诱导公式

奇变偶不变，符号看象限。

• $\sin (2k\pi +\alpha )=\phantom{0}$$\phantom{0}\sin \alpha$
• $\cos (2k\pi +\alpha )=\phantom{0}$$\phantom{0}\cos \alpha$
• $\tan (2k\pi +\alpha )=\phantom{0}$$\phantom{0}\tan \alpha$

• $\sin (2k\pi -\alpha )=\phantom{0}$$\phantom{0}-\phantom{0}$$\phantom{0}\sin \alpha$
• $\cos (2k\pi -\alpha )=\phantom{0}$$\phantom{0}\cos \alpha$
• $\tan (2k\pi -\alpha )=\phantom{0}$$\phantom{0}-\phantom{0}$$\phantom{0}\tan \alpha$

• $\sin (\pi +\alpha )=\phantom{0}$$\phantom{0}-\phantom{0}$$\phantom{0}\sin \alpha$
• $\cos (\pi +\alpha )=\phantom{0}$$\phantom{0}-\phantom{0}$$\phantom{0}\cos \alpha$
• $\tan (\pi +\alpha )=\phantom{0}$$\phantom{0}\tan \alpha$

• $\sin (\pi -\alpha )=\phantom{0}$$\phantom{0}\sin \alpha$
• $\cos (\pi -\alpha )=\phantom{0}$$\phantom{0}-\phantom{0}$$\phantom{0}\cos \alpha$
• $\tan (\pi -\alpha )=\phantom{0}$$\phantom{0}-\phantom{0}$$\phantom{0}\tan \alpha$

• $\sin ({\displaystyle \frac{\pi }{2}}+\alpha )=\phantom{0}$$\phantom{0}\cos \alpha$
• $\cos ({\displaystyle \frac{\pi }{2}}+\alpha )=\phantom{0}$$\phantom{0}-\phantom{0}$$\phantom{0}\sin \alpha$
• $\tan ({\displaystyle \frac{\pi }{2}}+\alpha )=\phantom{0}$$\phantom{0}-\phantom{0}$$\phantom{0}{\displaystyle \frac{1}{\tan \alpha }}$

• $\sin ({\displaystyle \frac{\pi }{2}}-\alpha )=\phantom{0}$$\phantom{0}\cos \alpha$
• $\cos ({\displaystyle \frac{\pi }{2}}-\alpha )=\phantom{0}$$\phantom{0}\sin \alpha$
• $\tan ({\displaystyle \frac{\pi }{2}}-\alpha )=\phantom{0}$$\phantom{0}{\displaystyle \frac{1}{\tan \alpha }}$

• $\sin ({\displaystyle \frac{3\pi }{2}}+\alpha )=\phantom{0}$$\phantom{0}-\phantom{0}$$\phantom{0}\cos \alpha$
• $\cos ({\displaystyle \frac{3\pi }{2}}+\alpha )=\phantom{0}$$\phantom{0}\sin \alpha$
• $\tan ({\displaystyle \frac{3\pi }{2}}+\alpha )=\phantom{0}$$\phantom{0}-\phantom{0}$$\phantom{0}{\displaystyle \frac{1}{\tan \alpha }}$

• $\sin ({\displaystyle \frac{3\pi }{2}}-\alpha )=\phantom{0}$$\phantom{0}-\phantom{0}$$\phantom{0}\cos \alpha$
• $\cos ({\displaystyle \frac{3\pi }{2}}-\alpha )=\phantom{0}$$\phantom{0}-\phantom{0}$$\phantom{0}\sin \alpha$
• $\tan ({\displaystyle \frac{3\pi }{2}}-\alpha )=\phantom{0}$$\phantom{0}{\displaystyle \frac{1}{\tan \alpha }}$

• $\sin (-\alpha )=\phantom{0}$$\phantom{0}-\phantom{0}$$\phantom{0}\sin \alpha$
• $\cos (-\alpha )=\phantom{0}$$\phantom{0}\cos \alpha$
• $\tan (-\alpha )=\phantom{0}$$\phantom{0}-\phantom{0}$$\phantom{0}\tan \alpha$

