三角函数公式总结

求导运算

$$(\sin x{)}^{\prime }=\cos x$$$$(\cos x{)}^{\prime }=\sin x$$$$(\tan x{)}^{\prime }=\frac{1}{{\cos }^{2}x}={\sec }^{2}x$$$$(\cot x{)}^{\prime }=-{\csc }^{2}x=-\frac{1}{{\sin }^{2}x}$$$$(\sec x{)}^{\prime }=\frac{\sin x}{{\cos }^{2}x}=\sec x\tan x$$$$(\arcsin x{)}^{\prime }=\frac{1}{\sqrt{1-x^2}}$$$$(\arccos x{)}^{\prime }=-\frac{1}{\sqrt{1-x^2}}$$$$(\arctan x{)}^{\prime }=\frac{1}{1+x^2}$$$$(\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{c}\mathrm{o}\mathrm{t}x{)}^{\prime }=-\frac{1}{1+x^2}$$$$(\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{s}\mathrm{e}\mathrm{c}x{)}^{\prime }=\frac{1}{|x|\sqrt{x^2-1}}$$$$(\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{c}\mathrm{s}\mathrm{c}x{)}^{\prime }=-\frac{1}{|x|\sqrt{x^2-1}}$$

和差角公式

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

和差化积

$$\sin \alpha \pm \sin \beta =2\sin \frac{\alpha \pm \beta }{2}\cos \frac{\alpha \mp \beta }{2}$$$$\cot \alpha +\cos \beta =2\cos \frac{\alpha +\beta }{2}\cos \frac{\alpha -\beta }{2}$$$$\cot \alpha -\cos \beta =-2\sin \frac{\alpha +\beta }{2}\sin \frac{\alpha -\beta }{2}$$$$\tan \alpha +\tan \beta =\frac{\sin (\alpha +\beta )}{\cos \alpha \cos \beta }$$

积化和差

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

二倍角公式

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

半角公式

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

万能公式

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

辅助角公式

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