常用数学公式

1. 三角函数

和角公式 & 差角公式	\(\cos (\alpha + \beta) = \cos (\alpha) \cos (\beta) - \sin (\alpha) \sin(\beta)\)，\(\cos (\alpha - \beta) = \cos (\alpha) \cos (\beta) + \sin (\alpha) \sin(\beta)\) \( \sin (\alpha + \beta) = \sin(\alpha)\cos(\beta) + \sin(\beta)\cos(\alpha) \)， \( \sin (\alpha - \beta) = \sin(\alpha)\cos(\beta) - \sin(\beta)\cos(\alpha) \) \( \tan (\alpha + \beta) = \cfrac{\tan (\alpha) + \tan (\beta)}{1 - \tan (\alpha) \tan (\beta)} \)， \( \tan (\alpha - \beta) = \cfrac{\tan (\alpha) - \tan (\beta)}{1 + \tan (\alpha) \tan (\beta)} \) \( \cot (\alpha + \beta) = \cfrac{\cot (\alpha) \cot (\beta) - 1}{\cot (\beta) + \cot(\alpha)} \)， \( \cot (\alpha + \beta) = \cfrac{\cot (\alpha) \cot (\beta) + 1}{\cot (\beta) - \cot(\alpha)} \)
倍角公式	\(\cos (2\alpha) = \cos^2(\alpha) - \sin^2(\alpha) = 2\cos^2(\alpha) - 1 = 1 - 2\sin^2(\alpha)\) \(\sin (2\alpha) = 2\cos(\alpha)\sin(\alpha)\) \(\tan (2\alpha) = \cfrac{2\tan(\alpha)}{1 - \tan^2(\alpha)}\) \(\cot (2\alpha) = \cfrac{\cot^2(\alpha) - 1}{2\cot(\alpha)}\)
和差化积	\(\sin(\alpha) + \sin(\beta) = 2\sin \left(\cfrac{\alpha + \beta}{2}\right)\cos \left(\cfrac{\alpha - \beta}{2}\right)\)， \(\sin(\alpha) - \sin(\beta) = 2\sin \left(\cfrac{\alpha - \beta}{2}\right)\cos \left(\cfrac{\alpha + \beta}{2}\right)\) \(\cos(\alpha) + \cos(\beta) = 2\cos \left(\cfrac{\alpha + \beta}{2}\right)\cos \left(\cfrac{\alpha - \beta}{2}\right)\)， \(\cos(\alpha) - \cos(\beta) = -2\sin \left(\cfrac{\alpha + \beta}{2}\right)\sin \left(\cfrac{\alpha - \beta}{2}\right)\)
欧拉公式	\(e^{ix} = \cos x + i \sin x\), \(\sin x = \cfrac{e^{ix} - e^{-ix}}{2i} \), \(\cos x = \cfrac{e^{ix} + e^{-ix}}{2} \)
正割余割	\(\sec x = \cfrac{1}{\cos x}\), \(\csc x = \cfrac{1}{\sin x}\)

2. 常见等式和不等式

平方和	\(1^2 + 2^2 + \cdots + n^2 = \cfrac{n(n+1)(2n+1)}{6}\)
立方和	\(1^3 + 2^3 + \cdots + n^3 = (1 + 2 + ... + n)^2 = \left[\cfrac{n(n+1)}{2}\right]^2\)
正弦绝对值	\(|\sin(x_2) - \sin(x_1)| ≤ |x_2 - x_1|\)

3. 微积分

\(f(x) = u(x)v(x)\)	\(f'(x) = u'(x)v(x) + u(x)v'(x)\)
\(f(x) = \cfrac{u(x)}{v(x)}\)	\(f'(x) = \cfrac{u'(x)v(x) - u(x)v'(x)}{v^2(x)}\)

4. 双曲函数

双曲正弦	\(\sinh x = \cfrac{e^x - e^{-x}}{2}\)	双曲余弦	\(\cosh x = \cfrac{e^x + e^{-x}}{2}\)
双曲正切	\(\tanh x = \cfrac{\sinh x}{\cosh x} = \cfrac{e^x - e^{-x}}{e^x + e^{-x}}\)	双曲余切	\(\cosh x = \cfrac{\cosh x}{\sinh x} = \cfrac{e^x + e^{-x}}{e^x - e^{-x}}\)
双曲正割	\(\text{sech} x = \cfrac{1}{\cosh x} = \cfrac{2}{e^x + e^{-x}}\)	双曲余割	\(\text{csch} x = \cfrac{1}{\sinh x} = \cfrac{2}{e^x - e^{-x}}\)
等式	\(\sinh (-x) = -\sinh x\) \(\cosh (-x) = \cosh x\) \(\cosh^2 x - \sinh^2 x = 1\)
加法公式	\(\sinh (x + y) = \sinh x \cosh y + \cosh x \sinh y\), \(\sinh (x - y) = \sinh x \cosh y - \cosh x\sinh y\) \(\cosh (x + y) = \cosh x \cosh y + \sinh x \sinh y\), \(\cosh (x - y) = \cosh x \cosh y - \sinh x\sinh y\)
倍角公式	\(\sinh (2x) = 2\sinh x \cosh y\) \(\cosh (2x) = \cosh^2 x + \sinh^2 x = 1 + 2\sinh^2 x\)
和差化积	\(\sinh x + \sinh y = 2\sinh \cfrac{x + y}{2}\cosh \cfrac{x - y}{2}\) \(\sinh x - \sinh y = 2\cosh \cfrac{x + y}{2}\sinh \cfrac{x - y}{2}\)

5. 高等代数

混合积	\(\boldsymbol{a} \times \boldsymbol{b} \cdot \boldsymbol{c} = \boldsymbol{b} \times \boldsymbol{c} \cdot \boldsymbol{a} = \boldsymbol{c} \times \boldsymbol{a} \cdot \boldsymbol{b}\)
矢量的连乘积	\(\boldsymbol{a} \cdot (\boldsymbol{b} \times \boldsymbol{c}) = \begin{vmatrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \\ \end{vmatrix}, \boldsymbol{a} \cdot (\boldsymbol{b} \times \boldsymbol{c}) = \boldsymbol{b} \cdot (\boldsymbol{c} \times \boldsymbol{a}) = \boldsymbol{c} \cdot (\boldsymbol{a} \times \boldsymbol{b}) \)