A Generalisation of Niven's Theorem for Trigonometric Functions

TL;DR

This paper extends Niven's Theorem by classifying rational values of powers of sine, cosine, and tangent functions at rational multiples of pi, using elementary methods, algebraic number theory, and Galois theory.

Contribution

It generalizes Niven's Theorem to classify all rational powers of sine, cosine, and tangent at rational multiples of pi, combining elementary and algebraic number theory techniques.

Findings

Classified rational values of cosine powers at rational multiples of pi.

Extended classification to tangent powers using algebraic number theory.

Provided a Galois theoretic explanation for the results.

Abstract

Niven's Theorem asserts that $\{\cos(r\pi)|r\in \mathbb{Q}\}\cap\mathbb{Q} = \{0, \pm 1, \pm\frac{1}{2}\}$. This paper uses elementary methods to classify all elements in the sets $\{\cos^n(r\pi)|r\in \mathbb{Q}, n \in \mathbb{N}\}\cap\mathbb{Q}$ and $\{\sin^n(r\pi)|r\in \mathbb{Q}, n \in \mathbb{N}\}\cap\mathbb{Q}$. Using some algebraic number theory, we extend this to a classification of all elements in $\{\tan^n(r\pi)|r\in \mathbb{Q}, n \in \mathbb{N}\}\cap\mathbb{Q}$. Finally, we present a short Galois theoretic argument to provide a more conceptual understanding of the results.