Trigonometric Cheat Sheet

$$ \begin{align*} sin^2(\theta) + \cos^2(\theta) &= 1 \\ tan^2(\theta) + 1 &= \sec^2(\theta) \\ 1 + cot^2(\theta) &= \csc^2(\theta) \end{align*} $$

$$ \begin{align*} \frac{sin(\theta)}{a} = \frac{\sin(\beta)}{b} = \frac{\sin(\gamma)}{c} \end{align*} $$

$$ \begin{align*} a^2 &= b^2 + c^2 - 2bc \cos(A) \\ b^2 &= a^2 + c^2 - 2ac \cos(B) \\ c^2 &= a^2 + b^2 - 2ab \cos(C) \\ \end{align*} $$

$$ \begin{align*} \sin(\alpha + \beta) &= \sin(\alpha)\cos(\beta) + \cos(\alpha)\sin(\beta) \\ \sin(\alpha - \beta) &= \sin(\alpha)\cos(\beta) - \cos(\alpha)\sin(\beta) \\ \cos(\alpha + \beta) &= \cos(\alpha)\cos(\beta) - \sin(\alpha)\sin(\beta) \\ \cos(\alpha - \beta) &= \cos(\alpha)\cos(\beta) + \sin(\alpha)\sin(\beta) \\ \tan(\alpha + \beta) &= \frac{\tan(\alpha) + \tan(\beta)}{1 - \tan(\alpha)\tan(\beta)} \\ \tan(\alpha - \beta) &= \frac{\tan(\alpha) - \tan(\beta)}{1 + \tan(\alpha)\tan(\beta)} \end{align*} $$

$$ \begin{align*} \sin(2x) &= 2\sin(x)\cos(x) \\ \cos(2x) &= \cos^2x - \sin^2(x) \\ &= 1 - 2sin^2x \\ &= 2cos^2x - 1 \\ \tan(2x) &= \frac{2\tan(x)}{1 - tan^2(x)} \end{align*} $$

$$ \begin{align*} \sin(x) = \cos\big(\frac{\pi}{2}-x\big) \\ \cos(x) = \sin\big(\frac{\pi}{2}-x\big) \\ \tan(x) = \cot\big(\frac{\pi}{2}-x\big) \\ \cot(x) = \tan\big(\frac{\pi}{2}-x\big) \\ \sec(x) = \csc\big(\frac{\pi}{2}-x\big) \\ \csc(x) = \sec\big(\frac{\pi}{2}-x\big) \\ \end{align*} $$

$$ \begin{align*} \sin(\alpha)\cos(\beta) = \frac{1}{2}\big[\sin(\alpha + \beta) + \sin(\alpha - \beta)\big] \\ \cos(\alpha)\sin(\beta) = \frac{1}{2}\big[\sin(\alpha + \beta) - \sin(\alpha - \beta)\big] \\ \sin(\alpha)\sin(\beta) = \frac{1}{2}\big[\cos(\alpha - \beta) - \cos(\alpha + \beta)\big] \\ \cos(\alpha)\cos(\beta) = \frac{1}{2}\big[\cos(\alpha - \beta) + \cos(\alpha + \beta)\big] \\ \end{align*} $$

$$ \begin{align*} \sin(\alpha) + \sin(\beta) = 2\sin\big(\frac{\alpha + \beta}{2}\big)\cos\big(\frac{\alpha - \beta}{2}\big) \\ \sin(\alpha) - \sin(\beta) = 2\cos\big(\frac{\alpha + \beta}{2}\big)\sin\big(\frac{\alpha - \beta}{2}\big) \\ \cos(\alpha) + \cos(\beta) = 2\cos\big(\frac{\alpha + \beta}{2}\big)\cos\big(\frac{\alpha - \beta}{2}\big) \\ \cos(\alpha) - \cos(\beta) = -2\sin\big(\frac{\alpha + \beta}{2}\big)\sin\big(\frac{\alpha - \beta}{2}\big) \\ \end{align*} $$

$$ \begin{align*} \sin(-\theta) = -\sin(\theta) \qquad \csc(-\theta) = -\sin(\theta) \\ \tan(-\theta) = -\tan(\theta) \qquad \cot(-\theta) = -\cot(\theta) \\ \cos(-\theta) = \cos(\theta) \qquad \sec(-\theta) = -\sec(\theta) \\ \end{align*} $$