三角恒等式

在数学中，三角恒等式是对出现的所有值都为實变量，涉及到三角函数的等式。这些恒等式在表达式中有些三角函数需要简化的时候是很有用的。一个重要应用是非三角函数的积分：一个常用技巧是首先使用使用三角函数的代换规则，则通过三角恒等式可简化结果的积分。

函数			反函數			倒数
中文	全寫	簡寫	中文	全寫	簡寫	中文	全寫	簡寫
正弦	sine	sin	反正弦	arcsine	arcsin	餘割	cosecant	csc
餘弦	cosine	cos	反餘弦	arccosine	arccos	正割	secant	sec
正切	tangent	tan	反正切	arctangent	arctan	餘切	cotangent	cot
餘切	cotangent	cot	反餘切	arccotangent	arccot	正切	tangent	tan
正割	secant	sec	反正割	arcsecant	arcsec	餘弦	cosine	cos
餘割	cosecant	csc	反餘割	arccosecant	arccsc	正弦	sine	sin

角度單位	值								計算機中代號
轉	$0$	${\frac {1}{12}}$	${\frac {1}{8}}$	${\frac {1}{6}}$	${\frac {1}{4}}$	${\frac {1}{2}}$	${\frac {3}{4}}$	$1$	無
角度	$0^{\circ }$	$30^{\circ }$	$45^{\circ }$	$60^{\circ }$	$90^{\circ }$	$180^{\circ }$	$270^{\circ }$	$360^{\circ }$	D
弧度	$0$	${\frac {\pi }{6}}$	${\frac {\pi }{4}}$	${\frac {\pi }{3}}$	${\frac {\pi }{2}}$	$\pi$	${\frac {3\pi }{2}}$	$2\pi$	R
梯度	$0^{g}$	$33{\frac {1}{3}}^{g}$	$50^{g}$	$66{\frac {2}{3}}^{g}$	$100^{g}$	$200^{g}$	$300^{g}$	$400^{g}$	G

函數	$\sin$	$\cos$	$\tan$	$\cot$	$\sec$	$\csc$
$\sin \theta$	$\sin \theta \$	${\sqrt {1-\cos ^{2}\theta }}$	${\frac {\tan \theta }{\sqrt {1+\tan ^{2}\theta }}}$	${\frac {1}{\sqrt {1+\cot ^{2}\theta }}}$	${\frac {\sqrt {\sec ^{2}\theta -1}}{\sec \theta }}$	${\frac {1}{\csc \theta }}$
$\cos \theta$	${\sqrt {1-\sin ^{2}\theta }}$	$\cos \theta \$	${\frac {1}{\sqrt {1+\tan ^{2}\theta }}}$	${\frac {\cot \theta }{\sqrt {1+\cot ^{2}\theta }}}$	${\frac {1}{\sec \theta }}$	${\frac {\sqrt {\csc ^{2}\theta -1}}{\csc \theta }}$
$\tan \theta$	${\frac {\sin \theta }{\sqrt {1-\sin ^{2}\theta }}}$	${\frac {\sqrt {1-\cos ^{2}\theta }}{\cos \theta }}$	$\tan \theta \$	${\frac {1}{\cot \theta }}$	${\sqrt {\sec ^{2}\theta -1}}$	${\frac {1}{\sqrt {\csc ^{2}\theta -1}}}$
$\cot \theta$	${{\sqrt {1-\sin ^{2}\theta }} \over \sin \theta }$	${\cos \theta \over {\sqrt {1-\cos ^{2}\theta }}}$	${1 \over \tan \theta }$	$\cot \theta \$	${1 \over {\sqrt {\sec ^{2}\theta -1}}}$	${\sqrt {\csc ^{2}\theta -1}}$
$\sec \theta$	${1 \over {\sqrt {1-\sin ^{2}\theta }}}$	${1 \over \cos \theta }$	${\sqrt {1+\tan ^{2}\theta }}$	${{\sqrt {1+\cot ^{2}\theta }} \over \cot \theta }$	$\sec \theta \$	${\csc \theta \over {\sqrt {\csc ^{2}\theta -1}}}$
$\csc \theta$	${1 \over \sin \theta }$	${1 \over {\sqrt {1-\cos ^{2}\theta }}}$	${{\sqrt {1+\tan ^{2}\theta }} \over \tan \theta }$	${\sqrt {1+\cot ^{2}\theta }}$	${\sec \theta \over {\sqrt {\sec ^{2}\theta -1}}}$	$\csc \theta \$

