Trig identities

 

 

Pythagorean identities

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

 

Parity identities

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

 

Sum angle identities

\begin{align*} \sin(\theta \pm \varphi) &= \sin\theta \cos\varphi \pm \cos\theta \sin\varphi \\ \cos(\theta \pm \varphi) &= \cos\theta \cos\varphi \mp \sin\theta \sin\varphi \\ \tan(\theta \pm \varphi) &= \frac{\tan\theta \pm \tan\varphi}{1 \mp \tan\theta \tan\varphi} \end{align*}

 

Double angle identities

\begin{align*} \sin 2\theta &= 2 \sin\theta \cos\theta \\ \cos 2\theta &= 2\cos^2\theta - 1 \\ \tan 2\theta &= \frac{2\tan\theta}{1 - \tan^2\theta} \end{align*}

 

Half angle identities

\begin{align*} \sin \frac{\theta}{2} &= \sqrt{\frac{1 - \cos\theta}{2}} \\ \cos \frac{\theta}{2} &= \sqrt{\frac{1 + \cos\theta}{2}} \\ \tan \frac{\theta}{2} &= \frac{1 - \cos\theta}{\sin} = \frac{\sin\theta}{1 + \cos\theta} \end{align*}

 

Sum identities

\begin{align*} \sin\theta + \sin\varphi &= \phantom{-}2 \sin\frac{\theta + \varphi}{2} \cos\frac{\theta - \varphi}{2} \\ \sin\theta - \sin\varphi &= \phantom{-}2 \cos\frac{\theta + \varphi}{2} \sin\frac{\theta - \varphi}{2} \\ \cos\theta + \cos\varphi &= \phantom{-}2 \cos\frac{\theta + \varphi}{2} \cos\frac{\theta - \varphi}{2} \\ \cos\theta - \cos\varphi &= -2 \sin\frac{\theta + \varphi}{2} \cos\frac{\theta - \varphi}{2} \\ \tan\theta \pm \tan\varphi &= \phantom{-2}\frac{\sin(\theta \pm \varphi)}{\cos\theta \cos\varphi} \end{align*}

 

Product identities

\begin{align*} 2\sin\theta \sin\varphi &= \cos(\theta - \varphi) - \cos(\theta + \varphi) \\ 2\cos\theta \cos\varphi &= \cos(\theta - \varphi) + \cos(\theta + \varphi) \\ 2\sin\theta \cos\varphi &= \sin(\theta - \varphi) + \sin(\theta + \varphi) \end{align*}

 

Finding all arc functions from arctan

\begin{align*} \text{arcsin}(x) &= \arctan(x / \sqrt{1 - x^2}) \\ \text{arccos}(x) &= \arctan(\sqrt{1 - x^2 }/ x) \\ \text{arccot}(x) &= \pi/2 - \arctan(x) \\ \text{arcsec}(x) &= \arctan(\sqrt{x^2 - 1}) \\ \text{arccsc}(x) &= \arctan(1/\sqrt{x^2 - 1}) \end{align*}

 

See also these notes on defining inverse functions.

 

Hyperbolic functions

\exp(i\theta) &= \cos\theta + i \sin\theta

\begin{align*} \sinh(x) &= \frac{\exp(x) - \exp(-x)}{2} \\ \cosh(x) &= \frac{\exp(x) + \exp(-x)}{2} \\ \tanh(x) &= \frac{\exp(x) - \exp(-x)}{\exp(x) + \exp(-x)} \\ \end{align*}

 

Relating circular and hyperbolic functions

\begin{align*} \sin\theta &= -i \sinh(i\theta) \\ \cos\theta &= \phantom{-i} \cosh(i\theta) \\ \tan\theta &= -i \tanh(i\theta) \\ \end{align*}