Math formulas - wikinder/wikinder GitHub Wiki

• Proofs

Euler's identities

$e^{i \pi} + 1 = 0$ (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)

Algebraic identities

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

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

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

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

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

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

Quadratic formula

$ax^2 + bx + c = 0$

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

Exponent rules

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

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

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

Logarithm rules

$\log_b (xy) = \log_b x + \log_b y$

$\log_b (x^k) = k \log_b x$

$\log_b x = \dfrac{\log_c x}{\log_c b}$

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

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

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