名稱	函數	值
正矢, versine	$\operatorname {versin} \theta$ $\operatorname {vers} \theta$ $\operatorname {ver} \theta$	$1-\cos \theta$
餘的正矢, vercosine	$\operatorname {vercosin} \theta$	$1+\cos \theta$
餘矢, coversine	$\operatorname {coversin} \theta$ $\operatorname {cvs} \theta$	$1-\sin \theta$
餘的餘矢, covercosine	$\operatorname {covercosin} \theta$	$1+\sin \theta$
半正矢, haversine	$\operatorname {haversin} \theta$	${\frac {1-\cos \theta }{2}}$
餘的半正矢, havercosine	$\operatorname {havercosin} \theta$	${\frac {1+\cos \theta }{2}}$
半餘矢, hacoversine cohaversine	$\operatorname {hacoversin} \theta$	${\frac {1-\sin \theta }{2}}$
餘的半餘矢, hacovercosine cohavercosine	$\operatorname {hacovercosin} \theta$	${\frac {1+\sin \theta }{2}}$
外正割，exsecant	$\operatorname {exsec} \theta$	$\sec \theta -1$
外餘割，excosecant	$\operatorname {excsc} \theta$	$\csc \theta -1$
弦函數, chord	$\operatorname {crd} \theta$	$2\sin \left({\frac {\theta }{2}}\right)$
純虛數指數函數, cosine and imaginary unit sine	$\operatorname {cis} \theta$	$\cos \theta +i\;\sin \theta$
輻角,Argument	$\arg x$	$\operatorname {Im} (\ln x)$
弧,Arc	arc x	$\theta$

移位 ${\tfrac {\pi }{2}}$	移位 $\pi$	移位 ${\tfrac {3\pi }{2}}$	移位 $2\pi$
移位 ${\tfrac {\pi }{2}}$	$\tan$ 和 $\cot$ 的周期	移位 ${\tfrac {3\pi }{2}}$	$\sin$ , $\cos$ , $\csc$ 和 $\sec$ 的周期
${\begin{aligned}\sin(\theta +{\tfrac {\pi }{2}})&=+\cos \theta \\\cos(\theta +{\tfrac {\pi }{2}})&=-\sin \theta \\\tan(\theta +{\tfrac {\pi }{2}})&=-\cot \theta \\\cot(\theta +{\tfrac {\pi }{2}})&=-\tan \theta \\\sec(\theta +{\tfrac {\pi }{2}})&=-\csc \theta \\\csc(\theta +{\tfrac {\pi }{2}})&=+\sec \theta \end{aligned}}$	${\begin{aligned}\sin(\theta +\pi )&=-\sin \theta \\\cos(\theta +\pi )&=-\cos \theta \\\tan(\theta +\pi )&=+\tan \theta \\\cot(\theta +\pi )&=+\cot \theta \\\sec(\theta +\pi )&=-\sec \theta \\\csc(\theta +\pi )&=-\csc \theta \end{aligned}}$	${\begin{aligned}\sin(\theta +{\tfrac {3\pi }{2}})&=-\cos \theta \\\cos(\theta +{\tfrac {3\pi }{2}})&=+\sin \theta \\\tan(\theta +{\tfrac {3\pi }{2}})&=-\cot \theta \\\cot(\theta +{\tfrac {3\pi }{2}})&=-\tan \theta \\\sec(\theta +{\tfrac {3\pi }{2}})&=+\csc \theta \\\csc(\theta +{\tfrac {3\pi }{2}})&=-\sec \theta \end{aligned}}$	${\begin{aligned}\sin(\theta +2\pi )&=+\sin \theta \\\cos(\theta +2\pi )&=+\cos \theta \\\tan(\theta +2\pi )&=+\tan \theta \\\cot(\theta +2\pi )&=+\cot \theta \\\sec(\theta +2\pi )&=+\sec \theta \\\csc(\theta +2\pi )&=+\csc \theta \end{aligned}}$

