Ultimate math formulas

Proofs

Expansion and factorization

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

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

$(a + b)^n = \displaystyle \sum_{k = 0}^n \binom{n}{k} a^{n - k} b^k$ (Binomial theorem)

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

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

$a^n - b^n = (a - b) \displaystyle \sum_{k = 0}^{n - 1} a^{n - 1 - k} b^k$

Quadratic formula

$ax^2 + bx + c = 0$

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

Exponent and logarithm rules

$a^m a^n = a^{m + n}$

$(a^m)^n = a^{mn}$

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

$\log_a(MN) = \log_a M + \log_a N$

$\log_a(M^n) = n \log_a M$

$\log_a M = \displaystyle \frac{\log_b M}{\log_b a}$

Fractions

$\displaystyle \frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}$

$\displaystyle \frac{a}{b} + \frac{c}{d} + \frac{e}{f} = \frac{adf + bcf + bde}{bdf}$

Systems of linear equations

$\begin{cases} ax + by = e \\ cx + dy = f \end{cases} \iff \begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} e \\ f \end{pmatrix}$

$x = \displaystyle \frac{ed - bf}{ad - bc} = \frac{\begin{vmatrix} e & b \\ f & d \end{vmatrix}}{\begin{vmatrix} a & b \\ c & d \end{vmatrix}}$

$y = \displaystyle \frac{af - ec}{ad - bc} = \frac{\begin{vmatrix} a & e \\ c & f \end{vmatrix}}{\begin{vmatrix} a & b \\ c & d \end{vmatrix}}$

Sequences

$\displaystyle \sum_{k = 1}^n k = 1 + 2 + \cdots + n = \frac{n(n + 1)}{2}$

$\displaystyle \sum_{k = 1}^n \bigl( a + (k - 1)d \bigr) = a + (a + d) + \cdots + \bigl( a + (n - 1)d \bigr) = \frac{n \bigl( 2a + (n - 1)d \bigr)}{2}$

$\displaystyle \sum_{k = 1}^n ar^{k - 1} = a + ar + \cdots + ar^{n - 1} = \frac{a(1 - r^n)}{1 - r}$

Derivatives

$(x^n)' = nx^{n - 1}$

Trigonometric identities

$\sin^2 \theta + \cos^2 \theta = 1$ (Pythagorean identity)

Addition formulas

$\sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta$

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

Double-angle formulas

$\sin 2\theta = 2 \sin \theta \cos \theta$

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

Multiple-angle formulas

$\sin n\theta = \displaystyle \sum_{k = 0}^{\left\lfloor \frac{n - 1}{2} \right\rfloor} (-1)^k \binom{n}{2k + 1} \cos^{n - (2k + 1)} \theta \sin^{2k + 1} \theta$

$\cos n\theta = \displaystyle \sum_{k = 0}^{\left\lfloor \frac{n}{2} \right\rfloor} (-1)^k \binom{n}{2k} \cos^{n - 2k} \theta \sin^{2k} \theta$

Euler's formula

$e^{i\pi} + 1 = 0$ (Euler's identity)

$e^{i\theta} = \cos \theta + i \sin \theta$ (Euler's formula)

$(\cos \theta + i \sin \theta)^n = \cos n\theta + i \sin n\theta$ (De Moivre's formula)

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