Trig Functions

\begin{align*} \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{y}{r} \\ \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{x}{r} \\ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{y}{x} \\ \end{align*}

\begin{align*} \sin^{-1}\left(\frac{y}{r}\right) = \theta \\ \cos^{-1}\left(\frac{x}{r}\right) = \theta \\ \tan^{-1}\left(\frac{y}{x}\right) = \theta \\ \end{align*}

\begin{align*} \csc(x) = \frac{1}{\sin(x)} \\ \sec(x) = \frac{1}{\cos(x)} \\ \cot(x) = \frac{1}{\tan(x)} \\ \end{align*}

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

\begin{align*} \sin(a + b) &= \sin(a)\cos(b) + \cos(a)\sin(b) \\ \cos(a + b) &= \cos(a)\cos(b) - \sin(a)\sin(b) \\ \tan(a + b) &= \frac{\tan(a) + \tan(b)}{1 - \tan(a)\tan(b)} \\ \end{align*}

\begin{align*} \sin(2x) &= 2\sin(x)\cos(x) \\ \cos(2x) &= \cos^2(x) - \sin^2(x) \\ &= 2\cos^2(x) - 1 \\ &= 1 - 2\sin^2(x) \\ \tan(2x) &= \frac{2\tan(x)}{1 - \tan^2(x)} \\ \\ \sin\left(\frac{x}{2}\right) &= \pm\sqrt{\frac{1 - \cos(x)}{2}} \\ \cos\left(\frac{x}{2}\right) &= \pm\sqrt{\frac{1 + \cos(x)}{2}} \\ \tan\left(\frac{x}{2}\right) &= \frac{\sin(x)}{1 + \cos(x)} = \frac{1 - \cos(x)}{\sin(x)} \end{align*}

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

\begin{align*} \sin(a) + \sin(b) &= 2\sin\left(\frac{a+b}{2}\right)\cos\left(\frac{a-b}{2}\right) \\ \sin(a) - \sin(b) &= 2\cos\left(\frac{a+b}{2}\right)\sin\left(\frac{a-b}{2}\right) \\ \cos(a) + \cos(b) &= 2\cos\left(\frac{a+b}{2}\right)\cos\left(\frac{a-b}{2}\right) \\ \cos(a) - \cos(b) &= -2\sin\left(\frac{a+b}{2}\right)\sin\left(\frac{a-b}{2}\right) \\ \\ \sin(a)\cos(b) &= \frac{1}{2}[\sin(a+b) + \sin(a-b)] \\ \cos(a)\cos(b) &= \frac{1}{2}[\cos(a+b) + \cos(a-b)] \\ \sin(a)\sin(b) &= \frac{1}{2}[\cos(a-b) - \cos(a+b)] \\ \end{align*}

\begin{align*} \sin\left(\frac{\pi}{2} - x\right) &= \cos(x) \\ \cos\left(\frac{\pi}{2} - x\right) &= \sin(x) \\ \tan\left(\frac{\pi}{2} - x\right) &= \cot(x) \\ \end{align*}

\begin{align*} \sin(-x) &= -\sin(x) \\ \cos(-x) &= \cos(x) \\ \tan(-x) &= -\tan(x) \\ \end{align*}

\begin{align*} \sin(x + 2\pi) &= \sin(x) \\ \cos(x + 2\pi) &= \cos(x) \\ \tan(x + \pi) &= \tan(x) \\ \end{align*}