重要公式

倍角公式

sin⁡2α=2sin⁡αcos⁡αcos⁡2α=cos⁡2α−sin⁡2αcos⁡2α=1−2sin⁡2αcos⁡2α=2cos⁡2α−1sin⁡3α=−4sin3α+3sin⁡αcos⁡3α=4cos3α−3cos⁡αtan⁡2α=2tan⁡α1−tan⁡2αcot⁡2α=cot⁡2α−12cot⁡α\sin2\alpha = 2\sin\alpha\cos\alpha \\ \cos2\alpha = \cos^2\alpha-\sin^2\alpha \\ \cos2\alpha = 1-2\sin^2\alpha \\ \cos2\alpha = 2\cos^2\alpha-1 \\ \sin3\alpha = -4sin^3\alpha + 3\sin\alpha \\ \cos3\alpha = 4cos^3\alpha - 3\cos\alpha \\ \tan2\alpha = \frac{2\tan\alpha}{1-\tan^2\alpha} \\ \cot2\alpha = \frac{\cot^2\alpha - 1}{2\cot\alpha}

半角公式

sin⁡2α2=12(1−cos⁡α)cos⁡2α2=12(1−cos⁡α)sin⁡α2=±1−cos⁡α2cos⁡α2=±1+cos⁡α2tan⁡α2=1−cos⁡αsin⁡αtan⁡α2=sin⁡α1+cos⁡αtan⁡α2=±1−cos⁡α1+cos⁡αcot⁡α2=1+cos⁡αsin⁡αcot⁡α2=sin⁡α1−cos⁡αcot⁡α2=±1+cos⁡α1−cos⁡α\sin^2 \frac{\alpha}{2} = \frac{1}{2} (1-\cos \alpha) \\ \cos^2 \frac{\alpha}{2} = \frac{1}{2} (1-\cos \alpha) \\ \sin \frac{\alpha}{2} = \pm \sqrt{\frac{1-\cos\alpha}{2}} \\ \cos \frac{\alpha}{2} = \pm \sqrt{\frac{1+\cos\alpha}{2}} \\ \tan \frac{\alpha}{2} = \frac{1-\cos\alpha}{\sin\alpha} \\ \tan \frac{\alpha}{2} = \frac{\sin\alpha}{1 + \cos\alpha} \\ \tan \frac{\alpha}{2} = \pm \sqrt{\frac{1-\cos\alpha}{1+\cos\alpha}} \\ \cot \frac{\alpha}{2} = \frac{1+\cos\alpha}{\sin\alpha} \\ \cot \frac{\alpha}{2} = \frac{\sin\alpha}{1 - \cos\alpha} \\ \cot \frac{\alpha}{2} = \pm \sqrt{\frac{1+\cos\alpha}{1-\cos\alpha}} \\

和差公式

sin⁡(α±β)=sin⁡αcos⁡β±cos⁡αsin⁡βcos⁡(α±β)=cos⁡αcos⁡β∓sin⁡αsin⁡βtan⁡(α±β)=tan⁡α±tan⁡β1∓tan⁡αtan⁡βcot⁡(α±β)=cot⁡αcot⁡β∓1cot⁡β±cot⁡α\sin(\alpha \pm \beta) = \sin\alpha\cos\beta \pm \cos\alpha\sin\beta \\ \cos(\alpha \pm \beta) = \cos\alpha\cos\beta \mp \sin\alpha\sin\beta \\ \tan(\alpha \pm \beta) = \frac{\tan\alpha \pm \tan\beta}{1 \mp \tan\alpha\tan\beta} \\ \cot(\alpha \pm \beta) = \frac{\cot\alpha\cot\beta \mp 1}{\cot\beta \pm \cot\alpha}

积化和差

sin⁡αcos⁡β=12[sin⁡(α+β)+sin⁡(α−β)]cos⁡αsin⁡β=12[sin⁡(α+β)−sin⁡(α−β)]cos⁡αcos⁡β=12[cos⁡(α+β)+cos⁡(α−β)]sin⁡αsin⁡β=−12[cos⁡(α+β)−cos⁡(α−β)]\sin\alpha\cos\beta = \frac{1}{2}[\sin(\alpha+\beta) + \sin(\alpha-\beta)] \\ \cos\alpha\sin\beta = \frac{1}{2}[\sin(\alpha + \beta) - \sin(\alpha - \beta)] \\ \cos\alpha\cos\beta = \frac{1}{2}[\cos(\alpha+\beta) + \cos(\alpha-\beta)] \\ \sin\alpha\sin\beta = - \frac{1}{2}[\cos(\alpha+\beta) - \cos(\alpha-\beta)]

和差化积

口口之和仍口口 赛赛之和赛口留 口口之差负赛赛 赛赛之差口塞收

cos⁡α+cos⁡β=2cos⁡α+β2cos⁡α−β2sin⁡α+sin⁡β=2sin⁡α+β2cos⁡α−β2cos⁡α−cos⁡β=−2sin⁡α+β2sin⁡α−β2sin⁡α−sin⁡β=2cos⁡α+β2sin⁡α−β2\cos\alpha + \cos\beta = 2\cos\frac{\alpha+\beta}{2}\cos\frac{\alpha-\beta}{2} \\ \sin\alpha + \sin\beta = 2\sin\frac{\alpha+\beta}{2}\cos\frac{\alpha-\beta}{2} \\ \cos\alpha - \cos\beta = -2\sin\frac{\alpha+\beta}{2}\sin\frac{\alpha-\beta}{2} \\ \sin\alpha - \sin\beta = 2\cos\frac{\alpha+\beta}{2}\sin\frac{\alpha-\beta}{2} \\

