trigonometric

和差角公式

$$\begin{aligned} \cos(a+b) = \cos a\cos b - \sin a\sin b \\ \cos(a-b) = \cos a\cos b + \sin a\sin b \\ \sin(a+b) = \sin a\cos b + \cos a\sin b \\ \sin(a-b) = \sin a\cos b - \cos a\sin b \\ \tan(a+b) = \frac{\tan a + \tan b}{1 - \tan a \tan b} \\ \tan(a-b) = \frac{\tan a - \tan b}{1 + \tan a \tan b} \end{aligned}$$

积分公式

$$\begin{aligned} \int_{-\frac{T}{2}}^{\frac{T}{2}} \cos(n\omega t)\sin(m\omega t)dt &=& 0 \\ \int_{-\frac{T}{2}}^{\frac{T}{2}} \cos(n\omega t)\cos(m\omega t)dt &=& \begin{cases} \frac{T}{2}, && n = m \\ 0, && n \neq m \end{cases} \\ \int_{-\frac{T}{2}}^{\frac{T}{2}} \sin(n\omega t)\sin(m\omega t)dt &=& \begin{cases} 0, && n = m \\ \frac{T}{2}, && n \neq m \end{cases} \end{aligned}$$