三角函数公式大全

一、特殊角的三角函数值#

基本特殊角#

角度	$0^\circ$	$30^\circ$	$45^\circ$	$60^\circ$	$90^\circ$	$180^\circ$	$270^\circ$	$360^\circ$
弧度	$0$	$\dfrac{\pi}{6}$	$\dfrac{\pi}{4}$	$\dfrac{\pi}{3}$	$\dfrac{\pi}{2}$	$\pi$	$\dfrac{3\pi}{2}$	$2\pi$
$\sin$	$0$	$\dfrac{1}{2}$	$\dfrac{\sqrt{2}}{2}$	$\dfrac{\sqrt{3}}{2}$	$1$	$0$	$-1$	$0$
$\cos$	$1$	$\dfrac{\sqrt{3}}{2}$	$\dfrac{\sqrt{2}}{2}$	$\dfrac{1}{2}$	$0$	$-1$	$0$	$1$
$\tan$	$0$	$\dfrac{\sqrt{3}}{3}$	$1$	$\sqrt{3}$	不存在	$0$	不存在	$0$
$\cot$	不存在	$\sqrt{3}$	$1$	$\dfrac{\sqrt{3}}{3}$	$0$	不存在	$0$	不存在

扩展特殊角#

角度	$15^\circ$	$75^\circ$	$105^\circ$	$120^\circ$	$135^\circ$	$150^\circ$
弧度	$\dfrac{\pi}{12}$	$\dfrac{5\pi}{12}$	$\dfrac{7\pi}{12}$	$\dfrac{2\pi}{3}$	$\dfrac{3\pi}{4}$	$\dfrac{5\pi}{6}$
$\sin$	$\dfrac{\sqrt{6}-\sqrt{2}}{4}$	$\dfrac{\sqrt{6}+\sqrt{2}}{4}$	$\dfrac{\sqrt{6}+\sqrt{2}}{4}$	$\dfrac{\sqrt{3}}{2}$	$\dfrac{\sqrt{2}}{2}$	$\dfrac{1}{2}$
$\cos$	$\dfrac{\sqrt{6}+\sqrt{2}}{4}$	$\dfrac{\sqrt{6}-\sqrt{2}}{4}$	$-\dfrac{\sqrt{6}-\sqrt{2}}{4}$	$-\dfrac{1}{2}$	$-\dfrac{\sqrt{2}}{2}$	$-\dfrac{\sqrt{3}}{2}$
$\tan$	$2-\sqrt{3}$	$2+\sqrt{3}$	$-2-\sqrt{3}$	$-\sqrt{3}$	$-1$	$-\dfrac{\sqrt{3}}{3}$

二、基本关系#

倒数关系#

$\sin x \cdot \csc x = 1 \qquad \cos x \cdot \sec x = 1 \qquad \tan x \cdot \cot x = 1$

商数关系#

$\tan x = \frac{\sin x}{\cos x} \qquad \cot x = \frac{\cos x}{\sin x}$

平方关系（同角三角函数）#

$\sin^2 x + \cos^2 x = 1$

$1 + \tan^2 x = \sec^2 x$

$1 + \cot^2 x = \csc^2 x$

三、诱导公式（奇变偶不变，符号看象限）#

基本诱导#

公式	$\sin$	$\cos$	$\tan$
$-\alpha$	$-\sin\alpha$	$\cos\alpha$	$-\tan\alpha$
$\pi/2 - \alpha$	$\cos\alpha$	$\sin\alpha$	$\cot\alpha$
$\pi/2 + \alpha$	$\cos\alpha$	$-\sin\alpha$	$-\cot\alpha$
$\pi - \alpha$	$\sin\alpha$	$-\cos\alpha$	$-\tan\alpha$
$\pi + \alpha$	$-\sin\alpha$	$-\cos\alpha$	$\tan\alpha$
$3\pi/2 - \alpha$	$-\cos\alpha$	$-\sin\alpha$	$\cot\alpha$
$3\pi/2 + \alpha$	$-\cos\alpha$	$\sin\alpha$	$-\cot\alpha$
$2\pi - \alpha$	$-\sin\alpha$	$\cos\alpha$	$-\tan\alpha$

