高中数学公式合集

高中数学公式合集

概率

排列组合

排列 组合
Ank=n!(nk)!A_n^k = \frac {n!}{(n-k)!} Cnk=n!(nk)!k!C_n^k = \frac {n!}{(n-k)!k!}

均值与方差

期望 方差
E(X)=i=1n(xipi)E(X) = \sum_{i=1}^{n} (x_i p_i) D(X)=i=1n(xiE(X))2piD(X) = \sum_{i=1}^{n} (x_i - E(X))^2 p_i
  1. 均值方差的性质
    • E(aX+b)=aE(X)+bE(aX+b) = aE(X) + b
    • D(aX+b)=a2D(X)D(aX+b) = a^2D(X)
  2. 两点分布与二项分布的均值、方差
    • XX服从两点分布,则E(X)=pE(X) = p,D(X)=p(1p)D(X) = p(1-p)
    • XB(n,p)X \sim B(n, p),则E(X)=npE(X) = npD(X)=np(1p)D(X) = np(1-p)

不等式

  1. a+b2ab\frac{a+b}{2}\geq\sqrt{ab}
  2. a2+b22aba^2+b^2\geq2ab
  3. a+b+c3(abc)13{a+b+c}{3}\geq{(abc)}^\frac{1}{3}
  4. a3+b3+c33abca^3+b^3+c^3\geq 3abc
  5. a1+a2++ann(a1a2an)1n\frac{a_1+a_2+\dots+a_n}{n} \geq {(a_1a_2…a_n)}^\frac{1}{n}
  6. 21a+1baba+b2a2+b22\frac{2}{\frac{1}{a}+\frac{1}{b}}\leq\sqrt{ab}\leq\frac{a+b}{2}\leq\sqrt{\frac{a^2+b^2}{2}}
  7. ALG 不等式 (Arithmetic-Logarithmic-Geometric mean inequalities)(对数均值不等式)
  8. 算术平均数a+b2\frac{a+b}{2}对数平均数balnblna\frac{b-a}{\ln{b}-\ln{a}}几何平均数ab\sqrt{ab}
  9. 均值不等式:
    • 两数均值不等式:a+b2ab\frac{a+b}{2} \geq \sqrt {ab}
    • n 数均值不等式:

      a1+a2++anna1a2ann\frac {a_1+a_2+ \cdots + a_n}{n} \geq \sqrt [n]{a_1a_2 \cdots a_n}

    • 调和平均数、几何平均数、算术平均数、平方平均数
    • 21a+1baba+b2a2+b22\frac {2}{\frac{1}{a}+\frac{1}{b}} \leq \sqrt{ab} \leq \frac {a+b}{2} \leq \sqrt {\frac{a^2 + b^2}{2}}
  10. 柯西不等式:(a2+b2)(c2+d2)(ab+cd)2(a^2 + b^2)(c^2 + d^2) \geq (ab + cd)^2
  11. 糖水不等式:ba<b+ca+c\frac {b}{a} < \frac {b+c}{a+c}

不等式拓展阅读

三角函数

特殊值

正弦 余弦 正切
sin(0)=0\sin (0) = 0 cos(0)=1\cos (0) = 1 tan(0)=0\tan (0) = 0
sin(π6)=12\sin (\frac {\pi}{6}) = \frac {1}{2} cos(π6)=32\cos (\frac {\pi}{6}) = \frac {\sqrt {3}}{2} tan(π6)=33\tan (\frac {\pi}{6}) = \frac {\sqrt{3}}{3}
sin(π4)=22\sin (\frac {\pi}{4}) = \frac {\sqrt {2}}{2} cos(π4)=22\cos (\frac {\pi}{4}) = \frac {\sqrt {2}}{2} tan(π4)=1\tan (\frac {\pi}{4}) = 1
sin(π3)=32\sin (\frac {\pi}{3}) = \frac {\sqrt {3}}{2} cos(π3)=12\cos (\frac {\pi}{3}) = \frac {1}{2} tan(π3)=3\tan (\frac {\pi}{3}) = \sqrt {3}
sin(π2)=1\sin (\frac {\pi}{2}) = 1 cos(π2)=0\cos (\frac {\pi}{2}) = 0 tan(π2)=+\tan (\frac {\pi}{2}) = +\infty

