# Expressing Trigonometric Functions Using The Exponential Function [[Trigonometric Functions]] can be expressed using [[The Exponential Function]] via [[Euler's Formula]]. ![[Readwise/Articles/Euler's Formula#^ptfdbg]] $\sin(\theta) = \frac 1 {2i} \left( e^{i\theta} - e^{-i\theta} \right)$ $\cos(\theta) = \frac 1 2 \left( e^{i\theta} + e^{-i\theta} \right)$ $\tan(\theta) = \frac 1 i \frac {e^{i\theta} - e^{-i\theta}}{e^{i\theta} + e^{-i\theta}}$ ^s1jedx If we then further plug in the series expansion for exp we get: $\sin(\theta) = \sum_{n=0}^\infty (-1)^n \frac {x^{2n+1}} {(2n+1)!}$ $\cos(\theta) = \sum_{n=0}^\infty (-1)^n \frac {x^{2n}} {(2n)!}$ ### Proof for Sine Into Euler's formula plugin in $\theta$ and $-\theta$ and subtract the two expressions. $ \begin{align} e^{i\theta} &= \cos \theta + i\sin \theta \\ e^{-i\theta} &= \cos (-\theta) + i\sin (-\theta) \\ e^{i\theta} - e^{-i\theta} &= \cos \theta + i\sin \theta - \cos (-\theta) -i\sin (-\theta) \\ e^{i\theta} - e^{-i\theta} &= 2i\sin \theta \end{align} $ ### Proof for Cosine Into Euler's formula plugin in $\theta$ and $-\theta$ and add the two expressions together. $ \begin{align} e^{i\theta} &= \cos \theta + i\sin \theta \\ e^{-i\theta} &= \cos (-\theta) + i\sin (-\theta) \\ e^{i\theta} + e^{-i\theta} &= \cos \theta + \cos (-\theta) + i(\sin \theta + \sin (-\theta)) \\ e^{i\theta} + e^{-i\theta} &= 2\cos \theta \end{align} $