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$$\begin{align*} e^x &= \sum_{k=0}^{\infty}\frac{x^k}{k!}=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\frac{x^4}{4!}+\frac{x^5}{5!}+\cdots\\ \cos x &= \sum_{k=0}^{\infty}\frac{(-1)^k x^{2k}}{(2k)!}=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+\cdots\\ \sin x &= \sum_{k=0}^{\infty}\frac{(-1)^k x^{2k+1}}{(2k+1)!}=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}+\cdots \end{align*}$$$$e^{i\theta}=1+(i\theta)+\frac{(i\theta)^2}{2!}+\frac{(i\theta)^3}{3!}+\frac{(i\theta)^4}{4!}+\frac{(i\theta)^5}{5!}+\cdots$$$$i^1=i,\ i^2=-1,\ i^3=-i,\ i^4=1,\ i^5=i,\dots$$ $$\begin{align*} e^{i\theta} &=1+i\theta+\frac{-1\cdot\theta^2}{2!}+\frac{-i\cdot\theta^3}{3!}+\frac{1\cdot\theta^4}{4!}+\frac{i\cdot\theta^5}{5!}+\cdots\\ &=\underbrace{\left(1-\frac{\theta^2}{2!}+\frac{\theta^4}{4!}-\cdots\right)}_{\cos\theta} +i\underbrace{\left(\theta-\frac{\theta^3}{3!}+\frac{\theta^5}{5!}-\cdots\right)}_{\sin\theta}\\ &=\cos\theta+i\sin\theta \end{align*}$$$$\theta=\pi： e^{i\pi}=\cos\pi+i\sin\pi=-1+0i \boldsymbol{e^{i\pi}+1=0}$$

$\boldsymbol{e^{i\theta}=\cos\theta+i\sin\theta}$

$e^{ix}=\cos x+i\sin x$