诱导公式

s

和差角公式

  • cos(a+b)=cosacosbsinasinb\cos (a+b) = \cos a \cos b - \sin a \sin b
  • cos(ab)=cosacosb+sinasinb\cos (a-b) = \cos a \cos b + \sin a \sin b
  • sin(a±b)=sinacosb±cosasinb\sin (a \pm b) = \sin a \cos b \pm \cos a \sin b
  • tan(a+b)=tana+tanb1tanatanb\tan (a+b)=\frac{\tan a+ \tan b}{1 - \tan a \cdot \tan b}
  • tan(ab)=tanatanb1+tanatanb\tan (a-b)=\frac{\tan a- \tan b}{1+ \tan a \cdot \tan b}

和差化积

  • sina+sinb=2sina+b2cosab2\sin a+ \sin b=2 \sin \frac{a+b}{2} \cos \frac{a-b}{2}
  • sinasinb=2cosa+b2sinab2\sin a- \sin b = 2 \cos \frac{a+b}{2} \sin \frac{a-b}{2}
  • cosa+cosb=2cosa+b2cosab2\cos a+ \cos b = 2 \cos \frac{a+b}{2} \cos \frac{a-b}{2}
  • cosacosb=2sina+b2sinab2\cos a- \cos b = -2 \sin \frac{a+b}{2} \sin \frac{a-b}{2}
  • tana±tanb=sin(a±b)cosacosb\tan a \pm \tan b = \frac {\sin (a \pm b)}{\cos a \cdot \cos b}
  • cota±cotb=±sin(a±b)sinasinb\cot a \pm \cot b = \pm \frac {\sin (a \pm b)}{\sin a \cdot \sin b}

积化和差

  • sinαcosβ=12[sin(α+β)+sin(αβ)]\sin \alpha \cos \beta = \frac{1}{2}[\sin (\alpha + \beta) + \sin(\alpha - \beta)]
  • cosαsinβ=12[sin(αβ)+sin(αβ)]\cos \alpha \sin \beta = \frac{1}{2}[\sin (\alpha - \beta) + \sin(\alpha - \beta)]
  • cosαcosβ=12[cos(α+β)+cos(αβ)]\cos \alpha \cos \beta = \frac{1}{2}[\cos (\alpha + \beta) + \cos(\alpha - \beta)]
  • sinαsinβ=12[cos(αβ)+cos(αβ)]\sin \alpha \sin \beta = - \frac{1}{2}[\cos (\alpha - \beta) + \cos(\alpha - \beta)]

二倍角

  • sin(2x)=2sin(x)cos(x)\sin (2x) = 2 \sin(x) \cos(x)
  • cos(2x)=cos2(x)sin2(x)=2cos2(x)1=12sin2(x)\cos (2x) = \cos^2(x) - \sin^2(x) = 2 \cos^2(x) - 1 = 1 - 2 \sin^2(x)
  • tan(2x)=2tana1tan2a\tan (2x) = \frac{2 \tan a}{1 - \tan^2 a}

正弦定理

asin(A)=csin(C)=csin(C)=2R\frac{a}{\sin(A)}=\frac{c}{sin(C)}=\frac{c}{sin(C)}=2R

余弦定理

cos(C)=a2+b2c22abcos(C) = \frac {a^2+b^2-c^2}{2ab}

升幂降角

升角降幂

SABC=12ABsinCS_{\triangle ABC} = \frac {1}{2}AB \sin C

向量

数列

an=a1+(n1)da_n = a_1 + (n-1)d

ana_n是等差数列,则有

Sn=(a1+an)n2S_n = \frac {(a_1 + a_n) * n}{2}

an=a1qn1a_n = a_1 q^{n-1}

ana_n是等比数列,则有

Sn=a1(1qn)1qS_n = \frac {a_1 * (1 - q^n)}{1-q}

an=SnSn1a_n = S_n - S_{n-1}

  1. 裂项相消
    1. an=1n(n+1)a_n=\frac{1}{n(n+1)}
    2. Sn=a1+a2+a3++an=11×2+12×3++1n(n+1=1112+1213++1n1n+1=11n+1\begin{aligned}S_n&=a_1+a_2+a_3+\cdots+a_n\\ &=\frac{1}{1\times 2}+\frac{1}{2\times 3}+\cdots+\frac{1}{n(n+1}\\ &=\frac{1}{1}-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\cdots+\frac{1}{n}-\frac{1}{n+1}\\ &=1-\frac{1}{n+1}\end{aligned}
  2. 错位相减

函数

奇函数:f(x)=f(x)f(x) = -f(x)

偶函数:f(x)=f(x)f(x)=f(-x)
ar×as=ar+sa^r \times a^s = a^{r+s}
(ab)r=arbr{(ab)}^r = a^rb^r
(ar)s=ars{(a^r)}^s=a^{rs}

二次函数

  1. Δ=b24ac\Delta=b^2-4ac
  2. x=b±Δ2abx = \frac{-b \pm \sqrt{\Delta}}{2ab}
  3. 韦达定理

{x=1y=2+x\left\{\begin{aligned}x&=1\\y&=2+x\end{aligned}\right.

