Linear independence of trigonometric numbers

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.