x=−b±b2−4ac2a{\displaystyle x={\frac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}}} a2−b2=(a+b)(a−b){\displaystyle a^{2}-b^{2}=\left(a+b\right)\left(a-b\right)} a3±b3=(a±b)(a2∓ab+b2){\displaystyle a^{3}\pm b^{3}=\left(a\pm b\right)\left(a^{2}\mp ab+b^{2}\right)} logaxy=logax+logay{\displaystyle \log _{a}xy=\log _{a}x+\log _{a}y} logaxy=logax−logay{\displaystyle \log _{a}{\frac {x}{y}}=\log _{a}x-\log _{a}y} logaxm=mlogax{\displaystyle \log _{a}x^{m}=m\log _{a}x} alogax=x{\displaystyle a^{\log _{a}x}=x} logax=logbxlogba{\displaystyle \log _{a}x={\frac {\log _{b}x}{\log _{b}a}}} a_⋅b_=|a_||b_|cosθ{\displaystyle {\underline {a}}\cdot {\underline {b}}=\left|{\underline {a}}\right|\left|{\underline {b}}\right|\cos {\theta }} (a+b)n=∑r=0nnCran−rbr{\displaystyle \left(a+b\right)^{n}=\sum _{r=0}^{n}{_{n}}C_{r}a^{n-r}b^{r}} (1+x)n=1+n1!x+n(n−1)2!x2+⋯+n(n−1)…(n−r+1)r!xr+…{\displaystyle \left(1+x\right)^{n}=1+{\frac {n}{1!}}x+{\frac {n\left(n-1\right)}{2!}}x^{2}+\dots +{\frac {n\left(n-1\right)\dots \left(n-r+1\right)}{r!}}x^{r}+\dots } [r(cosθ+isinθ)]n=rn(cosnθ+isinnθ){\displaystyle \left[r\left(\cos {\theta }+i\sin {\theta }\right)\right]^{n}=r^{n}\left(\cos {n\theta }+i\sin {n\theta }\right)} A−1=1det(A)adj(A){\displaystyle A^{-1}={\frac {1}{\det \left(A\right)}}{\text{adj}}\left(A\right)} 等差数列
等比数列
∑k=1nk=n(n+1)2{\displaystyle \sum _{k=1}^{n}k={\frac {n\left(n+1\right)}{2}}} ∑k=1nk2=n(n+1)(2n+1)6{\displaystyle \sum _{k=1}^{n}k^{2}={\frac {n\left(n+1\right)\left(2n+1\right)}{6}}} ∑k=1nk3=[n(n+1)2]2{\displaystyle \sum _{k=1}^{n}k^{3}=\left[{\frac {n\left(n+1\right)}{2}}\right]^{2}}
弧长=rθ{\displaystyle =r\theta } 扇形面积=12r2θ{\displaystyle ={\frac {1}{2}}r^{2}\theta } tanθ=sinθcosθ{\displaystyle \tan {\theta }={\frac {\sin {\theta }}{\cos {\theta }}}} cscθ=1sinθ{\displaystyle \csc {\theta }={\frac {1}{\sin {\theta }}}} secθ=1cosθ{\displaystyle \sec {\theta }={\frac {1}{\cos {\theta }}}} cotθ=1tanθ{\displaystyle \cot {\theta }={\frac {1}{\tan {\theta }}}} asinA=bsinB=csinC=2R{\displaystyle {\frac {a}{\sin {A}}}={\frac {b}{\sin {B}}}={\frac {c}{\sin {C}}}=2R} a2=b2+c2−2bccosA{\displaystyle a^{2}=b^{2}+c^{2}-2bc\cos {A}} Δ=12absinC{\displaystyle \Delta ={\frac {1}{2}}ab\sin {C}} Δ=s(s−a)(s−b)(s−c),s=a+b+c2{\displaystyle \Delta ={\sqrt {s\left(s-a\right)\left(s-b\right)\left(s-c\right)}},s={\frac {a+b+c}{2}}} 内切圆半径r=Δs{\displaystyle r={\frac {\Delta }{s}}} sin2θ+cos2θ=1{\displaystyle \sin ^{2}{\theta }+\cos ^{2}{\theta }=1} sec2θ=1+tan2θ{\displaystyle \sec ^{2}{\theta }=1+\tan ^{2}{\theta }} csc2θ=1+cot2θ{\displaystyle \csc ^{2}{\theta }=1+\cot ^{2}{\theta }} sin(A±B)=sinAcosB±cosAsinB{\displaystyle \sin {\left(A\pm B\right)}=\sin {A}\cos {B}\pm \cos {A}\sin {B}} cos(A±B)=cosAcosB∓sinAsinB{\displaystyle \cos {\left(A\pm B\right)}=\cos {A}\cos {B}\mp \sin {A}\sin {B}} tan(A±B)=tanA±tanB1∓tanAtanB{\displaystyle \tan {\left(A\pm B\right)}={\frac {\tan {A}\pm \tan {B}}{1\mp \tan {A}\tan {B}}}} sin2A=2sinAcosA{\displaystyle \sin {2A}=2\sin {A}\cos {A}} cos2A=cos2A−sin2A=2cos2A−1=1−2sin2A{\displaystyle {\begin{aligned}\cos {2A}&=\cos ^{2}{A}-\sin ^{2}{A}\\&=2\cos ^{2}{A}-1\\&=1-2\sin ^{2}{A}\\\end{aligned}}} tan2A=2tanA1−tan2A{\displaystyle \tan {2A}={\frac {2\tan {A}}{1-\tan ^{2}{A}}}} sinAcosB=sin(A+B)+sin(A−B)2{\displaystyle \sin {A}\cos {B}={\frac {\sin {\left(A+B\right)}+\sin {\left(A-B\right)}}{2}}} cosAcosB=cos(A+B)+cos(A−B)2{\displaystyle \cos {A}\cos {B}={\frac {\cos {\left(A+B\right)}+\cos {\left(A-B\right)}}{2}}} sinAsinB=cos(A−B)−cos(A+B)2{\displaystyle \sin {A}\sin {B}={\frac {\cos {\left(A-B\right)}-\cos {\left(A+B\right)}}{2}}} sinA+sinB=2sinA+B2cosA−B2{\displaystyle \sin {A}+\sin {B}=2\sin {\frac {A+B}{2}}\cos {\frac {A-B}{2}}} sinA−sinB=2cosA+B2sinA−B2{\displaystyle \sin {A}-\sin {B}=2\cos {\frac {A+B}{2}}\sin {\frac {A-B}{2}}} cosA+cosB=2cosA+B2cosA−B2{\displaystyle \cos {A}+\cos {B}=2\cos {\frac {A+B}{2}}\cos {\frac {A-B}{2}}} cosA−cosB=−2sinA+B2sinA−B2{\displaystyle \cos {A}-\cos {B}=-2\sin {\frac {A+B}{2}}\sin {\frac {A-B}{2}}}
d=(x2−x1)2+(y2−y1)2{\displaystyle d={\sqrt {\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}}}} 分比公式(mx2+nx1m+n,my2+ny1m+n){\displaystyle \left({\frac {mx_{2}+nx_{1}}{m+n}},{\frac {my_{2}+ny_{1}}{m+n}}\right)} 两直线的夹角θ{\displaystyle \theta } ，tanθ=|m2−m11+m2m1|{\displaystyle \tan {\theta }=\left|{\frac {m_{2}-m_{1}}{1+m_{2}m_{1}}}\right|} 直线方程是y−y1=m(x−x1){\displaystyle