100+ Maths Formulas

Algebra

$a^2 - b^2 = (a-b)(a+b)$

$(a+b)^2 = a^2 + 2ab + b^2$

$(a-b)^2 = a^2 - 2ab + b^2$

$(a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3$

$(a-b)^3 = a^3 - 3a^2b + 3ab^2 - b^3$

$a^3+b^3 = (a+b)(a^2-ab+b^2)$

$a^3-b^3 = (a-b)(a^2+ab+b^2)$

$\frac{a^m}{a^n} = a^{m-n}$

$(ab)^n = a^n b^n$

$a^{-n} = 1/a^n$

$\log(ab)=\log a + \log b$

$\log(a/b)=\log a - \log b$

$\log(a^n)=n\log a$

$\ln(e^x)=x$

Quadratic Equations

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

$D = b^2 - 4ac$

Sum of roots $= -b/a$

Product of roots $= c/a$

Geometry

Triangle area $= \frac{1}{2}bh$

Circle area $= \pi r^2$

Circle circumference $= 2\pi r$

Square area $= a^2$

Rectangle area $= lw$

Rectangle perimeter $= 2(l+w)$

Cube volume $= a^3$

Sphere volume $= \frac{4}{3}\pi r^3$

Sphere surface area $= 4\pi r^2$

Cylinder volume $= \pi r^2 h$

Cone volume $= \frac{1}{3}\pi r^2 h$

Trigonometry

$\sin^2\theta + \cos^2\theta = 1$

$\tan\theta = \frac{\sin\theta}{\cos\theta}$

$\sec\theta = 1/\cos\theta$

$\csc\theta = 1/\sin\theta$

$\cot\theta = 1/\tan\theta$

$\sin(2\theta)=2\sin\theta\cos\theta$

$\cos(2\theta)=\cos^2\theta-\sin^2\theta$

$\tan(2\theta)=\frac{2\tan\theta}{1-\tan^2\theta}$

$\sin(A+B)=\sin A\cos B+\cos A\sin B$

$\cos(A+B)=\cos A\cos B-\sin A\sin B$

Calculus

Derivatives

$\frac{d}{dx}(x^n)=nx^{n-1}$

$\frac{d}{dx}(\sin x)=\cos x$

$\frac{d}{dx}(\cos x)=-\sin x$

$\frac{d}{dx}(e^x)=e^x$

$\frac{d}{dx}(\ln x)=1/x$

$\frac{d}{dx}(\tan x)=\sec^2 x$

$\frac{d}{dx}(a^x)=a^x\ln a$

Integrals

$\int x^n dx = \frac{x^{n+1}}{n+1}+C$

$\int \frac{1}{x} dx = \ln|x| + C$

$\int e^x dx = e^x + C$

$\int \sin x dx = -\cos x + C$

$\int \cos x dx = \sin x + C$

$\int \sec^2 x dx = \tan x + C$

$\int a^x dx = \frac{a^x}{\ln a}+C$

Limits

$\lim_{x\to 0} \frac{\sin x}{x} = 1$

$\lim_{x\to \infty} 1/x = 0$

$\lim_{x\to 0}(1+x)^{1/x}=e$

Statistics & Probability

$\bar{x} = \frac{\sum x}{n}$

$\sigma^2=\frac{\sum (x-\bar{x})^2}{n}$

$\sigma=\sqrt{\sigma^2}$

$P(A)=\frac{\text{favourable}}{\text{total}}$

$P(A \cup B)=P(A)+P(B)-P(A\cap B)$

$P(A \cap B)=P(A)P(B|A)$

Combinatorics

$n! = n(n-1)\cdots1$

${}^nC_r=\frac{n!}{r!(n-r)!}$

${}^nP_r=\frac{n!}{(n-r)!}$

Matrices & Vectors

$A^{-1}=\frac{1}{\det(A)}\text{adj}(A)$

$\det\begin{pmatrix}a & b \\ c & d\end{pmatrix} = ad - bc$

$\vec{a}\cdot\vec{b}=|a||b|\cos\theta$

$\vec{a}\times\vec{b}=|a||b|\sin\theta$

$|\vec{a}|=\sqrt{a_x^2+a_y^2+a_z^2}$

Complex Numbers

$z=a+bi$

$|z|=\sqrt{a^2+b^2}$

$\bar{z}=a-bi$

$e^{i\theta}=\cos\theta+i\sin\theta$

$(re^{i\theta})^n = r^n e^{in\theta}$

Sequences & Series

$S_n=\frac{n}{2}(a+l)$

$S_n=\frac{n}{2}[2a+(n-1)d]$

$S_n=a\frac{r^n-1}{r-1}$

Infinite GP: $S=\frac{a}{1-r}$

$a_n=a+(n-1)d$

Physics (Maths-based)

$v = u + at$

$s = ut + \frac{1}{2}at^2$

$v^2 = u^2 + 2as$

$F = ma$

$W = Fd$

$P = W/t$

$\rho = m/V$

$p = F/A$

Advanced / Misc

$(a+b+c)^2 = a^2 + b^2 + c^2 + 2ab + 2ac + 2bc$

$(a+b+c)^3 = a^3+b^3+c^3+3(a+b)(b+c)(c+a)$

$\sqrt{a^2+b^2}$ (2D distance)

$\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$

$(x-h)^2+(y-k)^2=r^2$

$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$

$\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$

$E = mc^2$