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# 三角恒等式引用

## 倒数和商的恒等式

$\mathrm{sec}\left(\theta \right)=\frac{1}{\mathrm{cos}\left(\theta \right)}$

$\mathrm{csc}\left(\theta \right)=\frac{1}{\mathrm{sin}\left(\theta \right)}$

$\mathrm{cot}\left(\theta \right)=\frac{1}{\mathrm{tan}\left(\theta \right)}$

$\mathrm{tan}\left(\theta \right)=\frac{\mathrm{sin}\left(\theta \right)}{\mathrm{cos}\left(\theta \right)}$

$\mathrm{cot}\left(\theta \right)=\frac{\mathrm{cos}\left(\theta \right)}{\mathrm{sin}\left(\theta \right)}$

## 毕达哥拉斯恒等式

${\mathrm{sin}}^{2}\left(\theta \right)+{\mathrm{cos}}^{2}\left(\theta \right)={1}^{2}$
${\mathrm{tan}}^{2}\left(\theta \right)+{1}^{2}={\mathrm{sec}}^{2}\left(\theta \right)$
${\mathrm{cot}}^{2}\left(\theta \right)+{1}^{2}={\mathrm{csc}}^{2}\left(\theta \right)$

## 角的加减乘除恒等式

$\mathrm{sin}\left(\theta +\varphi \right)=\mathrm{sin}\theta \mathrm{cos}\varphi +\mathrm{cos}\theta \mathrm{sin}\varphi$
$\mathrm{sin}\left(\theta -\varphi \right)=\mathrm{sin}\theta \mathrm{cos}\varphi -\mathrm{cos}\theta \mathrm{sin}\varphi$
$\mathrm{cos}\left(\theta +\varphi \right)=\mathrm{cos}\theta \mathrm{cos}\varphi -\mathrm{sin}\theta \mathrm{sin}\varphi$
$\mathrm{cos}\left(\theta -\varphi \right)=\mathrm{cos}\theta \mathrm{cos}\varphi +\mathrm{sin}\theta \mathrm{sin}\varphi$
$\mathrm{tan}\left(\theta +\varphi \right)=\frac{\mathrm{tan}\theta +\mathrm{tan}\varphi }{1-\mathrm{tan}\theta \mathrm{tan}\varphi }$
$\mathrm{tan}\left(\theta -\varphi \right)=\frac{\mathrm{tan}\theta -\mathrm{tan}\varphi }{1+\mathrm{tan}\theta \mathrm{tan}\varphi }$

$\mathrm{sin}\left(2\theta \right)=2\mathrm{sin}\theta \mathrm{cos}\theta$
$\mathrm{cos}\left(2\theta \right)=2{\mathrm{cos}}^{2}\theta -1$
$\mathrm{tan}\left(2\theta \right)=\frac{2\mathrm{tan}\theta }{1-{\mathrm{tan}}^{2}\theta }$

$\mathrm{sin}\frac{\theta }{2}=±\sqrt{\frac{1-\mathrm{cos}\theta }{2}}$
$\mathrm{cos}\frac{\theta }{2}=±\sqrt{\frac{1+\mathrm{cos}\theta }{2}}$

## 对称与周期恒等式

$\mathrm{sin}\left(-\theta \right)=-\mathrm{sin}\left(\theta \right)$
$\mathrm{cos}\left(-\theta \right)=+\mathrm{cos}\left(\theta \right)$
$\mathrm{tan}\left(-\theta \right)=-\mathrm{tan}\left(\theta \right)$

$\mathrm{sin}\left(\theta +2\pi \right)=\mathrm{sin}\left(\theta \right)$
$\mathrm{cos}\left(\theta +2\pi \right)=\mathrm{cos}\left(\theta \right)$
$\mathrm{tan}\left(\theta +\pi \right)=\mathrm{tan}\left(\theta \right)$

## 余函数恒等式

$\mathrm{sin}\theta =\mathrm{cos}\left(\frac{\pi }{2}-\theta \right)$
$\mathrm{cos}\theta =\mathrm{sin}\left(\frac{\pi }{2}-\theta \right)$
$\mathrm{tan}\theta =\mathrm{cot}\left(\frac{\pi }{2}-\theta \right)$
$\mathrm{cot}\theta =\mathrm{tan}\left(\frac{\pi }{2}-\theta \right)$
$\mathrm{sec}\theta =\mathrm{csc}\left(\frac{\pi }{2}-\theta \right)$
$\mathrm{csc}\theta =\mathrm{sec}\left(\frac{\pi }{2}-\theta \right)$