin(x\pm y)}{\cos(x)\,\cos(y)}}\quad ,\quad \cot x\pm \cot y={\frac {\sin(y\pm x)}{\sin(x)\,\sin(y)}}$

Produkte der Winkelfunktionen

$\cos x\;\cos y={\frac {\cos(x-y)+\cos(x+y)}{2}}$

$\sin x\;\sin y={\frac {\cos(x-y)-\cos(x+y)}{2}}$

$\sin x\;\cos y={\frac {\sin(x-y)+\sin(x+y)}{2}}$

$\tan x\;\tan y={\frac {\tan x+\tan y}{\cot x+\cot y}}=-{\frac {\tan x-\tan y}{\cot x-\cot y}}$

$\cot x\;\cot y={\frac {\cot x+\cot y}{\tan x+\tan y}}=-{\frac {\cot x-\cot y}{\tan x-\tan y}}$

$\tan x\;\cot y={\frac {\tan x+\cot y}{\cot x+\tan y}}=-{\frac {\tan x-\cot y}{\cot x-\tan y}}$

$\sin x\;\sin y\;\sin z={\frac {1}{4}}\left[\sin(x+y-z)+\sin(y+z-x)+\sin(z+x-y)-\sin(x+y+z)\right]$

$\cos x\;\cos y\;\cos z={\frac {1}{4}}\left[\cos(x+y-z)+\cos(y+z-x)+\cos(z+x-y)+\cos(x+y+z)\right]$

$\sin x\;\sin y\;\cos z={\frac {1}{4}}\left[-\cos(x+y-z)+\cos(y+z-x)+\cos(z+x-y)-\cos(x+y+z)\right]$

$\sin x\;\cos y\;\cos z={\frac {1}{4}}\left[\sin(x+y-z)-\sin(y+z-x)+\sin(z+x-y)+\sin(x+y+z)\right]$

Potenzen der Winkelfunktionen

$(2\cos x)^{2n}=\sum _{k=-n}^{n}{2n \choose n-k}\cos 2kx$

$(2\sin x)^{2n}=\sum _{k=-n}^{n}(-1)^{k}\,{2n \choose n-k}\cos 2kx$

$(2\cos x)^{2n+1}=2\sum _{k=0}^{n}{2n+1 \choose n-k}\cos(2k+1)x$

$(2\sin x)^{2n+1}=2\sum _{k=0}^{n}(-1)^{k}\,{2n+1 \choose n-k}\sin(2k+1)x$