y-y_{1}=m\left(x-x_{1}\right)} 点到直线的距离=|Ax0+By0+CA2+B2|{\displaystyle =\left|{\frac {Ax_{0}+By_{0}+C}{\sqrt {A^{2}+B^{2}}}}\right|} 三角形的面积=12|(x1y2+x2y3+x3y1)−(x2y1+x3y2+x1y3)|{\displaystyle ={\frac {1}{2}}\left|\left(x_{1}y_{2}+x_{2}y_{3}+x_{3}y_{1}\right)-\left(x_{2}y_{1}+x_{3}y_{2}+x_{1}y_{3}\right)\right|} 圆的标准式(x−h)2+(y−k)2=r2{\displaystyle \left(x-h\right)^{2}+\left(y-k\right)^{2}=r^{2}} 平移{x=x′+hy=y′+k{\displaystyle {\begin{cases}x=x^{\prime }+h\\y=y^{\prime }+k\\\end{cases}}} 转轴{x=x′cosθ−y′sinθy=y′sinθ+y′cosθ{\displaystyle {\begin{cases}x=x^{\prime }\cos {\theta }-y^{\prime }\sin {\theta }\\y=y^{\prime }\sin {\theta }+y^{\prime }\cos {\theta }\\\end{cases}}} 抛物线
椭圆
双曲线
平均数x¯=∑fixi∑fi{\displaystyle {\bar {x}}={\frac {\sum {f_{i}x_{i}}}{\sum {f_{i}}}}} 平均差=∑|xi−x¯|fi∑fi{\displaystyle ={\frac {\sum {\left|x_{i}-{\bar {x}}\right|f_{i}}}{\sum {f_{i}}}}} 中位数M=L+(n2−Fmfm)Cm{\displaystyle M=L+\left({\frac {{\frac {n}{2}}-F_{m}}{f_{m}}}\right)C_{m}} 众数=L+(d1d1+d2)C{\displaystyle =L+\left({\frac {d_{1}}{d_{1}+d{2}}}\right)C} 上四分位数Q3=L3+(3n4−F3f3)C3{\displaystyle Q_{3}=L{3}+\left({\frac {{\frac {3n}{4}}-F_{3}}{f_{3}}}\right)C_{3}} 下四分位数Q1=L1+(n4−F1f1)C1{\displaystyle Q_{1}=L{1}+\left({\frac {{\frac {n}{4}}-F_{1}}{f_{1}}}\right)C_{1}} 四分位距=Q3−Q1{\displaystyle =Q_{3}-Q_{1}} 四分位差Q.D.=Q3−Q12{\displaystyle Q.D.={\frac {Q_{3}-Q_{1}}{2}}} 方差σ2=∑(xi−x¯)2fi∑fi=∑xi2fi∑fi−x¯2{\displaystyle \sigma ^{2}={\frac {\sum {\left(x_{i}-{\bar {x}}\right)^{2}f_{i}}}{\sum {f_{i}}}}={\frac {\sum {x_{i}^{2}f_{i}}}{\sum {f_{i}}}}-{\bar {x}}^{2}} 标准差σ=∑(xi−x¯)2fi∑fi=∑xi2fi∑fi−x¯2{\displaystyle \sigma ={\sqrt {\frac {\sum {\left(x_{i}-{\bar {x}}\right)^{2}f_{i}}}{\sum {f_{i}}}}}={\sqrt {{\frac {\sum {x_{i}^{2}f_{i}}}{\sum {f_{i}}}}-{\bar {x}}^{2}}}} 统计指数I=Q1Q0×100{\displaystyle I={\frac {Q_{1}}{Q_{0}}}\times 100} 综合指数=∑wixi∑wi{\displaystyle ={\frac {\sum {w_{i}x_{i}}}{\sum {w_{i}}}}} P(A∪B)=P(A)+P(B)−P(A∩B){\displaystyle P\left(A\cup B\right)=P\left(A\right)+P\left(B\right)-P\left(A\cap B\right)} P(A)=1−P(A′){\displaystyle P\left(A\right)=1-P\left(A^{\prime }\right)} nPr=n!(n−r)!{\displaystyle {_{n}}P_{r}={\frac {n!}{\left(n-r\right)!}}} nCr=n!(n−r)!r!{\displaystyle {_{n}}C_{r}={\frac {n!}{\left(n-r\right)!r!}}} 期望值E=x1p1+x2p2+⋯+xkpk{\displaystyle E=x_{1}p_{1}+x_{2}p_{2}+\dots +x_{k}p_{k}} 二项分配P(X=r)=nCrprqn−r{\displaystyle P\left(X=r\right)={_{n}}C_{r}p^{r}q^{n-r}}
