Complex numbers — FAQ

The roots of unity are the solutions to $z^n = 1$. Since $|1|=1$ and $\arg(1)=0$, the roots of unity all have a modulus of $1^{1/n}=1$ and are located on the unit circle. The method for finding roots of any complex number $w = re^{i\theta}$ is identical, but the roots will lie on a circle of radius $r^{1/n}$ and will be rotated by an angle of $\theta/n$ compared to the roots of unity.

If your target series involves cosines (e.g., $\sum \cos(k\theta)$), you will take the real part of the complex sum. If your target series involves sines (e.g., $\sum \sin(k\theta)$), you will take the imaginary part. Always define your target series as $C$ or $S$ at the start and form the complex sum $C+iS$.

If $n$ is even, the binomial expansion of $(z-z^{-1})^n$ will have an odd number of terms. The terms will pair up to form cosine terms, e.g., $z^k+z^{-k}=2\cos(k\theta)$. If $n$ is odd, the expansion will have an even number of terms, and they will pair up to form sine terms, e.g., $z^k-z^{-k}=2i\sin(k\theta)$. Pay close attention to the powers of $i$ from $(2i\sin\theta)^n$.