On Angles Whose Squared Trigonometric Functions are Rational

TL;DR

This paper investigates angles with rational squared trigonometric functions, introduces a basis for related rational linear relations, and explores their applications in Euclidean geometry, particularly in polyhedron equidecomposability.

Contribution

It constructs a basis for the vector space over Q generated by angles with rational squared trigonometric functions and applies it to geometric problems.

Findings

Established a basis for angles with rational squared trigonometric functions.

Connected geodetic angles to Euclidean geometry and polyhedron decompositions.

Provided a framework for analyzing rational relations among such angles.

Abstract

We consider the rational linear relations between real numbers whose squared trigonometric functions have rational values, angles we call ``geodetic''. We construct a convenient basis for the vector space over Q generated by these angles. Geodetic angles and rational linear combinations of geodetic angles appear naturally in Euclidean geometry; for illustration we apply our results to equidecomposability of polyhedra.