A New Approach to Rational Values of Trigonometric Functions

TL;DR

This paper explores the conditions under which trigonometric functions yield rational values, linking these to the algebraic properties of primitive roots of unity and their degrees over the rationals.

Contribution

It establishes a new equivalence between rational values of sine and cosine at rational multiples of pi and the degrees of primitive roots of unity over Q.

Findings

sin(aπ/b) and cos(aπ/b) are rational only for specific integer pairs

The rationality of these values is tied to primitive roots of unity of degrees 1, 2, 3, 4, and 6

Provides a new algebraic characterization of rational trigonometric values

Abstract

It is well known that sin(a\pi/b), cos(a\pi/b), etc., are only rational numbers for a few select integers a and b. We show that this is equivalent to the fact that only for d = 1,2,3,4, and 6 is the primitive dth root of unity of degree 2 over Q.