久久一区二区三区经典,夜夜av网

奇偶性

$\\ \sin(-x)=-\sin x \\ \cos(-x)=\cos x \\ \tan(-x)=-\tan x$

加減

$\\ \sin^2 x + cos^2 x=1 \\ \sin(x \pm y) = \sin x \cos y \pm \cos x \sin y \\ \cos(x \pm y) = \cos x \cos y \mp \sin x \sin y \\ \displaystyle \tan(x \pm y) = \frac{\tan x \pm \tan y}{1 \mp \tan x \tan y}$

$\displaystyle \sin(\frac\pi2+x)=\cos x,\quad \sin(\pi+x)=-\sin x,\quad \sin(\frac{3\pi}2+x)=-\cos x$

$\displaystyle \cos(\frac\pi2+x)=-\sin x,\quad \cos(\pi+x)=-\cos x,\quad \cos(\frac{3\pi}2+x)=\sin x$

$\displaystyle \tan(\frac\pi2 \pm x) = \mp \cot x,\quad \tan(\pi \pm x) = \pm \tan x,\quad \tan(\frac{3\pi}2 \pm x)= \mp \cot x$

倍角公式

$\displaystyle \sin2x = 2\sin x \cos x,\qquad \qquad \cos2x = \cos^2x - \sin^2x, \qquad \tan2x = \frac{2\tan x}{1-\tan^2x} = \frac2{\cot x-\tan x}$

$\displaystyle \sin3x = 3\sin x - 4\sin^3x,\qquad \cos3x = 4\cos^3x - 3\cos x,\quad \tan3x=\frac{3\tan x-tan^3x}{1-3\tan^2x}$

$\displaystyle \sin4x = 8\cos^3x\sin x - 4\cos x\sin x,\quad \cos4x = 8\cos^4x - 8\cos^2x+1, \quad \tan4x = \frac{4\tan x-4\tan^3x}{1-6\tan^2x+\tan^4x}$

萬能公式

$\displaystyle \sin2x = \frac{2\tan x}{1+\tan^2x},\quad \cos2x = \frac{1-\tan^2x}{1+\tan^2x},\quad \tan2x = \frac{2\tan x}{1-\tan^2x}$

半角公式

$\displaystyle \sin^2\frac x 2=\frac12(1-\cos x),\qquad \cos^2\frac x2=\frac12(1+\cos x) \\$

$\displaystyle \sin \frac x2 = \begin{cases} \displaystyle \sqrt{\frac {1}{2}(1-cos\ x)}, & 1 \le x \le \pi \\ \displaystyle -\sqrt{\frac {1}{2}(1-cos\ x)}, & \pi \le x \le 2\pi \end{cases}$

$\displaystyle \cos \frac x2 = \begin{cases} \displaystyle \sqrt{\frac {1}{2}(1+cos\ x)}, & -\pi \le x \le \pi \\ \displaystyle -\sqrt{\frac {1}{2}(1+cos\ x)}, & \pi \le x \le 3\pi \end{cases}$

$\displaystyle \tan\frac x2 = \frac{\sin x}{1+\cos x}=\frac{1-\cos x}{\sin x}$

和差化積

$\\ \displaystyle \sin x \pm \sin y = 2\sin\frac{x \pm y}2 \cos\frac{x \mp y}2 \\ \displaystyle \cos x + \cos y = 2\cos\frac{x+y}2 \cos\frac{x-y}2 \\ \displaystyle \cos x - \cos y = 2\sin\frac{x+y}2 \cos\frac{y-x}2 \\ \displaystyle \cos x \pm \sin x = \sqrt2 \sin(\frac\pi4 \pm x) \\ \displaystyle \tan x \pm \tan y = \frac{\sin(x \pm y)}{\cos x \cos y}$

