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常用三角函数公式表

时间:2026-03-20 03:11:08

三角函数的基本公式包括以下几类:

同角三角函数的基本关系

$\sin^2\alpha + \cos^2\alpha = 1$

$\tan\alpha = \frac{\sin\alpha}{\cos\alpha}$

$\cot\alpha = \frac{\cos\alpha}{\sin\alpha}$

诱导公式

$\sin(\pi + \alpha) = -\sin\alpha$

$\cos(\pi + \alpha) = -\cos\alpha$

$\tan(\pi + \alpha) = \tan\alpha$

$\sin(-\alpha) = -\sin\alpha$

$\cos(-\alpha) = \cos\alpha$

$\tan(-\alpha) = -\tan\alpha$

$\sin(\pi - \alpha) = \sin\alpha$

$\cos(\pi - \alpha) = -\cos\alpha$

$\tan(\pi - \alpha) = -\tan\alpha$

$\sin(\frac{\pi}{2} - \alpha) = \cos\alpha$

$\cos(\frac{\pi}{2} - \alpha) = \sin\alpha$

$\sin(\alpha + \frac{\pi}{2}) = \cos\alpha$

和差角公式

$\sin(A + B) = \sin A \cos B + \cos A \sin B$

$\sin(A - B) = \sin A \cos B - \cos A \sin B$

$\cos(A + B) = \cos A \cos B - \sin A \sin B$

$\cos(A - B) = \cos A \cos B + \sin A \sin B$

$\tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B}$

$\tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B}$

$\cot(A + B) = \frac{\cot A \cot B - 1}{\cot B + \cot A}$

$\cot(A - B) = \frac{\cot A \cot B + 1}{\cot B - \cot A}$

倍角公式

$\sin 2A = 2 \sin A \cos A$

$\cos 2A = \cos^2 A - \sin^2 A = 1 - 2 \sin^2 A = 2 \cos^2 A - 1$

$\tan 2A = \frac{2 \tan A}{1 - \tan^2 A}$

半角公式

$\sin\frac{A}{2} = \pm \sqrt{\frac{1 - \cos A}{2}}$

$\cos\frac{A}{2} = \pm \sqrt{\frac{1 + \cos A}{2}}$

$\tan\frac{A}{2} = \pm \sqrt{\frac{1 - \cos A}{1 + \cos A}}$

积化和差公式

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

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

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

$\cos A \cos B = \frac{1}{2}[\cos(A + B) + \cos(A - B)]$

和差化积公式

$\sin A + \sin B = 2 \sin\frac{A + B}{2} \cos\frac{A - B}{2}$

$\sin A - \sin B = 2 \cos\frac{A + B}{2} \sin\frac{A - B}{2}$

$\cos A + \cos B = 2 \cos\frac{A + B}{2} \cos\frac{A - B}{2}$

$\cos A - \cos B = -2 \sin\frac{A + B}{2} \sin\frac{A - B}{2}$

这些公式在解决三角函数问题时非常有用,建议熟练掌握