Complex numbers

De Moivre's theorem and roots of unity

$(\cos\theta+i\sin\theta)^n=\cos n\theta+i\sin n\theta$: to raise a complex number to a power, raise the modulus and multiply the argument. Run it backwards to find all $n$ $n$th roots, which sit evenly round a circle, and expand it to get identities for $\cos n\theta$ and $\sin n\theta$.