How to find each equation for a set of five equidistant points around a unit circle

Question

Given a unit circle I want to make five points around it that are the same distance from each other. I know that in order to make four I can do $(1,0),(0,1),(-1,0),(0,-1)$.

But how can I do this on a five point algorithm?

Answer 1

The $n$th roots of unity are $n$ equidistant points on the unit circle. The $k$th such point for $0\le k<n$ is given by $$e^{2\pi ik/n}=\cos\frac{2\pi k}n+i\sin\frac{2\pi k}n\implies\left(\cos\frac{2\pi k}n,\sin\frac{2\pi k}n\right)$$ For $n=4$ this reduces to the coordinates you gave.