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三倍角公式

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三倍角公式:

\[\sin 3\alpha  = 3\sin \alpha  - 4{\sin ^3}\alpha ,\cos 3\alpha  = 4{\cos ^3}\alpha  - 3\cos \alpha \]

\[\sin \alpha \sin ({60^0} + \alpha )\sin ({60^0} - \alpha ) = \frac{1}{4}\sin 3\alpha \]

\[\cos \alpha \cos ({60^0} + \alpha )\cos ({60^0} - \alpha ) = \frac{1}{4}\cos 3\alpha \]

\[\tan \alpha \tan ({60^0} + \alpha )\tan ({60^0} - \alpha ) = \tan 3\alpha \]


证明: `\sin 3\alpha  = 3\sin \alpha  - 4{\sin ^3}\alpha`.

证明: \begin{alignat}{2} \sin 3\alpha  & =  \sin (2\alpha + \alpha) \\ & = \sin 2\alpha \cos \alpha+ \cos 2\alpha \sin \alpha\\ & = 2\sin \alpha \cos ^2{\alpha} + \cos ^2{\alpha}\sin \alpha - \sin ^3{\alpha}\\ & = 3\sin \alpha \cos ^2 {\alpha}- \sin ^3{\alpha} \\ & = 3\sin \alpha (1-\sin ^2 {\alpha})- \sin ^3{\alpha} \\ & = 3\sin \alpha  - 4{\sin ^3}\alpha \end{alignat} .