三角恒等式

符号

基本關係

其他函數的基本關係

对称、移位和周期

對稱

移位和周期

角的和差恆等式

正弦與余弦的無限多項和

正切的有限多项和

多倍角公式

雙倍角、三倍角和半角公式

n倍角公式

其他函數的倍半角公式

幂简约公式

数值连乘

常見的恆等式

积化和差与和差化积恆等式

平方差公式

其他恆等式

托勒密定理

三角函數與雙曲函數的恆等式

线性组合

反三角函数

无限乘积公式

微積分

蘊涵

指数定义

参见

註釋

正弦	$\sin(\alpha \pm \beta )=\sin \alpha \cos \beta \pm \cos \alpha \sin \beta \,$
余弦	$\cos(\alpha \pm \beta )=\cos \alpha \cos \beta \mp \sin \alpha \sin \beta \,$
正切	$\tan(\alpha \pm \beta )={\frac {\tan \alpha \pm \tan \beta }{1\mp \tan \alpha \tan \beta }}$
余切	$\cot(\alpha \pm \beta )={\frac {\cot \alpha \cot \beta \mp 1}{\cot \beta \pm \cot \alpha }}$
正割	$\sec(\alpha \pm \beta )={\frac {\sec \alpha \sec \beta }{1\mp \tan \alpha \tan \beta }}$
余割	$\csc(\alpha \pm \beta )={\frac {\csc \alpha \csc \beta }{\cot \beta \pm \cot \alpha }}$
注意正負號的對應。 ${\begin{aligned}x\pm y=a\pm b&\Rightarrow \ x+y=a+b\\&{\mbox{and}}\ x-y=a-b\end{aligned}}$ ${\begin{aligned}x\pm y=a\mp b&\Rightarrow \ x+y=a-b\\&{\mbox{and}}\ x-y=a+b\end{aligned}}$

$T_{n}$ 是 $n$ 次切比雪夫多项式	$\cos n\theta =T_{n}\cos \theta \,$
$S_{n}$ 是 $n$ 次伸展多项式	$\sin ^{2}n\theta =S_{n}\sin ^{2}\theta \,$
棣莫弗定理， $i$ 是虚单位	$\cos n\theta +i\sin n\theta =(\cos \theta +i\sin \theta )^{n}\,$

