诱导公式

奇变偶不变，符号看象限。

• $\sin (2k\pi +\alpha )=\sin \alpha$
• $\cos (2k\pi +\alpha )=\cos \alpha$
• $\tan (2k\pi +\alpha )=\tan \alpha$

• $\sin (2k\pi -\alpha )=-\sin \alpha$
• $\cos (2k\pi -\alpha )=\cos \alpha$
• $\tan (2k\pi -\alpha )=-\tan \alpha$

• $\sin (\pi +\alpha )=-\sin \alpha$
• $\cos (\pi +\alpha )=-\cos \alpha$
• $\tan (\pi +\alpha )=\tan \alpha$

• $\sin (\pi -\alpha )=\sin \alpha$
• $\cos (\pi -\alpha )=-\cos \alpha$
• $\tan (\pi -\alpha )=-\tan \alpha$

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

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

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

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

• $\sin (-\alpha )=-\sin \alpha$
• $\cos (-\alpha )=\cos \alpha$
• $\tan (-\alpha )=-\tan \alpha$

和差角公式

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

二倍角公式

降幂扩角，升幂缩角。

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

三倍角公式

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

半角公式

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

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

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

和差化积

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

积化和差

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

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

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

三角形面积公式

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

其它公式

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