和差角公式

• $\sin (\alpha +\beta )=\phantom{0}$$\phantom{0}\sin \alpha \cos \beta +\phantom{0}$$\phantom{0}\cos \alpha \sin \beta$
• $\sin (\alpha -\beta )=\phantom{0}$$\phantom{0}\sin \alpha \cos \beta -\phantom{0}$$\phantom{0}\cos \alpha \sin \beta$
• $\cos (\alpha +\beta )=\phantom{0}$$\phantom{0}\cos \alpha \cos \beta -\phantom{0}$$\phantom{0}\sin \alpha \sin \beta$
• $\cos (\alpha -\beta )=\phantom{0}$$\phantom{0}\cos \alpha \cos \beta +\phantom{0}$$\phantom{0}\sin \alpha \sin \beta$
• $\tan (\alpha +\beta )=\phantom{0}$$\phantom{0}{\displaystyle \frac{\tan \alpha +\tan \beta }{1-\tan \alpha \tan \beta }}$
• $\tan (\alpha -\beta )=\phantom{0}$$\phantom{0}{\displaystyle \frac{\tan \alpha -\tan \beta }{1+\tan \alpha \tan \beta }}$

二倍角公式

降幂扩角，升幂缩角。

• $\sin 2\alpha =\phantom{0}$$\phantom{0}2\sin \alpha \cos \alpha$
• $\cos 2\alpha =\phantom{0}$$\phantom{0}{\cos }^2\alpha -\phantom{0}$$\phantom{0}{\sin }^2\alpha =\phantom{0}$$\phantom{0}1-\phantom{0}$$\phantom{0}2{\sin }^2\alpha =\phantom{0}$$\phantom{0}2{\cos }^2\alpha -\phantom{0}$$\phantom{0}1$
• $\tan 2\alpha =\phantom{0}$$\phantom{0}{\displaystyle \frac{2\tan \alpha }{1-{\tan }^2\alpha }}$
• $1+\phantom{0}$$\phantom{0}\sin 2\alpha =\phantom{0}$$\phantom{0}(\sin \alpha +\phantom{0}$$\phantom{0}\cos \alpha {)}^2$
• $1-\phantom{0}$$\phantom{0}\sin 2\alpha =\phantom{0}$$\phantom{0}(\sin \alpha -\phantom{0}$$\phantom{0}\cos \alpha {)}^2$
• $1+\phantom{0}$$\phantom{0}\cos 2\alpha =\phantom{0}$$\phantom{0}2{\cos }^2\alpha$
• $1-\phantom{0}$$\phantom{0}\cos 2\alpha =\phantom{0}$$\phantom{0}2{\sin }^2\alpha$

三倍角公式

• $\sin 3\alpha =\phantom{0}$$\phantom{0}3\sin \alpha -\phantom{0}$$\phantom{0}4{\sin }^3\alpha$
• $\cos 3\alpha =\phantom{0}$$\phantom{0}-\phantom{0}$$\phantom{0}3\cos \alpha +\phantom{0}$$\phantom{0}4{\cos }^3\alpha$
• $\tan 3\alpha =\phantom{0}$$\phantom{0}{\displaystyle \frac{3\tan \alpha -{\tan }^3\alpha }{1-3{\tan }^2\alpha }}=\phantom{0}$$\phantom{0}\tan \alpha \tan ({\displaystyle \frac{\pi }{3}}+\alpha )\tan ({\displaystyle \frac{\pi }{3}}-\alpha )$

