三角恒等式公式汇总¶
1. 和角公式¶
\[
\sin(A + B) = \sin A \cos B + \cos A \sin B
\]
\[
\cos(A + B) = \cos A \cos B - \sin A \sin B
\]
\[
\tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B}
\]
2. 差角公式¶
\[
\sin(A - B) = \sin A \cos B - \cos A \sin B
\]
\[
\cos(A - B) = \cos A \cos B + \sin A \sin B
\]
\[
\tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B}
\]
3. 倍角公式¶
\[
\sin(2a) = 2 \sin a \cos a
\]
\[
\cos(2a) = \cos^2 a - \sin^2 a = 2\cos^2 a - 1 = 1 - 2\sin^2 a
\]
\[
\tan(2a) = \frac{2 \tan a}{1 - \tan^2 a}
\]
4. 降幂公式¶
\[
\sin^2 a = \frac{1 - \cos 2a}{2}
\]
\[
\cos^2 a = \frac{1 + \cos 2a}{2}
\]
\[
\tan^2 a = \frac{1 - \cos 2a}{1 + \cos 2a}
\]
\[
\cot^2 a = \frac{1 + \cos 2a}{1 - \cos 2a}
\]
5. 辅助角公式¶
\[
a \sin x + b \cos x = \sqrt{a^2 + b^2} \cdot \sin(x + \psi)
\]
其中：
\[
\cos \psi = \frac{a}{\sqrt{a^2 + b^2}}, \quad \sin \psi = \frac{b}{\sqrt{a^2 + b^2}}
\]
6. 积化和差公式（积变和差）¶
\[
\sin A \cos B = \frac{1}{2} \left[ \sin(A + B) + \sin(A - B) \right]
\]
\[
\cos A \cos B = \frac{1}{2} \left[ \cos(A + B) + \cos(A - B) \right]
\]
\[
\sin A \sin B = \frac{1}{2} \left[ \cos(A - B) - \cos(A + B) \right]
\]
7. 和差化积公式（和差变积）¶
\[
\sin A + \sin B = 2 \sin\left(\frac{A + B}{2}\right) \cos\left(\frac{A - B}{2}\right)
\]
\[
\sin A - \sin B = 2 \cos\left(\frac{A + B}{2}\right) \sin\left(\frac{A - B}{2}\right)
\]
\[
\cos A + \cos B = 2 \cos\left(\frac{A + B}{2}\right) \cos\left(\frac{A - B}{2}\right)
\]
\[
\cos A - \cos B = -2 \sin\left(\frac{A + B}{2}\right) \sin\left(\frac{A - B}{2}\right)
\]