常见三角函数公式

两角和差

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

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

$\cos (\alpha+\beta)=\cos \alpha \cos \beta-\sin \alpha \sin \beta$

$\cos (\alpha-\beta)=\cos \alpha \cos \beta+\sin \alpha \sin \beta$

$\tan (\alpha+\beta)=\frac{\tan \alpha+\tan \beta}{1-\tan \alpha \tan \beta}$

$\tan (\alpha-\beta)=\frac{\tan \alpha-\tan \beta}{1+\tan \alpha \tan \beta}$

转换公式

$\sin^{2}\alpha+\cos^{2}\alpha=1$

$1+\tan^{2}\alpha=\sec^{2}\alpha$

$1+\cot^{2}\alpha=\csc^{2}\alpha$

和差化积

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

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

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

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

积化和差

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

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

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

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

欧拉公式

$e^{i x}=\cos x+i \sin x$

未完待续