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三角函数公式(二) 倍角公式 半角公式

[倍角公式] \[ \sin 2\alpha = 2\sin \alpha \cos \alpha = \frac{{2\tan \alpha }}{{1 + \tan ^2 \alpha }} \] \[ \begin{array}{l} \cos 2\alpha = \cos ^2 \alpha - \sin ^2 \alpha = 2\cos ^2 \alpha - 1 \\ = 1 - 2\sin ^2 \alpha = \frac{{1 - \tan ^2 \alpha }}{{1 + \tan ^2 \alpha }} \\ \end{array} \] \[ \tan 2\alpha = \frac{{2\tan \alpha }}{{1 - \tan ^2 \alpha }} \] \[ \cot 2\alpha = \frac{{\cot ^2 \alpha - 1}}{{2\cot \alpha }} \] \[ \sec 2\alpha = \frac{{\sec ^2 \alpha }}{{1 - \tan ^2 \alpha }} = \frac{{\cot \alpha + \tan \alpha }}{{\cot \alpha - \tan \alpha }} \] \[ \csc 2\alpha = \frac{1}{2}\sec \alpha \csc \alpha = \frac{1}{2}\left( {\tan \alpha + \cot \alpha } \right) \] \[ \begin{array}{l} \sin 3\alpha = - 4\sin ^3 \alpha + 3\sin \alpha \\ \cos 3\alpha = 4\cos ^3 \alpha - 3\cos \alpha \\ \end{array} \] \[ \tan 3\alpha = \frac{{3\tan \alpha - \tan ^3 \alpha }}{{1 - 3\tan ^2 \alpha }} \] \[ \cot 3\alpha = \frac{{\cot ^3 \alpha - 3\cot \alpha }}{{3\cot ^2 \alpha - 1}} \] \[ \begin{array}{l} \sin n\alpha = n\cos ^{n - 1} \alpha \sin \alpha - C_n^3 \cos ^{n - 3} \alpha \sin ^3 \alpha \\ + C_n^5 \cos ^{n - 5} \alpha \sin ^5 \alpha - \cdots \\ \end{array} \] \[ \begin{array}{l} \cos n\alpha = \cos ^n \alpha - C_n^2 \cos ^{n - 2} \alpha \sin ^2 \alpha \\ + C_n^4 \cos ^{n - 4} \alpha \sin ^4 \alpha - C_n^6 \cos ^{n - 6} \alpha \sin ^6 \alpha \cdots \\ \end{array} \] 式中n为正整数.

[半角公式] \[ \sin \frac{\alpha }{2} = \pm \sqrt {\frac{{1 - \cos \alpha }}{2}} \] \[ \cos \frac{\alpha }{2} = \pm \sqrt {\frac{{1 + \cos \alpha }}{2}} \] \[ \tan \frac{\alpha }{2} = \pm \sqrt {\frac{{1 - \cos \alpha }}{{1 + \cos \alpha }}} = \frac{{1 - \cos \alpha }}{{\sin \alpha }} = \frac{{\sin \alpha }}{{1 + \cos \alpha }} \] \[ \cot \frac{\alpha }{2} = \pm \sqrt {\frac{{1 + \cos \alpha }}{{1 - \cos \alpha }}} = \frac{{1 + \cos \alpha }}{{\sin \alpha }} = \frac{{\sin \alpha }}{{1 - \cos \alpha }} \] \[ \sec \frac{\alpha }{2} = \pm \sqrt {\frac{{2\sec \alpha }}{{\sec \alpha + 1}}} \] \[ \csc \frac{\alpha }{2} = \pm \sqrt {\frac{{2\sec \alpha }}{{\sec \alpha - 1}}} \]




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