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]
\(\tan \alpha = \Large \frac{\sin \alpha}{\cos \alpha}\) ： 对边除以邻边, 音标[ˈtændʒənt]
\(\cot \alpha = \Large \frac{1}{\tan \alpha} = \frac{\cos \alpha}{\sin \alpha}\) : 邻边除以对边, 音标['kəʊ'tændʒənt]
\(\sec \alpha = \Large \frac{1}{\cos \alpha}\) : 斜边除以邻边, 音标['si:kənt]
\(\csc \alpha = \Large \frac{1}{\sin \alpha}\) : 斜边除以对边, 音标['kəʊ'si:kənt]
和差角公式
\(\sin(\alpha \pm \beta)=\sin \alpha\cos \beta \pm \cos \alpha \sin \beta\)
\(\cos(\alpha \pm \beta)=\cos \alpha \cos \beta \mp \sin \alpha\sin \beta\)
\(\tan(\alpha \pm \beta)= \Large \frac{\tan \alpha \pm tan \beta}{1 \mp \tan \alpha \cdot \tan \beta}\)
\(\cot(\alpha \pm \beta)= \Large \frac{\cot \alpha \cdot \cot \beta \mp 1}{\cot \beta \pm \cot \alpha}\)
和差化积公式
\(\sin \alpha + \sin \beta = 2\sin{\Large \frac{\alpha + \beta}{2}}\cdot \cos {\Large \frac{\alpha - \beta}{2}}\)
\(\sin \alpha - \sin \beta = 2\cos{\Large \frac{\alpha + \beta}{2}}\cdot \sin {\Large \frac{\alpha - \beta}{2}}\)
\(\cos \alpha + \cos \beta = 2\cos{\Large \frac{\alpha + \beta}{2}}\cdot \cos {\Large \frac{\alpha - \beta}{2}}\)
\(\cos \alpha - \cos \beta = -2\sin{\Large \frac{\alpha + \beta}{2}}\cdot \sin {\Large \frac{\alpha - \beta}{2}}\)
积化和差公式
\(\sin \alpha \cos \beta = {\Large\frac{1}{2}}[\sin(\alpha + \beta)+\sin(\alpha - \beta)]\)
\(\cos \alpha \sin \beta = {\Large\frac{1}{2}}[\sin(\alpha + \beta)-\sin(\alpha - \beta)]\)
\(\cos \alpha \cos \beta = {\Large\frac{1}{2}}[\cos(\alpha + \beta)+\cos(\alpha - \beta)]\)
\(\sin \alpha \sin \beta = {\Large\frac{1}{2}}[\cos(\alpha + \beta)-\cos(\alpha - \beta)]\)
倍角公式
\(\sin 2 \alpha = 2\sin \alpha \cos \alpha=\Large \frac{2\tan \alpha}{1 + \tan^2 \alpha}\)
\(\cos 2 \alpha = \cos^2 \alpha - \sin^2 \alpha \\ \qquad = 2\cos^2 \alpha - 1 \\\qquad = 1 - 2\sin^2 \alpha \\\qquad = \Large \frac{1-\tan^2 \alpha}{1 + \tan^2 \alpha}\)
\(\tan 2 \alpha = {\Large \frac{2\tan \alpha}{1-\tan^2 \alpha}}\cdot \cot 2 \alpha = \Large \frac{\cot^2 \alpha - 1}{2\cot \alpha}\)
\(\sin 3 \alpha = 3\sin \alpha - 4\sin^3 \alpha\)
\(\cos 3 \alpha = 4\cos^3 \alpha - 3\cos \alpha\)
