# 高中数学/必修四/第三章：三角恒等变换

## 3.1 两角和与差的正弦、余弦和正切公式

### 3.1.1 两角差的余弦公式

${\displaystyle \cos(\alpha -\beta )=\cos \alpha \cos \beta +\sin \alpha \sin \beta }$


### 3.1.2 两角和与差的正余弦、正切公式

${\displaystyle \sin(\alpha \pm \beta )=\sin \alpha \cos \beta \pm \cos \alpha \sin \beta }$

${\displaystyle \cos(\alpha \pm \beta )=\cos \alpha \cos \beta \mp \sin \alpha \sin \beta }$

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

### 3.1.3 二倍角的正余弦、正切公式

${\displaystyle \sin 2\theta =2\sin \theta \cos \theta }$

${\displaystyle \cos 2\theta =\cos ^{2}\theta -\sin ^{2}\theta =2\cos ^{2}\theta -1=1-2\sin ^{2}\theta }$

${\displaystyle \tan 2\theta ={\frac {2\tan \theta }{1-\tan ^{2}\theta }}}$

## 3.2 简单的三角恒等变换

### 3.2.1 半角公式

${\displaystyle \sin 2\theta =2\sin \theta \cos \theta }$

${\displaystyle \cos 2\theta =\cos ^{2}\theta -\sin ^{2}\theta =2\cos ^{2}\theta -1=1-2\sin ^{2}\theta }$

${\displaystyle \tan 2\theta ={\frac {2\tan \theta }{1-\tan ^{2}\theta }}}$

### 3.2.2 万能公式

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

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

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

### 3.2.4 降幂公式

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

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

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

### 3.2.5 积化和差公式

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

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

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

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

### 3.2.6 和差化积公式

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

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

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

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

### 3.2.7 辅助角公式

${\displaystyle a\sin \alpha +b\cos \alpha ={\sqrt {a^{2}+b^{2}}}\cdot \sin \left(\alpha +\arctan {\frac {b}{a}}\right)}$