limx→0sinxx=1{\displaystyle \lim _{x\to 0}{\frac {\sin {x}}{x}}=1} limx→∞(1+1x)x=e{\displaystyle \lim _{x\to \infty }{\left(1+{\frac {1}{x}}\right)^{x}}=e} ddx(uv)=udvdx+vdudx{\displaystyle {\frac {d}{dx}}\left(uv\right)=u{\frac {dv}{dx}}+v{\frac {du}{dx}}} ddx(uv)=vdudx−udvdxv2{\displaystyle {\frac {d}{dx}}\left({\frac {u}{v}}\right)={\frac {v{\frac {du}{dx}}-u{\frac {dv}{dx}}}{v^{2}}}} dydx=dydududx{\displaystyle {\frac {dy}{dx}}={\frac {dy}{du}}{\frac {du}{dx}}} ddxf(g(x))=f′(g(x))g′(x){\displaystyle {\frac {d}{dx}}f\left(g\left(x\right)\right)=f^{\prime }\left(g\left(x\right)\right)g^{\prime }\left(x\right)} ddxxn=nxx−1{\displaystyle {\frac {d}{dx}}x^{n}=nx^{x-1}} ddxsinx=cosx{\displaystyle {\frac {d}{dx}}\sin {x}=\cos {x}} ddxcosx=−sinx{\displaystyle {\frac {d}{dx}}\cos {x}=-\sin {x}} ddxtanx=sec2x{\displaystyle {\frac {d}{dx}}\tan {x}=\sec ^{2}{x}} ddxcotx=−csc2x{\displaystyle {\frac {d}{dx}}\cot {x}=-\csc ^{2}{x}} ddxsecx=secxtanx{\displaystyle {\frac {d}{dx}}\sec {x}=\sec {x}\tan {x}} ddxcscx=−cscxcotx{\displaystyle {\frac {d}{dx}}\csc {x}=-\csc {x}\cot {x}} ddxlnx=1x{\displaystyle {\frac {d}{dx}}\ln {x}={\frac {1}{x}}} ddxlogax=1xlna{\displaystyle {\frac {d}{dx}}\log _{a}x={\frac {1}{x\ln {a}}}} ddxex=ex{\displaystyle {\frac {d}{dx}}e^{x}=e^{x}} ddxax=axlna{\displaystyle {\frac {d}{dx}}a^{x}=a^{x}\ln {a}} ddxsin−1x=11−x2{\displaystyle {\frac {d}{dx}}\sin ^{-1}{x}={\frac {1}{\sqrt {1-x^{2}}}}} ddxtan−1x=11+x2{\displaystyle {\frac {d}{dx}}\tan ^{-1}{x}={\frac {1}{1+x^{2}}}} ∫xndx=xn+1n+1+C,n≠−1{\displaystyle \int x^{n}dx={\frac {x^{n+1}}{n+1}}+C,n\neq -1} ∫cosxdx=sinx+C{\displaystyle \int \cos {x}dx=sin{x}+C} ∫sinxdx=−cosx+C{\displaystyle \int \sin {x}dx=-\cos {x}+C} ∫sec2xdx=tanx+C{\displaystyle \int \sec ^{2}{x}dx=\tan {x}+C} ∫csc2xdx=cotx+C{\displaystyle \int \csc ^{2}{x}dx=\cot {x}+C} ∫secxtanxdx=secx+C{\displaystyle \int \sec {x}\tan {x}dx=\sec {x}+C} ∫cscxcotxdx=−cscx+C{\displaystyle \int \csc {x}\cot {x}dx=-\csc {x}+C} ∫1xdx=lnx+C{\displaystyle \int {\frac {1}{x}}dx=\ln {x}+C} ∫exdx=ex+C{\displaystyle \int e^{x}dx=e^{x}+C} ∫axdx=axlna+C{\displaystyle \int a^{x}dx={\frac {a^{x}}{\ln {a}}}+C} ∫dx1−x2=sin−1x+C{\displaystyle \int {\frac {dx}{\sqrt {1-x^{2}}}}=\sin ^{-1}{x}+C} ∫dx1+x2=tan−1x+C{\displaystyle \int {\frac {dx}{1+x^{2}}}=\tan ^{-1}{x}+C} 面积∫abydx{\displaystyle \int _{a}^{b}ydx} 或∫abxdy{\displaystyle \int _{a}^{b}xdy} 或∫αβ12[r(θ)]2dθ{\displaystyle \int _{\alpha }^{\beta }{\frac {1}{2}}\left[r\left(\theta \right)\right]^{2}d\theta } 体积π∫aby2dx{\displaystyle \pi \int _{a}^{b}y^{2}dx} 或π∫abx2dy{\displaystyle \pi \int _{a}^{b}x^{2}dy} 牛顿法xn=xn−1=f(xn−1)f′(xn−1){\displaystyle x_{n}=x_{n-1}={\frac {f\left(x_{n-1}\right)}{f^{\prime }\left(x_{n-1}\right)}}} 梯形法∫abf(x)dx≈b−an(y0+yn2+y1+y2+⋯+yn−1){\displaystyle \int _{a}^{b}f\left(x\right)dx\approx {\frac {b-a}{n}}\left({\frac {y_{0}+y_{n}}{2}}+y_{1}+y_{2}+\dots +y_{n-1}\right)} 辛普逊法∫abf(x)dx≈b−a6n[(y0+y2n)+4(y1+y3+⋯+y2n−1)+2(y2+y4+⋯+y2n−2)]{\displaystyle \int _{a}^{b}f\left(x\right)dx\approx {\frac {b-a}{6n}}\left[\left(y_{0}+y_{2n}\right)+4\left(y_{1}+y_{3}+\dots +y_{2n-1}\right)+2\left(y_{2}+y_{4}+\dots +y_{2n-2}\right)\right]}
![{\displaystyle \left[r\left(\cos {\theta }+i\sin {\theta }\right)\right]^{n}=r^{n}\left(\cos {n\theta }+i\sin {n\theta }\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/10066ef7e351e077c55e08f0af2b659095bc7c3b)
![{\displaystyle S_{n}={\frac {n}{2}}\left[2a+\left(n-1\right)d\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a41bc9111209f6863b31d836c3692ed1f031a0bf)
![{\displaystyle \sum _{k=1}^{n}k^{3}=\left[{\frac {n\left(n+1\right)}{2}}\right]^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/af8029449361675d9166b08c08a17e5fadfb6443)
![{\displaystyle \int _{\alpha }^{\beta }{\frac {1}{2}}\left[r\left(\theta \right)\right]^{2}d\theta }](https://wikimedia.org/api/rest_v1/media/math/render/svg/26d6b25564ce82f20f4c078759a57ed6140b39f6)
![{\displaystyle \int _{a}^{b}f\left(x\right)dx\approx {\frac {b-a}{6n}}\left[\left(y_{0}+y_{2n}\right)+4\left(y_{1}+y_{3}+\dots +y_{2n-1}\right)+2\left(y_{2}+y_{4}+\dots +y_{2n-2}\right)\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3359bdc6031defd22661505c23ae1dbfd98d910e)