四、和差角公式#

$\sin(\alpha \pm \beta) = \sin\alpha\cos\beta \pm \cos\alpha\sin\beta$

$\cos(\alpha \pm \beta) = \cos\alpha\cos\beta \mp \sin\alpha\sin\beta$

$\tan(\alpha \pm \beta) = \frac{\tan\alpha \pm \tan\beta}{1 \mp \tan\alpha\tan\beta}$

五、倍角公式#

二倍角#

$\sin 2\alpha = 2\sin\alpha\cos\alpha$

$\cos 2\alpha = \cos^2\alpha - \sin^2\alpha = 2\cos^2\alpha - 1 = 1 - 2\sin^2\alpha$

$\tan 2\alpha = \frac{2\tan\alpha}{1 - \tan^2\alpha}$

三倍角#

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

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

六、半角公式#

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

$\cos\frac{\alpha}{2} = \pm\sqrt{\frac{1 + \cos\alpha}{2}}$

$\tan\frac{\alpha}{2} = \pm\sqrt{\frac{1 - \cos\alpha}{1 + \cos\alpha}} = \frac{\sin\alpha}{1 + \cos\alpha} = \frac{1 - \cos\alpha}{\sin\alpha}$

七、和差化积#

$\sin\alpha + \sin\beta = 2\sin\frac{\alpha+\beta}{2}\cos\frac{\alpha-\beta}{2}$

$\sin\alpha - \sin\beta = 2\cos\frac{\alpha+\beta}{2}\sin\frac{\alpha-\beta}{2}$

$\cos\alpha + \cos\beta = 2\cos\frac{\alpha+\beta}{2}\cos\frac{\alpha-\beta}{2}$

$\cos\alpha - \cos\beta = -2\sin\frac{\alpha+\beta}{2}\sin\frac{\alpha-\beta}{2}$

八、积化和差#

$\sin\alpha\cos\beta = \frac{1}{2}[\sin(\alpha+\beta) + \sin(\alpha-\beta)]$

$\cos\alpha\sin\beta = \frac{1}{2}[\sin(\alpha+\beta) - \sin(\alpha-\beta)]$

$\cos\alpha\cos\beta = \frac{1}{2}[\cos(\alpha+\beta) + \cos(\alpha-\beta)]$

$\sin\alpha\sin\beta = -\frac{1}{2}[\cos(\alpha+\beta) - \cos(\alpha-\beta)]$

九、万能公式#

设 $t = \tan\dfrac{\alpha}{2}$ ，则：

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

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

$\tan\alpha = \frac{2t}{1 - t^2}$

十、辅助角公式#

$a\sin x + b\cos x = \sqrt{a^2 + b^2}\,\sin(x + \varphi)$

其中 $\varphi$ 满足 $\displaystyle\sin\varphi = \frac{b}{\sqrt{a^2+b^2}}$ ， $\displaystyle\cos\varphi = \frac{a}{\sqrt{a^2+b^2}}$ 。

十一、反三角函数#

函数	定义域	值域	性质
$y = \arcsin x$	$[-1, 1]$	$\left[-\dfrac{\pi}{2}, \dfrac{\pi}{2}\right]$	奇函数
$y = \arccos x$	$[-1, 1]$	$[0, \pi]$	$\arccos(-x) = \pi - \arccos x$
$y = \arctan x$	$\mathbb{R}$	$\left(-\dfrac{\pi}{2}, \dfrac{\pi}{2}\right)$	奇函数
$y = \operatorname{arccot} x$	$\mathbb{R}$	$(0, \pi)$	$\operatorname{arccot}(-x) = \pi - \operatorname{arccot} x$

恒等式#

$\arcsin x + \arccos x = \frac{\pi}{2}$

$\arctan x + \operatorname{arccot} x = \frac{\pi}{2}$

十二、正余弦定理（三角形中）#

正弦定理#

$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R$

其中 $R$ 为外接圆半径。

余弦定理#

$a^2 = b^2 + c^2 - 2bc\cos A$

$b^2 = a^2 + c^2 - 2ac\cos B$

$c^2 = a^2 + b^2 - 2ab\cos C$

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