和差化积

$ \sin\alpha \sin\beta = -\frac{1}{2}[\cos(\alpha + \beta) - \cos(\alpha - \beta)] $

$ \cos\alpha \cos\beta = \frac{1}{2}[\cos(\alpha + \beta) + \cos(\alpha - \beta)] $

$ \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)] $

积化和差

$ \sin\alpha + \sin\beta = 2 \sin\left(\frac{\alpha + \beta}{2}\right) \cos\left(\frac{\alpha - \beta}{2}\right) $

$ \sin\alpha - \sin\beta = 2 \cos\left(\frac{\alpha + \beta}{2}\right) \sin\left(\frac{\alpha - \beta}{2}\right) $

$ \cos\alpha + \cos\beta = 2 \cos\left(\frac{\alpha + \beta}{2}\right) \cos\left(\frac{\alpha - \beta}{2}\right) $

$ \cos\alpha - \cos\beta = -2 \sin\left(\frac{\alpha + \beta}{2}\right) \sin\left(\frac{\alpha - \beta}{2}\right) $

万能公式

$ \sin\alpha = \frac{2\tan(\alpha/2)}{1 + \tan^2(\alpha/2)} $

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

$ \tan\alpha = \frac{2\tan(\alpha/2)}{1 - \tan^2(\alpha/2)} $