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半角公式

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

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

$\cos \alpha=2 \cos ^{2} \frac{\alpha}{2}-1=1-2 \sin ^{2} \frac{\alpha}{2}$

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

$\cot \frac{\alpha}{2}=\frac{1+\cos \alpha}{\sin \alpha}=\frac{\sin \alpha}{1-\cos \alpha}$

$\sec \frac{\alpha}{2}=\frac{\pm \sqrt{\frac{\sec \alpha+1}{2 \sec \alpha}} 2 \sec \alpha}{\sec \alpha+1}=\frac{\pm \sqrt{\frac{4 \sec ^{3} \alpha+\sec ^{2} \alpha}{2 \sec \alpha}}}{\sec \alpha+1}$

$\csc \frac{\alpha}{2}=\frac{\pm \sqrt{\frac{\sec \alpha-1}{2 \sec \alpha}} 2 \sec \alpha}{\sec \alpha-1}=\frac{\pm \sqrt{\frac{4 \sec ^{3} \alpha-\sec ^{2} \alpha}{2 \sec \alpha}}}{\sec \alpha-1}$

倍角公式

$\sin 2 \alpha=2 \sin \alpha \cos \alpha$

$\cos 2 \alpha=\cos ^{2} \alpha-\sin ^{2} \alpha=2 \cos ^{2} \alpha-1=1-2 \sin ^{2} \alpha$

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

$\cot 2 \alpha=\frac{\cot ^{2} \alpha-1}{2 \cot \alpha}$

$\sec 2 \alpha=\frac{\sec ^{2} \alpha+\csc ^{2} \alpha}{\csc ^{2} \alpha-\sec ^{2} \alpha}=\frac{\sec ^{2} \alpha \csc ^{2} \alpha}{\csc ^{2} \alpha-\sec ^{2} \alpha}$

$\csc 2 \alpha=\frac{\sec ^{2} \alpha+\csc ^{2} \alpha}{2 \sec \alpha \csc \alpha}=\frac{\sec ^{2} \alpha \csc \alpha}{2}$

曲率公式

$曲率 K=\frac{\left|y^{\prime \prime}\right|}{\left(1+y^{\prime 2}\right)^{\frac{3}{2}}}$

$曲率半径 \rho=\frac{1}{K}=\frac{\left(1+y^{\prime 2}\right)^{\frac{3}{2}}}{\left|y^{\prime \prime}\right|}$

点到直线距离公式

设直线 $\mathrm{L}$ 的方程为 $\mathrm{Ax}+\mathrm{By}+\mathrm{C}=0$ ，点 $\mathrm{P}$ 的坐标为 $(x 0, y 0)$ ，则点 $\mathrm{P}$ 到直线 $\mathrm{L}$ 的距离为: $\frac{\left|A x_{0}+B y_{0}+C\right|}{\sqrt{A^{2}+B^{2}}}$

常用求导公式

$\left(x^{\alpha}\right)^{\prime}=\alpha x^{\alpha-1}, \quad\left(a^{x}\right)^{\prime}=a^{x} \ln a, \quad\left(e^{x}\right)^{\prime}=e^{x}, \quad\left(\log _{a} x\right)^{\prime}=\frac{1}{x \ln a}, \quad(\ln x)^{\prime}=\frac{1}{x}$

$(\sin x)^{\prime}=\cos x, \quad(\cos x)^{\prime}=-\sin x, \quad(\arcsin x)^{\prime}=\frac{1}{\sqrt{1-x^{2}}}, \quad(\arccos x)^{\prime}=-\frac{1}{\sqrt{1-x^{2}}}$

$(\tan x)^{\prime}=\sec ^{2} x, \quad(\cot x)^{\prime}=-\csc ^{2} x, \quad(\arctan x)^{\prime}=\frac{1}{1+x^{2}}, \quad(\operatorname{arccot} x)^{\prime}=-\frac{1}{1+x^{2}}$

$(\sec x)^{\prime}=\sec x \tan x, \quad(\csc x)^{\prime}=-\csc x \cot x$

常用等价无穷小

$a^{x}-1 \sim x \ln a$

$\arcsin (a) x \sim \sin (a) x \sim(a) x$

$\arctan (a) x \sim \tan (a) x \sim(a) x$

$\ln (1+x) \sim x$

$\sqrt{1+x}-\sqrt{1-x} \sim x$

$(1+a x)^{b}-1 \sim a b x$

$\sqrt[b]{1+a x}-1 \sim \frac{a}{b} x$

$1-\cos x \sim \frac{x^{2}}{2}$

$x-\ln (1+x) \sim \frac{x^{2}}{2}$

$\tan x-\sin x \sim \frac{x^{3}}{2}$

$\tan x-x \sim \frac{x^{3}}{3}$

$x-\arctan x \sim \frac{x^{3}}{3}$

$x-\sin x \sim \frac{x^{3}}{6}$

$\arcsin x-x \sim \frac{x^{3}}{6}$

常用麦克劳林公式

$e^{x}=1+x+\frac{x^{2}}{2 !}+\cdots+\frac{x^{n}}{n !}+\cdots=\sum_{n=0}^{\infty} \frac{x^{n}}{n !}$

