From Taylor Series to Euler Formula ex=∑k=0∞xkk!e^x=\sum_{k=0}^\infin \frac{x^k}{k!} ex=k=0∑∞k!xk Plug in x=iθx=i\thetax=iθ eiθ=1+iθ+(iθ)22!+(iθ)33!+⋯e^{i\theta}=1+i\theta+\frac{(i\theta)^2}{2!}+\frac{(i\theta)^3}{3!}+\cdots eiθ=1+iθ+2!(iθ)2+3!(iθ)3+⋯ Separate real part and imaginary part, we can get eiθ=cosθ+isinθe^{i\theta}=\cos\theta + i\sin\theta eiθ=cosθ+isinθ DeMoivre’s Theorem [cosθ+isinθ]n=cos(nθ)+isin(nθ)[\cos\theta+i\sin\theta]^n=\cos(n\theta)+i\sin(n\theta) [cosθ+isinθ]n=cos(nθ)+isin(nθ)