De Moivre's theorem

For powers of polar-form complex numbers for $n\in \mathbb{N}$, we have the following theorem.

$$z^n=[r(\cos x+i\sin x){]}^n=r^n(\cos nx+i\sin nx)=r^n{e}^{i\theta }=r^n\text{cis}n\theta$$ $${}^n\sqrt{z}={}^n\sqrt{r}\text{cis}\frac{\theta +2k\pi }{n};\{k<n,k\in \mathbb{N}\}$$

We can also draw these two conclusions:

$$z^n+\frac{1}{z^n}=2\cos (n\theta )$$ $$z^n-\frac{1}{z^n}=2i\sin (n\theta )$$

See also

• Multiple angle identities, a nice application of De Moivre’s theorem