積化和差

$\\ \displaystyle \sin x \sin y = \frac12 (\cos(x-y)-\cos(x+y)) \\ \displaystyle \cos x \cos y = \frac12 (\cos(x-y)+\cos(x+y)) \\ \displaystyle \sin x \cos y = \frac12 (\sin(x-y)+\sin(x+y)) \\ \displaystyle \tan x \tan y =\ \frac{\tan x + \tan y}{\cot x + \cot y} =\ -\frac{\tan x - \tan y}{\cot x - \cot y}$

方冪公式

$\\ \displaystyle \sin^2x = \frac12 (1-\cos2x) = 1-\cos^2x = \frac{\tan^2x}{1+\tan^2x}\\ \displaystyle \cos^2x = \frac12 (1+\cos2x) = 1-\sin^2x = \frac1{1+\tan^2x}\\ \displaystyle \tan^2x = \frac{\sin^2x}{1-\sin^2x} = \frac{1-\cos^2x}{\cos^2x}$

輔助角公式

$\\ \displaystyle a\sin\alpha + b\cos\alpha = \sqrt{a^2+b^2}\sin(\alpha+\varphi), \quad 0 \le \varphi \le 2\pi$

$\displaystyle \sin\alpha \pm \cos\alpha = \sqrt2\sin(\alpha\pm\frac\pi4)$

$\displaystyle \sqrt3\sin\alpha \pm \cos\alpha = 2\sin(\alpha\pm\frac\pi6)$

$\displaystyle \sin\alpha \pm \sqrt3\cos\alpha = 2\sin(\alpha\pm\frac\pi3)$

特殊三角等式

$\sin(\alpha+\beta)\sin(\alpha-\beta) = \sin^2\alpha-\sin^2\beta$

$\cos(\alpha+\beta)\cos(\alpha-\beta) = \cos^2\alpha-\sin^2\beta=\cos^2\beta-\sin^2\alpha$

$\cot\alpha-\tan\alpha = 2\cot2\alpha$

常見公式變形

$\displaystyle \sqrt{1+\cos\alpha} = \sqrt2\cos\frac\alpha2,\quad \sqrt{1-\cos\alpha} = \sqrt2\sin\frac\alpha2$

$\displaystyle 1 \pm \sin2\alpha =(\sin\alpha \pm \cos\alpha)^2,\quad \frac{1\pm\tan\alpha}{1\mp\tan\alpha} = \tan(\frac\pi4 \pm \alpha)$

$\displaystyle \tan\alpha \pm \tan\beta = \tan(\alpha \pm \beta)(1 \mp \tan\alpha\tan\beta)$

常見角的變換

$\displaystyle \alpha = (\alpha + \beta) - \beta, \quad \alpha = 2\frac\alpha2, \quad \frac\pi2 = (\frac\pi4+\alpha)+(\frac\pi4-\alpha)$

$\displaystyle 2\alpha = (\alpha+\beta) + (\alpha-\beta), \\ 2\beta = (\alpha+\beta) - (\alpha-\beta) \qquad$
$\displaystyle \frac{\alpha+\beta}2 = (\alpha-\frac\beta2) - (\frac\alpha2-\beta) \\ \displaystyle \frac{\alpha-\beta}2 = (\alpha+\frac\beta2) - (\frac\alpha2+\beta)$

冪指數(shù)運算法則

$\displaystyle a^ra^s=a^{r+s},\quad (a^r)^s=a^{rs},\quad (ab)^r=a^rb^r \\ a^{\frac m n}=\sqrt[n]{a^m},\quad (a>0, m,n\in\mathbb{N},且n>1)$

對數(shù)運算法則

$a^{\log_aN} = N, \quad \log_aa = 1, \quad \log_a 1 = 0 \quad (其中N>0,a>0且a \ne 1)$

$\displaystyle \log_a(MN) = \log_aM + \log_aN, \quad \log_a(\frac MN) = \log_aM - \log_aN$

$\displaystyle \log_aN^m = m\log_aN, \quad \log_a b = \frac{\log_cb}{\log_ca} (a>0且a\ne1,c>0且c\ne1,b>0)$