$\sin x=x-\frac{1}{3 !} x^{3}+\cdots+(-1)^{n} \frac{1}{(2 n+1) !} x^{2 n+1}+\cdots=\sum_{n=0}^{\infty}(-1)^{n} \frac{x^{2 n+1}}{(2 n+1) !}$

$\cos x=1-\frac{1}{2 !} x^{2}+\cdots+(-1)^{n} \frac{1}{(2 n) !} x^{2 n}+\cdots=\sum_{n=0}^{\infty}(-1)^{n} \frac{x^{2 n}}{(2 n) !}$

$\ln (1+x)=x-\frac{1}{2} x^{2}+\cdots+(-1)^{n-1} \frac{x^{n}}{n}+\cdots=\sum_{n=0}^{\infty}(-1)^{n-1} \frac{x^{n}}{n},-1<x \leq 1$

$\frac{1}{1-x}=1+x+x^{2}+\cdots+x^{n}+\cdots=\sum_{n=0}^{\infty} x^{n},|x|<1$

$\frac{1}{1+x}=1-x+x^{2}-\cdots+(-1)^{n} x^{n}+\cdots=\sum_{n=0}^{\infty}(-1)^{n} x^{n},|x|<1$

$(1+x)^{\alpha}=1+\alpha x+\frac{\alpha(\alpha-1)}{2} x^{2}+o\left(x^{2}\right)(x \rightarrow 0, \alpha \neq 0)$

$\tan x=x+\frac{1}{3} x^{3}+o\left(x^{3}\right)(x \rightarrow 0)$

$\arcsin x=x+\frac{1}{6} x^{3}+o\left(x^{3}\right)(x \rightarrow 0)$

$\arctan x=x-\frac{1}{3} x^{3}+o\left(x^{3}\right)(x \rightarrow 0)$

常用无穷级数

$e^{x}=1+x+\frac{x^{2}}{2 !}+\frac{x^{3}}{3 !}+\ldots+\frac{x^{k}}{k !}+\ldots(-\infty<x<\infty)$

$\ln (1+x)=x-\frac{x^{2}}{2}+\frac{x^{3}}{3}-\cdots+\frac{(-1)^{k-1} x^{k}}{k}+\cdots(-1<x \leq 1)$

$\sin x=x-\frac{x^{3}}{3 !}+\frac{x^{5}}{5 !}-\ldots+\frac{(-1)^{k-1} x^{2 k-1}}{(2 k-1) !}+\ldots(-\infty<x<\infty)$

$\cos x=1-\frac{x^{2}}{2 !}+\frac{x^{4}}{4 !}-\ldots+\frac{(-1)^{k} x^{2 k}}{(2 k) !}+\ldots(-\infty<x<\infty)$

$\arcsin x=x+\frac{1}{2} \cdot \frac{x^{3}}{3}+\frac{1 \cdot 3}{2 \cdot 4} \cdot \frac{x^{5}}{5}+\frac{1 \cdot 3 \cdot 5}{2 \cdot 4 \cdot 6} \cdot \frac{x^{7}}{7} \cdots+\frac{\left(\begin{array}{c}2 k \\ k\end{array}\right) x^{2 k+1}}{4^{k}(2 k+1)}+\cdots(|x|<1)$

$\arccos x=\frac{\pi}{2}-\arcsin x=\frac{\pi}{2}-\left(x+\frac{1}{2} \cdot \frac{x^{3}}{3}+\frac{1 \cdot 3}{2 \cdot 4} \cdot \frac{x^{5}}{5}+\cdots\right)=\frac{\pi}{2}-\sum_{k=0}^{\infty} \frac{\left(\begin{array}{c}2 k \\ k\end{array}\right) x^{2 k+1}}{4^{k}(2 k+1)}(|x|<1)$

$\arctan x=x-\frac{x^{3}}{3}+\frac{x^{5}}{5}-\cdots+\frac{(-1)^{k} x^{2 k+1}}{2 k+1}+\cdots(|x| \leq 1)$

$\sinh x=x+\frac{x^{3}}{3 !}+\frac{x^{5}}{5 !}+\ldots+\frac{x^{2 k-1}}{(2 k-1) !}+\ldots(-\infty<x<\infty)$

$\cosh x=1+\frac{x^{2}}{2 !}+\frac{x^{4}}{4 !}+\ldots+\frac{x^{2 k}}{(2 k) !}+\ldots(-\infty<x<\infty)$

$\operatorname{arcsinh} x=x-\left(\frac{1}{2}\right) \frac{x^{3}}{3}+\left(\frac{1 \cdot 3}{2 \cdot 4}\right) \frac{x^{5}}{5}-\left(\frac{1 \cdot 3 \cdot 5}{2 \cdot 4 \cdot 6}\right) \frac{x^{7}}{7}+\cdots+\left(\frac{(-1)^{k}(2 k) !}{2^{2 k} k !^{2}}\right) \cdot \frac{x^{2 k+1}}{2 k+1}+\cdots(|x|<1)$

$\operatorname{arctanh} x=x+\frac{x^{3}}{3}+\frac{x^{5}}{5}+\cdots+\frac{x^{2 k+1}}{2 k+1}+\cdots(|x|<1)$

基本积分表

基本积分表的扩充