A simple proof of the transcendence of the trigonometric functions

TL;DR

This paper presents a straightforward proof demonstrating that the values of trigonometric functions at any nonzero rational number are transcendental, contributing to the understanding of their algebraic properties.

Contribution

It provides a simple and accessible proof of the transcendence of trigonometric function values at nonzero rational points, which was previously more complex.

Findings

Values of trigonometric functions at nonzero rational numbers are transcendental.

The proof simplifies understanding of algebraic properties of these values.

Supports further research in transcendence theory.

Abstract

In this note, we give a simple proof that the values of the trigonometric functions at any nonzero rational number are transcendental numbers.