半角公式

• $\sin {\displaystyle \frac{\alpha }{2}}=\phantom{0}$$\phantom{0}\pm \sqrt{{\displaystyle \frac{1-\cos \alpha }{2}}}$
• $\cos {\displaystyle \frac{\alpha }{2}}=\phantom{0}$$\phantom{0}\pm \sqrt{{\displaystyle \frac{1+\cos \alpha }{2}}}$
• $\tan {\displaystyle \frac{\alpha }{2}}=\phantom{0}$$\phantom{0}{\displaystyle \frac{\sin \alpha }{1+\cos \alpha }}=\phantom{0}$$\phantom{0}{\displaystyle \frac{1-\cos \alpha }{\sin \alpha }}=\phantom{0}$$\phantom{0}\pm \sqrt{{\displaystyle \frac{1-\cos \alpha }{1+\cos \alpha }}}$

辅助角公式

$$a\sin \theta +b\cos \theta =\sqrt{a^2+b^2}\cdot \sin (\theta +\phi )$$$$\sin \phi ={\displaystyle \frac{b}{\sqrt{a^2+b^2}}},\cos \phi ={\displaystyle \frac{a}{\sqrt{a^2+b^2}}},\tan \phi ={\displaystyle \frac{b}{a}}$$

万能公式

• $\sin \alpha =\phantom{0}$$\phantom{0}{\displaystyle \frac{2\tan {\displaystyle \frac{\alpha }{2}}}{1+{\tan }^2{\displaystyle \frac{\alpha }{2}}}}$
• $\cos \alpha =\phantom{0}$$\phantom{0}{\displaystyle \frac{1-{\tan }^2{\displaystyle \frac{\alpha }{2}}}{1+{\tan }^2{\displaystyle \frac{\alpha }{2}}}}$
• $\tan \alpha =\phantom{0}$$\phantom{0}{\displaystyle \frac{2\tan {\displaystyle \frac{\alpha }{2}}}{1-{\tan }^2{\displaystyle \frac{\alpha }{2}}}}$

和差化积

• $\sin \alpha +\phantom{0}$$\phantom{0}\sin \beta =\phantom{0}$$\phantom{0}2\sin {\displaystyle \frac{\alpha +\beta }{2}}\cos {\displaystyle \frac{\alpha -\beta }{2}}$
• $\sin \alpha -\phantom{0}$$\phantom{0}\sin \beta =\phantom{0}$$\phantom{0}2\sin {\displaystyle \frac{\alpha -\beta }{2}}\cos {\displaystyle \frac{\alpha +\beta }{2}}$
• $\cos \alpha +\phantom{0}$$\phantom{0}\cos \beta =\phantom{0}$$\phantom{0}2\cos {\displaystyle \frac{\alpha +\beta }{2}}\cos {\displaystyle \frac{\alpha -\beta }{2}}$
• $\cos \alpha -\phantom{0}$$\phantom{0}\cos \beta =\phantom{0}$$\phantom{0}-\phantom{0}$$\phantom{0}2\sin {\displaystyle \frac{\alpha +\beta }{2}}\sin {\displaystyle \frac{\alpha -\beta }{2}}$
• $\tan \alpha +\phantom{0}$$\phantom{0}\tan \beta =\phantom{0}$$\phantom{0}{\displaystyle \frac{\sin (\alpha +\beta )}{\cos \alpha \cos \beta }}$
• $\tan \alpha -\phantom{0}$$\phantom{0}\tan \beta =\phantom{0}$$\phantom{0}{\displaystyle \frac{\sin (\alpha -\beta )}{\cos \alpha \cos \beta }}$

积化和差

• $\sin \alpha \cos \beta =\phantom{0}$$\phantom{0}{\displaystyle \frac{1}{2}}[\sin (\alpha +\beta )+\phantom{0}$$\phantom{0}\sin (\alpha -\beta )]$
• $\cos \alpha \sin \beta =\phantom{0}$$\phantom{0}{\displaystyle \frac{1}{2}}[\sin (\alpha +\beta )-\phantom{0}$$\phantom{0}\sin (\alpha -\beta )]$
• $\cos \alpha \cos \beta =\phantom{0}$$\phantom{0}{\displaystyle \frac{1}{2}}[\cos (\alpha +\beta )+\phantom{0}$$\phantom{0}\cos (\alpha -\beta )]$
• $\sin \alpha \sin \beta =\phantom{0}$$\phantom{0}-\phantom{0}$$\phantom{0}{\displaystyle \frac{1}{2}}[\cos (\alpha +\beta )-\phantom{0}$$\phantom{0}\cos (\alpha -\beta )]$

