Euler Form

Euler, another person, said that since Taylor Series is a thing, we can say that

eiθ=1+iθ+(iθ)22!+(iθ)33!+⋯=1+iθ−θ22!−iθ33!+θ44!+iθ55!\begin{gather*} e^{i\theta}=1+i\theta+\frac{(i\theta)^2}{2!}+\frac{(i\theta)^3}{3!}+\cdots\\ =1+i\theta-\frac{\theta^2}{2!}-i\frac{\theta^3}{3!}+\frac{\theta^4}{4!}+i\frac{\theta^5}{5!} \end{gather*}

If we rearrange the terms we get

eiθ=(1−θ22!+θ44!−⋯ )+i(θ−θ33!+θ55!+⋯ )=cos⁡θ+isin⁡θ\begin{gather*} e^{i\theta}=\left(1-\frac{\theta^2}{2!}+\frac{\theta^4}{4!}-\cdots\right)+i\left(\theta -\frac{\theta^3}{3!}+\frac{\theta^5}{5!}+\cdots\right)\\ =\cos\theta+i\sin\theta \end{gather*}

This means we can represent Complex Number in this weird form too:

z=reiθ\begin{gather*} z=re^{i\theta} \end{gather*}

also trigometry can use complex numbers too

cos⁡θ=12(eiθ+e−iθ)\cos\theta= \frac{1}{2}(e^{i\theta}+e^{-i\theta}) sin⁡θ=12i(eiθ−e−iθ)\sin\theta= \frac{1}{2i}(e^{i\theta}-e^{-i\theta})