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三角恒等变换

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  • 两角和与差的正弦、余弦和正切、余切公式 

正弦 `\sin(\theta\pm\psi)=\sin\theta\cos\psi\pm\cos\theta\sin\psi`;

余弦 `\cos(\theta\pm\psi)=\cos\theta\cos\psi\mp\sin\theta\sin\psi`;

正切 `\tan(\theta\pm\psi)=\frac{\tan\theta\pm\tan\psi}{1\mp\tan\theta\tan\psi}`;

余切 `\cot(\theta\pm\psi)=\frac{\cot\theta\cot\psi\mp1}{\cot\psi\pm\cot\theta}`;

  • 二倍角公式

正弦 `\sin 2\alpha = 2 \sin \alpha \cos \alpha = \frac{2 \tan \alpha} {1 + \tan^2 \alpha}`;

余弦 `\cos 2\theta = \cos^2 \theta - \sin^2 \theta = 2 \cos^2 \theta - 1 = 1 - 2 \sin^2 \theta = \frac{1 - \tan^2 \theta} {1 + \tan^2 \theta}`;

正切 `\tan 2\theta = \frac{2 \tan \theta} {1 - \tan^2 \theta} = \frac{1}{1-\tan\theta}-\frac{1}{1+\tan\theta}`; 

余弦 `\cot 2\theta = \frac{\cot \theta - \tan \theta}{2}`.

`\sin^2\theta = \frac{1 - \cos 2\theta}{2}`;

`\cos^2\theta = \frac{1 + \cos 2\theta}{2}`.

  • 半角公式

正弦 `\sin \frac{\theta}{2} = \pm\, \sqrt{\frac{1 - \cos \theta}{2}}`;

余弦 `\cos \frac{\theta}{2} = \pm\, \sqrt{\frac{1 + \cos\theta}{2}}`.

  • 积化和差与和差化积

积化和差

`\sin\Theta\sin\Phi={\cos(\Theta-\Phi)-\cos(\Theta+\Phi)\over2}`;

`\cos\Theta\cos\Phi={\cos(\Theta-\Phi)+\cos(\Theta+\Phi)\over2}`;

`\sin\Theta\cos\Phi={\sin(\Theta+\Phi)+\sin(\Theta-\Phi)\over2}`;

`\cos\Theta\sin\Phi={\sin(\Theta+\Phi)-\sin(\Theta-\Phi)\over2}`.

和差化积

`\sin\Theta+\sin\Phi=2\sin\frac{\Theta+\Phi}{2}\cos\frac{\Theta-\Phi}{2}`;

`\cos\Theta+\cos\Phi=2\cos\frac{\Theta+\Phi}{2}\cos\frac{\Theta-\Phi}{2}`;

`\cos\Theta-\cos\Phi=-2\sin{\Theta+\Phi\over2}\sin{\Theta-\Phi\over2}`;

`\sin\Theta-\sin\Phi=2\cos{\Theta+\Phi\over2}\sin{\Theta-\Phi\over2}`.