正弦定理

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

其中 $R$ 为 $\Delta ABC$ 外接圆半径。

余弦定理

• $a^2=\phantom{0}$$\phantom{0}{b}^2+\phantom{0}$$\phantom{0}{c}^2-\phantom{0}$$\phantom{0}2bc\cos A$
• $b^2=\phantom{0}$$\phantom{0}{a}^2+\phantom{0}$$\phantom{0}{c}^2-\phantom{0}$$\phantom{0}2ac\cos B$
• $c^2=\phantom{0}$$\phantom{0}{a}^2+\phantom{0}$$\phantom{0}{b}^2-\phantom{0}$$\phantom{0}2ab\cos B$

三角形面积公式

• ${\displaystyle S_{\Delta ABC}=\phantom{0}}$$\phantom{0}{\displaystyle \frac{1}{2}}\cdot a\cdot h$
• ${\displaystyle S_{\Delta ABC}=\phantom{0}}$$\phantom{0}{\displaystyle \frac{1}{2}}ab\sin C=\phantom{0}$$\phantom{0}{\displaystyle \frac{1}{2}}bc\sin A=\phantom{0}$$\phantom{0}{\displaystyle \frac{1}{2}}ac\sin B$
• ${\displaystyle S_{\Delta ABC}=\phantom{0}}$$\phantom{0}{\displaystyle \frac{abc}{4R}}$（$R$ 为 $\Delta ABC$ 外接圆半径）
• ${\displaystyle S_{\Delta ABC}=\phantom{0}}$$\phantom{0}{\displaystyle \frac{a+b+c}{2}}\cdot r$（$r$ 为 $\Delta ABC$ 内接圆半径）
• ${\displaystyle S_{\Delta ABC}=\phantom{0}}$$\phantom{0}\sqrt{p(p-a)(p-b)(p-c)},\phantom{0}$$\phantom{0}p=\phantom{0}$$\phantom{0}{\displaystyle \frac{a+b+c}{2}}$

其它公式

• ${\displaystyle {\sin }^2\theta +\phantom{0}}$$\phantom{0}{\cos }^2\theta =\phantom{0}$$\phantom{0}1$
• ${\displaystyle 1+\phantom{0}}$$\phantom{0}{\tan }^2\theta =\phantom{0}$$\phantom{0}{\displaystyle \frac{1}{{\cos }^2\theta }}$
• ${\displaystyle 1+\phantom{0}}$$\phantom{0}{\displaystyle \frac{1}{{\tan }^2\theta }}=\phantom{0}$$\phantom{0}{\displaystyle \frac{1}{{\sin }^2\theta }}$
• ${\displaystyle \tan A+\phantom{0}}$$\phantom{0}\tan B+\phantom{0}$$\phantom{0}\tan C=\phantom{0}$$\phantom{0}\tan A\tan B\tan C$（$\Delta ABC$ 非 $Rt$ 三角形）
• ${\displaystyle {\sin }^{2}A+\phantom{0}}$$\phantom{0}{\sin }^{2}B+\phantom{0}$$\phantom{0}{\sin }^{2}C=\phantom{0}$$\phantom{0}2+\phantom{0}$$\phantom{0}2\cos A\cos B\cos C$（在 $\Delta ABC$ 中）
• ${\displaystyle {\cos }^{2}A+\phantom{0}}$$\phantom{0}{\cos }^{2}B+\phantom{0}$$\phantom{0}{\cos }^{2}C=\phantom{0}$$\phantom{0}1-\phantom{0}$$\phantom{0}2\cos A\cos B\cos C$（在 $\Delta ABC$ 中）