		弦	切	割
雙倍角公式	正	${\begin{aligned}\sin 2\theta &=2\sin \theta \cos \theta \ \\&={\frac {2\tan \theta }{1+\tan ^{2}\theta }}\end{aligned}}$	${\begin{aligned}\tan 2\theta &={\frac {2\tan \theta }{1-\tan ^{2}\theta }}\ \\&={\frac {1}{1-\tan \theta }}-{\frac {1}{1+\tan \theta }}\end{aligned}}$	${\begin{aligned}\sec 2\theta &={\frac {\sec ^{2}\theta }{1-\tan ^{2}\theta }}\\&={\frac {\sec ^{2}\theta }{2-\sec ^{2}\theta }}\end{aligned}}$
雙倍角公式	餘	${\begin{aligned}\cos 2\theta &=\cos ^{2}\theta -\sin ^{2}\theta \\&=2\cos ^{2}\theta -1\\&=1-2\sin ^{2}\theta \\&={\frac {1-\tan ^{2}\theta }{1+\tan ^{2}\theta }}\end{aligned}}$	${\begin{aligned}\cot 2\theta &={\frac {\cot ^{2}\theta -1}{2\cot \theta }}\\&={\frac {\cot \theta -\tan \theta }{2}}\end{aligned}}$	${\begin{aligned}\csc 2\theta &={\frac {\csc ^{2}\theta }{2\cot \theta }}\\&={\frac {\sec \theta \csc \theta }{2}}\end{aligned}}$
降次公式	正	$\sin ^{2}\theta ={\frac {1-\cos 2\theta }{2}}$	$\tan ^{2}\theta ={\frac {1-\cos 2\theta }{1+\cos 2\theta }}$
降次公式	餘	$\cos ^{2}\theta ={\frac {1+\cos 2\theta }{2}}$	$\cot ^{2}\theta ={\frac {1+\cos 2\theta }{1-\cos 2\theta }}$
三倍角公式	正	${\begin{aligned}\sin 3\theta &=3\sin \theta -4\sin ^{3}\theta \\&=4\sin \theta \sin \left({\frac {\pi }{3}}-\theta \right)\sin \left({\frac {\pi }{3}}+\theta \right)\end{aligned}}$	${\begin{aligned}\tan 3\theta &={\frac {3\tan \theta -\tan ^{3}\theta }{1-3\tan ^{2}\theta }}\\&=\tan \theta \tan \left({\frac {\pi }{3}}-\theta \right)\tan \left({\frac {\pi }{3}}+\theta \right)\end{aligned}}$	${\begin{aligned}\sec 3\theta &={\frac {\sec ^{3}\theta }{4-3\sec ^{2}\theta }}\\&={\dfrac {1}{4\cos \theta \cos \left({\dfrac {\pi }{3}}-\theta \right)\cos \left({\dfrac {\pi }{3}}+\theta \right)}}\end{aligned}}$
三倍角公式	餘	${\begin{aligned}\cos 3\theta &=4\cos ^{3}\theta -3\cos \theta \\&=4\cos \theta \cos \left({\frac {\pi }{3}}-\theta \right)\cos \left({\frac {\pi }{3}}+\theta \right)\end{aligned}}$	${\begin{aligned}\cot 3\theta &={\frac {\cot ^{3}\theta -3\cot \theta }{3\cot ^{2}\theta -1}}\\&=\cot \theta \cot \left({\frac {\pi }{3}}-\theta \right)\cot \left({\frac {\pi }{3}}+\theta \right)\end{aligned}}$	${\begin{aligned}\csc 3\theta &={\frac {\csc ^{3}\theta }{3\csc ^{2}\theta -4}}\\&={\dfrac {1}{4\sin \theta \sin \left({\dfrac {\pi }{3}}-\theta \right)\sin \left({\dfrac {\pi }{3}}+\theta \right)}}\end{aligned}}$
半角公式	正	$\sin {\frac {\theta }{2}}=\pm \,{\sqrt {\frac {1-\cos \theta }{2}}}$	${\begin{aligned}\tan {\frac {\theta }{2}}&=\csc \theta -\cot \theta \\&=\pm \,{\sqrt {1-\cos \theta \over 1+\cos \theta }}\\&={\frac {\sin \theta }{1+\cos \theta }}\\&={\frac {1-\cos \theta }{\sin \theta }}\\&=\pm {\sqrt {\cot ^{2}\theta +1}}-\cot \theta \end{aligned}}$	$\sec {\frac {\theta }{2}}=\pm \,{\sqrt {\frac {2\sec \theta }{\sec \theta +1}}}$
半角公式	餘	$\cos {\frac {\theta }{2}}=\pm \,{\sqrt {\frac {1+\cos \theta }{2}}}$	${\begin{aligned}\cot {\frac {\theta }{2}}&=\csc \theta +\cot \theta \\&=\pm \,{\sqrt {1+\cos \theta \over 1-\cos \theta }}\\&={\frac {\sin \theta }{1-\cos \theta }}\\&={\frac {1+\cos \theta }{\sin \theta }}\\&=\pm {\sqrt {\cot ^{2}\theta +1}}+\cot \theta \end{aligned}}$	$\csc {\frac {\theta }{2}}=\pm \,{\sqrt {\frac {2\sec \theta }{\sec \theta -1}}}$

