Linear independence of trigonometric numbers

TL;DR

This paper establishes a precise criterion for when the numbers 1, cos(πr₁), and cos(πr₂) are rationally independent for rational r₁, r₂, and extends the result to larger fields, with applications to classifying certain rational triangles.

Contribution

It provides a necessary and sufficient condition for rational independence of 1 and two cosine values at rational multiples of π, extending classical results and applying to geometric classifications.

Findings

Criteria for rational independence of cosine values

Extension of classical results to larger number fields

Complete classification of rational triangles with specific properties

Abstract

Given any two rational numbers $r_1$ and $r_2$, a necessary and sufficient condition is established for the three numbers $1$, $\cos (\pi r_1)$, and $\cos (\pi r_2)$ to be rationally independent. Extending a classical fact sometimes attributed to I. Niven, the result even yields linear independence over larger number fields. The tools employed in the proof are applicable also in the case of more than two trigonometric numbers. As an application, a complete classification is given of all planar triangles with rational angles and side lengths each containing at most one square root. Such a classification was hitherto known only in the special case of right triangles.