基础数学-三角函数
基础数学-三角函数
诱导公式
| 公式一 | 公式二 |
|---|---|
| sin（2kπ+α）=sin α cos（2kπ+α）=cos α tan（2kπ+α）=tan α cot（2kπ+α）=cot α sec（2kπ+α）=sec α csc（2kπ+α）=csc α | sin（π+α）=-sin α cos（π+α）=-cos α tan（π+α）=tan α cot（π+α）=cot α sec（π+α）=-sec α csc（π+α）=-csc α |
| 公式三 | 公式四 |
| sin（-α）=-sin α cos（-α）=cos α tan（-α）=-tan α cot（-α）=-cot α sec（-α）=sec α csc（-α）=-csc α | sin（π-α）=sin α cos（π-α）=-cos α tan（π-α）=-tan α cot（π-α）=-cot α sec（π-α）=-sec α csc（π-α）=csc α |
| 公式五 | 公式六 |
| sin（α-π）=-sin α cos（α-π）=-cos α tan（α-π）=tan α cot（α-π）=cot α sec（α-π）=-sec α csc（α-π）=-csc α | sin（2π-α）=-sin α cos（2π-α）=cos α tan（2π-α）=-tan α cot（2π-α）=-cot α sec（2π-α）=sec α csc（2π-α）=-csc α |
| 公式七 | 公式八 |
| sin（π/2+α）=cosα cos（π/2+α）=−sinα tan（π/2+α）=-cotα cot（π/2+α）=-tanα sec（π/2+α）=-cscα csc（π/2+α）=secα | sin（π/2-α）=cosα cos（π/2-α）=sinα tan（π/2-α）=cotα cot（π/2-α）=tanα sec（π/2-α）=cscα csc（π/2-α）=secα |
| 公式九 | 公式十 |
| sin（3π/2+α）=-cosα cos（3π/2+α）=sinα tan（3π/2+α）=-cotα cot（3π/2+α）=-tanα sec（3π/2+α）=cscα csc（3π/2+α）=-secα | sin（3π/2-α）=-cosα cos（3π/2-α）=-sinα tan（3π/2-α）=cotα cot（3π/2-α）=tanα sec（3π/2-α）=-cscα csc（3π/2-α）=-secα |
倍角公式
\(\begin{aligned} \sin 2 \alpha &=2 \sin \alpha \cdot \cos \alpha \\ \cos 2 \alpha &=\cos ^{2} \alpha-\sin ^{2} \alpha=2 \cos ^{2} \alpha-1=1-2 \sin ^{2} \alpha \\ \tan 2 \alpha &=\frac{2 \tan \alpha}{1-\tan ^{2} \alpha} \end{aligned}\)
半角公式
\(\sin ^{2} \frac{\alpha}{2}=\frac{1-\cos \alpha}{2}\) \(\cos ^{2} \frac{\alpha}{2}=\frac{1+\cos \alpha}{2}\) \(\tan ^{2} \frac{\alpha}{2}=\frac{1-\cos \alpha}{1+\cos \alpha}\) \(\tan \frac{\alpha}{2}=\frac{\sin \alpha}{1+\cos \alpha}=\frac{1-\cos \alpha}{\sin \alpha}=\csc \alpha-\cot \alpha\) \(\cot \frac{\alpha}{2}=\frac{\sin \alpha}{1-\cos \alpha}=\frac{1+\cos \alpha}{\sin \alpha}=\csc \alpha+\cot \alpha\)
和角公式
\(\sin (\alpha+\beta)=\sin \alpha \cdot \cos \beta+\cos \alpha \cdot \sin \beta\) \(\sin (\alpha-\beta)=\sin \alpha \cdot \cos \beta-\cos \alpha \cdot \sin \beta\) \(\cos (\alpha+\beta)=\cos \alpha \cdot \cos \beta-\sin \alpha \cdot \sin \beta\) \(\cos (\alpha-\beta)=\cos \alpha \cdot \cos \beta+\sin \alpha \cdot \sin \beta\) \(\tan (\alpha+\beta)=\frac{\tan \alpha+\tan \beta}{1-\tan \alpha \cdot \tan \beta}\) \(\tan (\alpha-\beta)=\frac{\tan \alpha-\tan \beta}{1+\tan \alpha \cdot \tan \beta}\)
积化和差
\(\sin \alpha \cos \beta=\frac{1}{2}[\sin (\alpha+\beta)+\sin (\alpha-\beta)]\) \(\cos \alpha \sin \beta=\frac{1}{2}[\sin (\alpha+\beta)-\sin (\alpha-\beta)]\) \(\cos \alpha \cos \beta=\frac{1}{2}[\cos (\alpha+\beta)+\cos (\alpha-\beta)]\) \(\sin \alpha \sin \beta=-\frac{1}{2}[\cos (\alpha+\beta)-\cos (\alpha-\beta)]\)
和差化积
\(\sin \alpha+\sin \beta=2 \sin \frac{\alpha+\beta}{2} \cos \frac{\alpha-\beta}{2} \cdots \cdots(1)\) \(\sin \alpha-\sin \beta=2 \cos \frac{\alpha+\beta}{2} \sin \frac{\alpha-\beta}{2} \ldots \ldots\) \(\cos \alpha+\cos \beta=2 \cos \frac{\alpha+\beta}{2} \cos \frac{\alpha-\beta}{2} \cdots \cdots(3)\) \(\cos \alpha-\cos \beta=-2 \sin \frac{\alpha+\beta}{2} \sin \frac{\alpha-\beta}{2} \ldots \ldots(4)\) \(\tan \alpha+\tan \beta=\frac{\sin (\alpha+\beta)}{\cos \alpha \cos \beta} \cdots \cdots(5)\) \(\tan \alpha-\tan \beta=\frac{\sin (\alpha-\beta)}{\cos \alpha \cos \beta} \cdots \cdots(6)\) \(\cot \alpha+\cot \beta=\frac{\sin (\alpha+\beta)}{\sin \alpha \sin \beta} \cdots \cdots(7)\) \(\cot \alpha-\cot \beta=-\frac{\sin (\alpha-\beta)}{\sin \alpha \sin \beta} \cdots \cdots(8)\) \(\tan \alpha+\cot \beta=\frac{\cos (\alpha-\beta)}{\cos \alpha \sin \beta} \cdots \cdots(9)\) \(\tan \alpha-\cot \beta=-\frac{\cos (\alpha+\beta)}{\cos \alpha \sin \beta} \cdots \cdots(10)\)
万能公式
\(\sin a=\frac{2 \tan \frac{a}{2}}{1+\tan ^{2} \frac{a}{2}}\) \(\cos a=\frac{1-\tan ^{2} \frac{a}{2}}{1+\tan ^{2} \frac{a}{2}}\) \(\tan a=\frac{2 \tan \frac{a}{2}}{1-\tan ^{2} \frac{a}{2}}\)
\(\cot \alpha=\frac{1-\tan ^{2} \frac{\alpha}{2}}{2 \tan \frac{\alpha}{2}}\) \(\sec \alpha=\frac{1+\tan ^{2} \frac{\alpha}{2}}{1-\tan ^{2} \frac{\alpha}{2}}\) \(\csc \alpha=\frac{1+\tan ^{2} \frac{\alpha}{2}}{2 \tan \frac{\alpha}{2}}\)