When is the (co)sine of a rational angle equal to a rational number?

TL;DR

This paper investigates when the cosine of rational multiples of pi is rational, proving it must be an integer multiple of 1/2, and explores which irrationalities can occur at rational multiples of pi.

Contribution

It provides an accessible proof for a specific rational cosine value condition and analyzes possible quadratic and cubic irrationalities of cosine at rational multiples of pi.

Findings

Cosine of a rational multiple of pi is rational only if it is an integer multiple of 1/2.

Identifies which quadratic irrationalities can be cosine values at rational multiples of pi.

Discusses the occurrence of cubic irrationalities in cosine values at rational multiples of pi.

Abstract

If the cosine of a rational multiple of $\pi$ is a rational number then it is an integral multiple of $\frac12$. For this fact, we give a proof accessible to an interested school student. We then discuss which quadratic and cubic irrationalities are values of cosine at ratinal multiples of $\pi$.