Statement
Write a complex number in polar form as
\[z=r(\cos\theta+i\sin\theta),\]where $r\ge 0$ is its modulus and $\theta$ is an argument measured in radians. For every integer $n$,
\[z^n=r^n\bigl(\cos(n\theta)+i\sin(n\theta)\bigr).\]For $n<0$, require $z\ne 0$; for $n=0$, the expression is defined when $z\ne 0$ as well. For positive $n$, the identity also holds at $z=0$.
Proof
Euler’s formula gives $z=re^{i\theta}$. Raising both sides to an integer power gives
\[z^n=(re^{i\theta})^n=r^n e^{in\theta} =r^n\bigl(\cos(n\theta)+i\sin(n\theta)\bigr),\]using $e^{i\phi}=\cos\phi+i\sin\phi$. Equivalently, each multiplication by $z$ multiplies the modulus by $r$ and adds $\theta$ to the argument; after $n$ multiplications, these become $r^n$ and $n\theta$. The same rule extends to negative powers by taking reciprocals when $z\ne0$.
Example
Taking $z=\cos(\pi/6)+i\sin(\pi/6)$ and $n=3$,
\[z^3=\cos(3\pi/6)+i\sin(3\pi/6) =\cos(\pi/2)+i\sin(\pi/2)=i.\]The angles here are in radians.