Trigonometric Identity

三角恒等式

两角和差
$\cos (\alpha \pm \beta )=\cos \alpha \cos \beta \mp \sin \alpha \sin \beta \\ \sin (\alpha \pm \beta )=\sin \alpha \cos \beta \pm \cos \alpha \sin \beta \\ \tan (\alpha \pm \beta )=\frac{\tan \alpha \pm \tan \beta } {1\mp \tan \alpha \tan \beta}$

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

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

倍角公式
$\sin 2\alpha=2\sin \alpha \cos \alpha =\frac{2}{\tan \alpha +\cot \alpha} \\ \cos 2\alpha=\cos^2 \alpha-\sin^2 \alpha \\ \tan 2\alpha =\frac{2\tan \alpha}{1-\tan^2 \alpha} \\ \cot 2\alpha=\frac{\cot^2\alpha -1}{2\cot \alpha} \\ \sin 3\alpha=3\sin \alpha -4\sin^3 \alpha \\ \cos 3\alpha=4\cos^3 \alpha-3\cos \alpha \\ \tan 3\alpha=\frac{3\tan \alpha -\tan^3 \alpha}{1-3\tan^2 \alpha} \\ \cot 3\alpha=\frac{\cot^3 \alpha -3\cot \alpha}{3\cot \alpha -1}$

半角公式 （正负由 $\frac{\alpha}{2}$所在的象限决定）
$\sin \frac{\alpha}{2}=\pm \sqrt{\frac{1-\cos \alpha }{2}} \\ \cos\frac{\alpha}{2}=\pm \sqrt{\frac{1+\cos \alpha }{2}} \\ \tan \frac{\alpha}{2}=\pm \sqrt {\frac{1-\cos \alpha }{1+\cos \alpha }}=\frac{\sin \alpha}{1+\cos \alpha}=\frac{1-\cos \alpha}{\sin \alpha} \\ \cot\frac{\alpha}{2}=\pm \sqrt {\frac{1+\cos \alpha}{1-\cos \alpha}}=\frac{1+\cos \alpha}{\sin \alpha} =\frac{\sin \alpha}{1-\cot \alpha}$

辅助角公式
$a\sin \alpha +b\cos \alpha =\sqrt{a^2+b^2}\sin (\alpha +\arctan\frac{b}{a}) \\ a\sin \alpha +b\cos \alpha =\sqrt{a^2+b^2}\cos (\alpha -\arctan\frac{a}{b})$

万能公式
$\sin\alpha=\frac{2\tan\frac{\alpha}{2}}{1+\tan ^2\frac{\alpha}{2}} \\ \cos\alpha=\frac{1-\tan ^2\frac{\alpha}{2}}{1+\tan ^2\frac{\alpha}{2}} \\ \tan\alpha=\frac{2\tan\frac{\alpha}{2}}{1-\tan ^2\frac{\alpha}{2}}$

降幂公式
$\sin^2 \alpha=\frac{1-\cos 2\alpha}{2} \\ \cos^2 \alpha=\frac{1+\cos 2\alpha}{2} \\ \tan^2 \alpha=\frac{1-\cos 2\alpha}{1+\cos 2\alpha}$

正弦定理(R为外接圆半径)
$\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}=2R$

余弦定理
$c^2=a^2+b^2-2ab\cos C$