Algebraic relations between sine and cosine values

TL;DR

This paper demonstrates that all algebraic relations between sine and cosine values at algebraic points follow from fundamental identities, using the Lindemann-Weierstrass theorem.

Contribution

It establishes that the algebraic relations among sine and cosine values are generated solely by basic identities, linking classical trigonometry with transcendence theory.

Findings

All algebraic relations derive from Pythagorean and addition formulas.

The result relies on the Lindemann-Weierstrass theorem.

Provides a complete characterization of algebraic relations between sine and cosine at algebraic points.

Abstract

The aim of this note is to show that any algebraic relation over $\overline{\mathbb{Q}}$ between the values of the trigonometric functions sine and cosine at algebraic points can be derived from the Pythagorean identity and the angle addition formulas. This result is obtained as a consequence of the Lindemann-Weierstrass theorem.