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Abstract

Niven’s theorem states that the sine, cosine, and tangent functions are rational for only a few rational multiples of π. Specifically, for angles θ that are rational multiples of π, the only rational values of sin(θ) and cos(θ) are 0, ±½, and ±1. For tangent, the only rational values are 0 and ±1. We present a proof of this fact, along with a generalization, using the structure of ideals in imaginary quadratic rings. We first show that the theorem holds for the tangent function using elementary properties of Gaussian integers, before extending the approach to other imaginary quadratic rings. We then show for which rational multiples of π the squares of the sine, cosine, and tangent functions are rational, providing a generalized form of Niven’s theorem. We end with a discussion of a few related combinatorial identities.

Author Bio

Caroline Nunn is an undergraduate student at the University of Maryland, College Park majoring in math and minoring in physics. Her major interests are currently algebraic and analytic number theory, but she is also interested in a variety of other mathematical topics. The research for this paper was completed in the first part of 2021. Caroline plans to begin graduate school in the fall of 2022.

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