复数

a+bia+bi

空间几何

S=2πr(r+l)S_{圆柱体}=2\pi r(r+l)
V=ShV_{柱体}=Sh
SS_{圆锥}
VV_{锥}
SS_{圆台}
VV_{台}
SS_{球}
VV_{球}

解析几何

直线y=kx+by = kx + bAx+By+C=0Ax + By + C = 0
y=1xy = \frac {1}{x}
y=ax2y = ax^2

圆锥曲线

x2+y2=r2x^2 + y^2 = r^2
椭圆x2a2+y2b2=1\frac {x^2}{a^2} + \frac {y^2}{b^2} = 1
与椭圆相交的直线,交点线段长:AB=1+k2x1x2=1+1k2y1y2|AB| = \sqrt{1+k^2}|x_1 - x_2| = \sqrt{1+\frac{1}{k^2}}|y_1 - y_2|
抛物线x=2pyx = 2py
抛物线焦点弦长:AB=x1+x2+p=2psin2θ2p|AB| = x_1 + x_2 + p = \frac{2p}{\sin^2 \theta} \geq 2p

1AF+1BF=2p\frac{1}{|AF|}+\frac{1}{|BF|} = \frac{2}{p}

x1x2=p24x_1x_2 = \frac{p^2}{4} y1y2=p2y_1y_2 = -p^2

AF=p1cosθ|AF| = \frac{p}{1- \cos \theta}BF=p1+cosθ|BF| = \frac{p}{1+ \cos \theta}

SABC=p22sinθS_{\triangle ABC} = \frac{p^2}{2 \sin \theta}

双曲线x2a2y2b2=1\frac {x^2}{a^2} - \frac {y^2}{b^2} = 1

渐近线y=±baxy = \pm \frac {b}{a} x

斜率公式:kp1p2=y1y2x1x2k_{p_1p_2} = \frac{y_1 - y_2}{x_1 - x_2}

倾斜角α\alphak=tanα(απ2)k = \tan \alpha (\alpha \neq \frac{\pi}{2})

点到直线距离公式:d=Ax0+By0+CA2+B2d = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2+B^2}}

中点公式:x=x1+x22x = \frac{x_1+x_2}{2}x=y1+y22x = \frac{y_1+y_2}{2}

重心公式:x=x1+x2+x33x = \frac{x_1+x_2+x_3}{3}y=y1+y2+y33y = \frac{y_1+y_2+y_3}{3}

线段长度:s=(x1x2)2+(y1y2)2s=\sqrt{(x_1-x_2)^2+(y_1-y_2)^2}

对数

(N>0,a>0,a1)(N>0,a>0,a\neq 1)
logaMN=logaM+logaN\log_a{MN}=\log_aM+\log_aN
logaMN=logaMlogaN\log^a{\frac{M}{N}}=\log_aM-\log_aN
logaNn=nlogaN\log_a{N^n}=n\log_aN
alogaN=Na^{\log_a{N}} = N
logaa=1\log_a{a} = 1
loga1=0\log_a{1} = 0
(a>0a1,c>0c1)(a>0且a \neq 1, c>0 且 c \neq 1)
logab=logcblogca\log_a{b} = \frac{\log_c{b}}{\log_c{a}}

导数

原函数 导函数
kx+bkx + b kk
xax^a axa1ax^{a-1}
1x\frac{1}{x} 1x2- \frac{1}{x^2}
lnx\ln{x} 1x\frac{1}{x}
axa^x axlnaa^x \ln{a}
logaxlog_a{x} 1xlna\frac{1}{x \ln{a}}
sinx\sin x cosx\cos x
cosx\cos x sinx-\sin x
uvuv uv+uvuv'+u'v
u+vu+v u+vu'+v'

https://zhuanlan.zhihu.com/p/41855459
https://www.mohu.org/info/symbols/symbols.htm
https://texwiki.texjp.org/?LaTeX%E5%85%A5%E9%96%80%2F%E7%B0%A1%E5%8D%98%E3%81%AA%E6%95%B0%E5%BC%8F%282%29#ma22efee