常用的三角函数公式

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• ${\sin }^{2}x+{\cos }^{2}x=1$

• $\tan x=\frac{\sin x}{\cos x}$

• $\cot x=\frac{1}{\tan x}=\frac{\cos x}{\sin x}$

• $\sec x=\frac{1}{\cos x}$

• $\csc x=\frac{1}{\sin x}$

• ${\tan }^{2}x={\sec }^2-1=\frac{1}{{\cos }^{2}x}-1=\frac{1-{\cos }^{2}x}{{\cos }^{2}x}=\frac{{\sin }^{2}x}{{\cos }^{2}x}$

• ${\cot }^2={\csc }^{2}x-1=\frac{1}{{\sin }^{2}x}-1=\frac{1-{\sin }^{2}x}{{\sin }^{2}x}=\frac{{\cos }^{2}x}{{\sin }^{2}x}$

• $\cos x=\sin (x+\frac{\pi }{2})$，$\sin x$ 向左平移 $\frac{\pi }{2}$. (左加右减)

• $\sin x=\cos (x-\frac{\pi }{2})$

• $\cos x=\cos (-x)$，偶函数

• $\sin x=-\sin (-x)$，奇函数

• $\sin x=-\sin (x\pm \pi )$，$\sin x$无论是向左、还是向右平移 $\pi$ 个单位后，乘以-1，关于x轴对称之后函数图像不变.

• $\cos x=-\cos (x\pm \pi )$

• $\arcsin x+\arccos x=\frac{\pi }{2}$.

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倍(半)角公式

• $\cos (A\pm B)=\cos A\cdot \cos B\mp \sin A\cdot \sin B$.

• $\sin (A\pm B)=\sin A\cdot \cos B\pm \cos A\cdot \sin B$.

• $\cos (2A)={\cos }^{2}A-{\sin }^{2}A=1-2{\sin }^{2}A=2{\cos }^{2}A-1$.

• $\cos A={\cos }^2\frac{A}{2}-{\sin }^2\frac{A}{2}=1-2{\sin }^2\frac{A}{2}=2{\cos }^2\frac{A}{2}-1$.

• $\sin (2A)=2\sin A\cdot \cos A$.

• $\sin A=2\sin \frac{A}{2}\cdot \cos \frac{A}{2}$.

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• $\tan 2\alpha =\frac{2\tan \alpha }{1-{\tan }^2\alpha }$.

• $\tan \alpha =\frac{2\tan \frac{\alpha }{2}}{1-{\tan }^2\frac{\alpha }{2}}$.

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

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

$$\tan \alpha =\frac{2\tan \frac{\alpha }{2}}{1-{\tan }^2\frac{\alpha }{2}}$$
$$\tan \alpha -\tan \alpha {\tan }^2\frac{\alpha }{2}=2\tan \frac{\alpha }{2}$$
$$\tan \alpha \cdot {\tan }^2\frac{\alpha }{2}+2\tan \frac{\alpha }{2}-\tan \alpha =0$$
求根公式：
$$\tan \frac{\alpha }{2}=\frac{-2\pm \sqrt{4+4{\tan }^2\alpha }}{2\tan \alpha }=\frac{-1\pm \sec \alpha }{\tan \alpha }=\frac{-\cos \alpha \pm 1}{\sin \alpha }$$
当 $\alpha \in (0,\pi )$ 时，$\tan \frac{\alpha }{2}>0$，而 $-\frac{\cos \alpha +1}{\sin \alpha }<0$.

$\therefore \tan \frac{\alpha }{2}=-\frac{\cos \alpha +1}{\sin \alpha }$ 不成立.

$$\therefore \tan \frac{\alpha }{2}=\frac{1-\cos \alpha }{\sin \alpha }=\frac{(1-\cos \alpha )\cdot (1+\cos \alpha )}{\sin \alpha +\sin \alpha \cdot \cos \alpha }=\frac{1-{\cos }^2\alpha }{\sin \alpha +\sin \alpha \cdot \cos \alpha }=\frac{\sin \alpha }{1+\cos \alpha }$$

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正、余弦化切弦

• $\sin 2x=2\sin x\cdot \cos x=\frac{2\tan x}{{\sec }^{2}x}=\frac{2\tan x}{1+{\tan }^{2}x}$.

• $\cos 2x={\cos }^{2}x-{\sin }^{2}x=\frac{1-{\tan }^{2}x}{{\sec }^{2}x}=\frac{1-{\tan }^{2}x}{1+{\tan }^{2}x}$.

• $\sin x=\frac{2\tan \frac{x}{2}}{1+{\tan }^2\frac{x}{2}}$.

• $\cos x=\frac{1-{\tan }^2\frac{x}{2}}{1+{\tan }^2\frac{x}{2}}$.

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辅助角公式

• $\sin \alpha \cdot \frac{a}{\sqrt{a^2+b^2}}-\cos \alpha \cdot \frac{b}{\sqrt{a^2+b^2}}=\sin \alpha \cdot \cos \beta -\cos \alpha \cdot \sin \beta =\sin (\alpha -\beta )$.

其中，令 $\cos \beta =\frac{a}{\sqrt{a^2+b^2}}$，$\sin \beta =\frac{b}{\sqrt{a^2+b^2}}$，

则有 ${\cos }^2\beta +{\sin }^2\beta =(\frac{a}{\sqrt{a^2+b^2}}{)}^2+(\frac{b}{\sqrt{a^2+b^2}}{)}^2=1$.

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$\frac{E_m}{L^2{\omega }^2+R^2}\cdot (R\cdot \sin \omega t-L\omega \cdot \cos \omega t)$

$=\frac{E_m}{\sqrt{L^2{\omega }^2+R^2}}\cdot (\sin \omega t\cdot \frac{R}{\sqrt{L^2{\omega }^2+R^2}}-\cos \omega t\cdot \frac{L\omega }{\sqrt{L^2{\omega }^2+R^2}})$

$=\frac{E_m}{\sqrt{L^2{\omega }^2+R^2}}\cdot \sin (\omega t-\phi )$.

其中 $\cos \phi =\frac{R}{\sqrt{L^2{\omega }^2+R^2}}$，$\sin \phi =\frac{L\omega }{\sqrt{L^2{\omega }^2+R^2}}$.

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积化和差、和差化积

参考往期文章，点击跳转