$n$ 倍角公式
$\sin n\theta =\sum _{k=0}^{n}{\binom {n}{k}}\cos ^{k}\theta \,\sin ^{n-k}\theta \,\sin \left[{\frac {1}{2}}(n-k)\pi \right]=\sin \theta \sum _{k=0}^{\lfloor {\frac {n-1}{2}}\rfloor }(-1)^{k}{\binom {n-1-k}{k}}~(2\cos \theta )^{n-1-2k}$ （第二类切比雪夫多项式）
$\cos n\theta =\sum _{k=0}^{n}{\binom {n}{k}}\cos ^{k}\theta \,\sin ^{n-k}\theta \,\cos \left[{\frac {1}{2}}(n-k)\pi \right]={\frac {1}{2}}\sum _{k=0}^{\lfloor {\frac {n}{2}}\rfloor }(-1)^{k}{\frac {n}{n-k}}{\binom {n-k}{k}}~(2\cos \theta )^{n-2k}$ （第一类切比雪夫多项式）
$\tan n\theta ={\frac {\displaystyle \sum _{k=1}^{\left[{\frac {n}{2}}\right]}(-1)^{k+1}{\binom {n}{2k-1}}\tan ^{2k-1}\theta }{\displaystyle \sum _{k=1}^{\left[{\frac {n+1}{2}}\right]}(-1)^{k+1}{\binom {n}{2(k-1)}}\tan ^{2(k-1)}\theta }}$
$n$ 倍遞迴公式
$\tan \,n\theta ={\frac {\tan(n{-}1)\theta +\tan \theta }{1-\tan(n{-}1)\theta \,\tan \theta }}$ 、 $\cot \,n\theta ={\frac {\cot(n{-}1)\theta \,\cot \theta -1}{\cot(n{-}1)\theta +\cot \theta }}$ 。(遞迴關係)

正弦	餘弦	其他
$\sin ^{2}\theta ={\frac {1-\cos 2\theta }{2}}$	$\cos ^{2}\theta ={\frac {1+\cos 2\theta }{2}}$	$\sin ^{2}\theta \cos ^{2}\theta ={\frac {1-\cos 4\theta }{8}}$
$\sin ^{3}\theta ={\frac {3\sin \theta -\sin 3\theta }{4}}$	$\cos ^{3}\theta ={\frac {3\cos \theta +\cos 3\theta }{4}}$	$\sin ^{3}\theta \cos ^{3}\theta ={\frac {3\sin 2\theta -\sin 6\theta }{32}}$
$\sin ^{4}\theta ={\frac {3-4\cos 2\theta +\cos 4\theta }{8}}$	$\cos ^{4}\theta ={\frac {3+4\cos 2\theta +\cos 4\theta }{8}}$	$\sin ^{4}\theta \cos ^{4}\theta ={\frac {3-4\cos 4\theta +\cos 8\theta }{128}}$
$\sin ^{5}\theta ={\frac {10\sin \theta -5\sin 3\theta +\sin 5\theta }{16}}$	$\cos ^{5}\theta ={\frac {10\cos \theta +5\cos 3\theta +\cos 5\theta }{16}}$	$\sin ^{5}\theta \cos ^{5}\theta ={\frac {10\sin 2\theta -5\sin 6\theta +\sin 10\theta }{512}}$

	餘弦	正弦
如果 $n$ 是奇數	$\cos ^{n}\theta ={\frac {2}{2^{n}}}\sum _{k=0}^{\frac {n-1}{2}}{\binom {n}{k}}\cos {[(n-2k)\theta ]}$	$\sin ^{n}\theta ={\frac {2}{2^{n}}}\sum _{k=0}^{\frac {n-1}{2}}(-1)^{\left({\frac {n-1}{2}}-k\right)}{\binom {n}{k}}\sin {[(n-2k)\theta ]}$
如果 $n$ 是偶數	$\cos ^{n}\theta ={\frac {1}{2^{n}}}{\binom {n}{\frac {n}{2}}}+{\frac {2}{2^{n}}}\sum _{k=0}^{{\frac {n}{2}}-1}{\binom {n}{k}}\cos {[(n-2k)\theta ]}$	$\sin ^{n}\theta ={\frac {1}{2^{n}}}{\binom {n}{\frac {n}{2}}}+{\frac {2}{2^{n}}}\sum _{k=0}^{{\frac {n}{2}}-1}(-1)^{\left({\frac {n}{2}}-k\right)}{\binom {n}{k}}\cos {[(n-2k)\theta ]}$

