A Generalisation of Niven's Theorem for Trigonometric Functions

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.

Based on second author's bachelor's thesis. Comments welcome