万能公式

sin⁡2x=2tan⁡x1+tan⁡2xcos⁡2x=1−tan⁡2x1+tan⁡2x\sin 2x = \frac{2\tan x}{1+\tan^2 x} \\ \cos 2x = \frac{1-\tan^2 x}{1+\tan^2 x}

对数运算法则

log⁡aMN=log⁡aM+log⁡aNlog⁡aMN=log⁡aM−log⁡aNlog⁡aMn=nlog⁡aMlog⁡aMn=1nlog⁡aM\log_a{MN} = \log_aM + \log_aN \\ \log_a{\frac{M}{N}} = \log_aM - \log_aN \\ \log_a{M^n} = n\log_aM \\ \log_a{\sqrt[n]{M}} = \frac{1}{n}\log_aM

一元二次方程

根的公式−b±b2−4ac2a\frac{-b \pm \sqrt{b^2-4ac}}{2a}

韦达定理

x1+x2=−bax1x2=cax_1+x_2=-\frac{b}{a} \\ x_1x_2=\frac{c}{a}

抛物线顶点 (−b2a,c−b24a)(-\frac{b}{2a}, c-\frac{b^2}{4a})

因式分解

(a+b)2=a2+2ab+b2(a−b)2=a2−2ab+b2(a+b)3=a3+3a2b+3ab2+b3(a−b)3=a3−3a2b+3ab2−b3a2−b2=(a+b)(a−b)a3−b3=(a−b)(a2+b2+ab)a3+b3=(a+b)(a2+b2−ab)an−bn=(a−b)(an−1+an−2b+⋯+abn−2+bn−1)a2n−b2n=(a+b)(a2n−1−a2n−2b+⋯+ab2n−2−b2n−1)a2n+1+b2n+1=(a+b)(a2n−a2n−1b+⋯−ab2n−1+b2n)(a+b)n=∑k=0nCnkan−kbk(a+b)^2 = a^2 + 2ab + b^2 \\ (a-b)^2 = a^2 - 2ab + b^2 \\ (a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3 \\ (a-b)^3 = a^3 - 3a^2b + 3ab^2 - b^3 \\ a^2 - b^2 = (a+b)(a-b) \\ a^3 - b^3 = (a-b)(a^2 + b^2 + ab) \\ a^3 + b^3 = (a+b)(a^2 + b^2 - ab) \\ a^n - b^n = (a-b)(a^{n-1} + a^{n-2}b + \cdots + ab^{n-2} + b^{n-1}) \\ a^{2n} - b^{2n} = (a+b)(a^{2n-1} - a^{2n-2}b + \cdots + ab^{2n-2} - b^{2n-1}) \\ a^{2n+1} + b^{2n+1} = (a+b)(a^{2n} - a^{2n-1}b + \cdots - ab^{2n-1} + b^{2n}) \\ (a+b)^n = \sum^n_{k=0} C^k_n a^{n-k}b^k

杨辉三角

C32=3C42=6C^2_3 = 3 \\ C^2_4 = 6 \\

阶乘双阶乘

0!=1(2n)!!=2nn!(2n−1)!!=1⋅3⋅5⋅…⋅(2n−1)∫0π2sin⁡10xdx=910⋅78⋅56⋅34⋅12⋅π2∫0π2cos⁡9xdx=89⋅67⋅45⋅230! = 1 \\ (2n)!! = 2^nn! \\ (2n-1)!! = 1 \cdot 3 \cdot 5 \cdot \ldots \cdot (2n-1) \\ \int^{\frac{\pi}{2}}_0 \sin^10 x dx = \frac{9}{10} \cdot \frac{7}{8} \cdot \frac{5}{6} \cdot \frac{3}{4} \cdot \frac{1}{2} \cdot \frac{\pi}{2} \\ \int^{\frac{\pi}{2}}_0 \cos^9 x dx = \frac{8}{9} \cdot \frac{6}{7} \cdot \frac{4}{5} \cdot \frac{2}{3}

常用不等式

∣a±b∣≤∣a∣+∣b∣∣∣a∣−∣b∣∣≤∣a−b∣∣∫abf(x)dx∣≤∫ab∣f(x)∣dxab≤a+b2≤a2+b22a>b,n>0,an>bna>b,n<0,an<bnsin⁡x<x<tan⁡xarctan⁡x≤x≤arcsin⁡xex≥x+1x−1≥ln⁡x11+x<ln⁡(1+1x)<1x|a \pm b| \le |a| + |b| \\ ||a| - |b|| \le |a-b| \\ |\int^b_a f(x) dx| \le \int^b_a |f(x)| dx \\ \sqrt{ab} \le \frac{a+b}{2} \le \sqrt{\frac{a^2+b^2}{2}} \\ a>b,n>0,a^n > b^n \\ a>b,n<0,a^n < b^n \\ \sin x < x < \tan x \\ \arctan x \le x \le \arcsin x \\ e^x \ge x+1 \\ x-1 \ge \ln x \\ \frac{1}{1+x} < \ln(1 + \frac{1}{x}) < \frac{1}{x}