积化和差	和差化积
$\sin \alpha \cos \beta ={\sin(\alpha +\beta )+\sin(\alpha -\beta ) \over 2}$	$\sin \alpha +\sin \beta =2\sin {\frac {\alpha +\beta }{2}}\cos {\frac {\alpha -\beta }{2}}$
$\cos \alpha \sin \beta ={\sin(\alpha +\beta )-\sin(\alpha -\beta ) \over 2}$	$\sin \alpha -\sin \beta =2\cos {\alpha +\beta \over 2}\sin {\alpha -\beta \over 2}$
$\cos \alpha \cos \beta ={\cos(\alpha +\beta )+\cos(\alpha -\beta ) \over 2}$	$\cos \alpha +\cos \beta =2\cos {\frac {\alpha +\beta }{2}}\cos {\frac {\alpha -\beta }{2}}$
$\sin \alpha \sin \beta =-{\cos(\alpha +\beta )-\cos(\alpha -\beta ) \over 2}$	$\cos \alpha -\cos \beta =-2\sin {\alpha +\beta \over 2}\sin {\alpha -\beta \over 2}$

三角函數	雙曲函數
$\sin \theta =-i\sinh {i\theta }\,$	$\sinh {\theta }=i\sin {(-i\theta )}\,$
$\cos {\theta }=\cosh {i\theta }\,$	$\cosh {\theta }=\cos {(-i\theta )}\,$
$\tan \theta =-i\tanh {i\theta }\,$	$\tanh {\theta }=i\tan {(-i\theta )}\,$
$\cot {\theta }=i\coth {i\theta }\,$	$\coth \theta =-i\cot {(-i\theta )}\,$
$\sec {\theta }=\operatorname {sech} {\,i\theta }\,$	$\operatorname {sech} {\theta }=\sec {(-i\theta )}\,$
$\csc {\theta }=i\;\operatorname {csch} {\,i\theta }\,$	$\operatorname {csch} \theta =-i\csc {(-i\theta )}\,$

函数	反函数
$\sin \theta ={\frac {e^{{i}\theta }-e^{-{i}\theta }}{2{i}}}\,$	$\arcsin x=-{i}\ln \left({i}x+{\sqrt {1-x^{2}}}\right)\,$
$\cos \theta ={\frac {e^{{i}\theta }+e^{-{i}\theta }}{2}}\,$	$\arccos x=-{i}\ln \left(x+{\sqrt {x^{2}-1}}\right)\,$
$\tan \theta ={\frac {e^{{i}\theta }-e^{-{i}\theta }}{{i}(e^{{i}\theta }+e^{-{i}\theta })}}\,$	$\arctan x={\frac {i}{2}}\ln \left({\frac {{i}+x}{{i}-x}}\right)\,$
$\csc \theta ={\frac {2{i}}{e^{{i}\theta }-e^{-{i}\theta }}}\,$	$\operatorname {arccsc} x=-{i}\ln \left({\tfrac {i}{x}}+{\sqrt {1-{\tfrac {1}{x^{2}}}}}\right)\,$
$\sec \theta ={\frac {2}{e^{{i}\theta }+e^{-{i}\theta }}}\,$	$\operatorname {arcsec} x=-{i}\ln \left({\tfrac {1}{x}}+{\sqrt {1-{\tfrac {i}{x^{2}}}}}\right)\,$
$\cot \theta ={\frac {{i}(e^{{i}\theta }+e^{-{i}\theta })}{e^{{i}\theta }-e^{-{i}\theta }}}\,$	$\operatorname {arccot} x={\frac {i}{2}}\ln \left({\frac {{i}-x}{{i}+x}}\right)\,$

$\operatorname {cis} \,\theta =e^{{i}\theta }\,$	$\operatorname {arccis} \,x=-{i}\ln x\,$

$\sinh \theta ={\frac {e^{\theta }-e^{-\theta }}{2}}\,$	$\operatorname {arcsinh} \,x=\ln \left(x\pm {\sqrt {x^{2}+1}}\right)\,$
$\cosh \theta ={\frac {e^{\theta }+e^{-\theta }}{2}}\,$	$\operatorname {arccosh} \,x=\ln \left(x\pm {\sqrt {x^{2}-1}}\right)=\pm \ln \left(x+{\sqrt {x^{2}-1}}\right)\,$
$\tanh \theta ={\frac {\sinh \theta }{\cosh \theta }}={\frac {e^{\theta }-e^{-\theta }}{e^{\theta }+e^{-\theta }}}\,$	$\operatorname {arctanh} \,x={\frac {1}{2}}\ln \left({\frac {1+x}{1-x}}\right)\,$