三角函数公式

和差角公式

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

倍角公式

\(\sin 2\theta=2\sin\theta \cos\theta\)
\(\cos 2\theta=\cos^2\theta-\sin^2 \theta=2\cos^2\theta-1=1-2\sin^2\theta\)

平方化倍角

\(\sin^2\theta=\frac{1-\cos2\theta}{2}\)
\(\cos^2\theta=\frac{1+\cos2\theta}{2}\)

半角公式

\(\sin\frac{\theta}{2}=\pm\sqrt{\frac{1-\cos\theta}{2}}\)
\(\cos\frac{\theta}{2}=\pm\sqrt{\frac{1+\cos\theta}{2}}\)

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

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

辅助角公式

\(A \sin \alpha + B \cos \alpha = \sqrt{A^2+B^2} \sin (\alpha + \varphi)\)，其中 \(\varphi = \arctan \frac{B}{A}\)。

万能公式

\(\sin \alpha = \frac{2\tan \frac{\alpha}{2} }{1+ \tan^2 \frac{\alpha}{2}}\)
\(\tan \alpha=\frac{2 \tan \frac{\alpha}{2}}{ 1- \tan^2 \frac{\alpha}{2}}\)
\(\cos \alpha= \frac{ 1- \tan^2 \frac{\alpha}{2}}{1+ \tan^2 \frac{\alpha}{2}}\)
其中 \(\alpha \neq 2k \pi + \pi, k\in \mathbb{Z}\)。

正弦定理

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

余弦定理

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