數(shù)列

	等差	等比
定義	$a_{n+1}-a_n = d$	$\frac{a_{n+1}}{a_n} = q$
通項公式	$a_n=a_1+(n-1)d\\a_n=a_m+(n-m)d$	$a_n=a_1q^{n-1}\\a_n=a_mq^{n-m}$
公差(比)	$\displaystyle d=\frac{a_n-a_1}{n-1}(n\ne1)\\\displaystyle d=\frac{a_n-a_m}{n-m}(n\ne m)$	$\displaystyle q^{n-1}=\frac{a_n}{a_1} \\ q^{n-m}=\frac{a_n}{a_m}$
前n項和	$\displaystyle S_n=\frac{n(a_1+a_n)}{2} \\ \displaystyle =na_1+\frac{n(n+1)}{2}d$	$\displaystyle S_n=\frac{a_1(1-q^n)}{1-q}=\frac{a_1-a_nq}{1-q} \quad (q\ne1) \\ S_n=na_1 \quad(q=1)$
中項公式	$\displaystyle A=\frac a b$	$G=\pm\sqrt{ab} \ (ab>0)$

$\displaystyle 1+2+3+\cdots+n \qquad\qquad = \frac{n(n+2)}2 \\ \displaystyle 1+3+5+\cdots+(2n-1) \quad =n^2 \\ \displaystyle 1^2+2^2+3^2+\cdots+n^2 \qquad=\frac{n(n+1)(2n+1)}6 \\ \displaystyle 1^3+2^3+3^3+\cdots+n^3 \qquad=[\frac{n(n+1)}2]^2$

導(dǎo)數(shù)

法則	公式
加法	$(f+g)'=f'+g'$
乘以常數(shù)	$(Cf)'=Cf'$
乘法	$(f \cdot g)'=f' \cdot g+f \cdot g'$
除法	$\displaystyle (\frac f g)'= \frac{f' \cdot g - f \cdot g'}{g^2}$
復(fù)合函數(shù)	$設(shè)\ y=f(u),u=g(x)\\\displaystyle f(g(x))'=\frac{df}{du}\frac{du}{dx}$

函數(shù)	導(dǎo)數(shù)	取值范圍
C常數(shù)	$0$	$x\in \mathbb{R}$
x	$1$	$x\in \mathbb{R}$
$x^2$	$2x$	$x\in \mathbb{R}$
$x^n(n=1,2\cdots)$	$nx^{n-1}$	$x\in \mathbb{R}$
$\displaystyle \frac1x$	$\displaystyle -\frac1{x^2}$	$x\ne0$
$\displaystyle \frac1{x^n}(n=1,2\cdots)$	$\displaystyle -\frac{n}{x^{n+1}}$	$x\ne0$
$\displaystyle x^a=e^{a\cdot \ln x}(a為實數(shù))$	$ax^{a-1}$	$x>0$
$\displaystyle \sqrt x=x^{\frac12}$	$\displaystyle \frac1{2\sqrt x}$	$x>0$
$\displaystyle \sqrt[n]x=x^{\frac1n}(n=2,3\cdots)$	$\displaystyle \frac{\sqrt[n]x}{nx}$	$x>0$
$\ln x$	$\displaystyle \frac1x$	$x>0$
$\displaystyle \log_ax=\frac{\ln x}{\ln a}\\(a>0,a\ne1)$	$\displaystyle \frac1{x\ln a}$	$x>0$
$e^x$	$e^x$	$x\in \mathbb{R}$
$a^x=e^{x\cdot\ln a}\\(a>0,a\ne1)$	$a^x\ln a$	$x\in \mathbb{R}$
$\sin x$	$\cos x$	$x\in \mathbb{R}$
$\cos x$	$-\sin x$	$x\in \mathbb{R}$
$\tan x$	$\displaystyle \frac1{\cos^2x}$	$\displaystyle x \ne k\pi+\frac\pi2,k\in\mathbb{Z}$