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三角函数公式 1. 加法公式 和差与积互化公式

[加法公式] \[ \sin \left( {\alpha \pm \beta } \right) = \sin \alpha \cos \beta \pm \cos \alpha \sin \beta \] \[ \cos \left( {\alpha \pm \beta } \right) = \cos \alpha \cos \beta \mp \sin \alpha \sin \beta \] \[ \tan \left( {\alpha \pm \beta } \right) = \frac{{\tan \alpha \pm \tan \beta }}{{1 \mp \tan \alpha \tan \beta }} \] \[ \cot \left( {\alpha \pm \beta } \right) = \frac{{\cot \alpha \cot \beta \mp 1}}{{\cot \beta \pm \cot \alpha }} \] \[ \begin{array}{l} \sin \left( {\alpha + \beta + \gamma } \right) = \sin \alpha \cos \beta \cos \gamma + \cos \alpha \sin \beta \cos \gamma \\ + \cos \alpha \cos \beta \sin \gamma - \sin \alpha \sin \beta \sin \gamma \\ = \sin \alpha \sin \beta \sin \gamma \left( {\cot \beta \cot \gamma + \cot \gamma \cot \alpha + \cot \alpha \cot \beta - 1} \right) \\ = \cos \alpha \cos \beta \cos \gamma \left( {\tan \alpha + \tan \beta + \tan \gamma - \tan \alpha \tan \beta \tan \gamma } \right) \\ \end{array} \] \[ \begin{array}{l} \cos \left( {\alpha + \beta + \gamma } \right) = \cos \alpha \cos \beta \cos \gamma - \cos \alpha \sin \beta \sin \gamma \\ - \sin \alpha \cos \beta \sin \gamma - \sin \alpha \sin \beta \cos \gamma \\ = \cos \alpha \cos \beta \cos \gamma \left( {1 - \tan \beta \tan \gamma - \tan \gamma \tan \alpha - \tan \alpha \tan \beta } \right) \\ = - \sin \alpha \sin \beta \sin \gamma \left( {\cot \alpha + \cot \beta + \cot \gamma - \cot \alpha \cot \beta \cot \gamma } \right) \\ \end{array} \] [和差与积互化公式] \[ \sin \alpha + \sin \beta = 2\sin \frac{{\alpha + \beta }}{2}\cos \frac{{\alpha - \beta }}{2} \] \[ \sin \alpha - \sin \beta = 2\cos \frac{{\alpha + \beta }}{2}\sin \frac{{\alpha - \beta }}{2} \] \[ \cos \alpha + \cos \beta = 2\cos \frac{{\alpha + \beta }}{2}\cos \frac{{\alpha - \beta }}{2} \] \[ \cos \alpha - \cos \beta = - 2\sin \frac{{\alpha + \beta }}{2}\sin \frac{{\alpha - \beta }}{2} \] \[ \tan \alpha \pm \tan \beta = \frac{{\sin \left( {\alpha \pm \beta } \right)}}{{\cos \alpha \cos \beta }} \] \[ \cot \alpha \pm \cot \beta = \pm \frac{{\sin \left( {\alpha \pm \beta } \right)}}{{\sin \alpha \sin \beta }} \] \[ \tan \alpha \pm \cot \beta = \pm \frac{{\cos \left( {\alpha \mp \beta } \right)}}{{\cos \alpha \sin \beta }} \] \[ \sin \alpha \sin \beta = - \frac{1}{2}\left[ {\cos \left( {\alpha + \beta } \right) - \cos \left( {\alpha - \beta } \right)} \right] \] \[ \cos \alpha \cos \beta = \frac{1}{2}\left[ {\cos \left( {\alpha + \beta } \right) + \cos \left( {\alpha - \beta } \right)} \right] \] \[ \sin \alpha \cos \beta = \frac{1}{2}\left[ {\sin \left( {\alpha + \beta } \right) + \sin \left( {\alpha - \beta } \right)} \right] \] 


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