数学三角函数公式表

时间:2025-02-13 21:46:58 单机游戏

三角函数公式表包括以下主要内容:

基本三角函数定义

正弦函数(sin):对边/斜边

余弦函数(cos):邻边/斜边

正切函数(tan):对边/邻边

余切函数(cot):邻边/对边

正割函数(sec):斜边/邻边

余割函数(csc):斜边/对边

三角函数的基本关系

平方关系:$\sin^2\alpha + \cos^2\alpha = 1$

商数关系:$\tan\alpha = \frac{\sin\alpha}{\cos\alpha}$,$\cot\alpha = \frac{\cos\alpha}{\sin\alpha}$

倒数关系:$\frac{1}{\tan\alpha} = \cot\alpha$,$\frac{1}{\cot\alpha} = \tan\alpha$

诱导公式

终边相同的角的三角函数值相等:$\sin(2k\pi + \alpha) = \sin\alpha$,$\cos(2k\pi + \alpha) = \cos\alpha$,$\tan(2k\pi + \alpha) = \tan\alpha$,$\cot(2k\pi + \alpha) = \cot\alpha$($k \in \mathbb{Z}$)

$\pi + \alpha$的三角函数值:$\sin(\pi + \alpha) = -\sin\alpha$,$\cos(\pi + \alpha) = -\cos\alpha$,$\tan(\pi + \alpha) = \tan\alpha$,$\cot(\pi + \alpha) = \cot\alpha$

$-\alpha$的三角函数值:$\sin(-\alpha) = -\sin\alpha$,$\cos(-\alpha) = \cos\alpha$,$\tan(-\alpha) = -\tan\alpha$,$\cot(-\alpha) = -\cot\alpha$

$\pi - \alpha$的三角函数值:$\sin(\pi - \alpha) = \sin\alpha$,$\cos(\pi - \alpha) = -\cos\alpha$,$\tan(\pi - \alpha) = -\tan\alpha$,$\cot(\pi - \alpha) = -\cot\alpha$

$2\pi - \alpha$的三角函数值:$\sin(2\pi - \alpha) = -\sin\alpha$,$\cos(2\pi - \alpha) = \cos\alpha$,$\tan(2\pi - \alpha) = -\tan\alpha$,$\cot(2\pi - \alpha) = -\cot\alpha$

$\frac{\pi}{2} \pm \alpha$及$\frac{3\pi}{2} \pm \alpha$的三角函数值:

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

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

$\sin(\frac{3\pi}{2} + \alpha) = -\cos\alpha$,$\cos(\frac{3\pi}{2} + \alpha) = -\sin\alpha$,$\tan(\frac{3\pi}{2} + \alpha) = \cot\alpha$,$\cot(\frac{3\pi}{2} + \alpha) = \tan\alpha$

$\sin(\frac{3\pi}{2} - \alpha) = -\cos\alpha$,$\cos(\frac{3\pi}{2} - \alpha) = -\sin\alpha$,$\tan(\frac{3\pi}{2} - \alpha) = -\cot\alpha$,$\cot(\frac{3\pi}{2} - \alpha) = -\tan\alpha$

和差化积公式

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