三角函数公式

三角函数公式

在直角三角形中：

$\sin \alpha$ ：对边除以斜边, 音标[saɪn]

$\cos \alpha$ ：邻边除以斜边, 音标[ˈkəʊsaɪn]

$\tan \alpha =\frac{\sin \alpha }{\cos \alpha }$ ： 对边除以邻边, 音标[ˈtændʒənt]

$\cot \alpha =\frac{1}{\tan \alpha }=\frac{\cos \alpha }{\sin \alpha }$ : 邻边除以对边, 音标['kəʊ'tændʒənt]

$\sec \alpha =\frac{1}{\cos \alpha }$ : 斜边除以邻边, 音标['si:kənt]

$\csc \alpha =\frac{1}{\sin \alpha }$ : 斜边除以对边, 音标['kəʊ'si:kənt]

和差角公式

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

$\cos (\alpha \pm \beta )=\cos \alpha \cos \beta \mp \sin \alpha \sin \beta$

$\tan (\alpha \pm \beta )=\frac{\tan \alpha \pm tan\beta }{1\mp \tan \alpha \cdot \tan \beta }$

$\cot (\alpha \pm \beta )=\frac{\cot \alpha \cdot \cot \beta \mp 1}{\cot \beta \pm \cot \alpha }$

和差化积公式

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

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

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

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

积化和差公式

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

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

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

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

倍角公式

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

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

$\tan 2\alpha =\frac{2\tan \alpha }{1-{\tan }^2\alpha }\cdot \cot 2\alpha =\frac{{\cot }^2\alpha -1}{2\cot \alpha }$

$\sin 3\alpha =3\sin \alpha -4{\sin }^3\alpha$

$\cos 3\alpha =4{\cos }^3\alpha -3\cos \alpha$