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.
Lawrence C. Washington
"A Proof of a Generalization of Niven's Theorem Using Algebraic Number Theory,"
Rose-Hulman Undergraduate Mathematics Journal: Vol. 22:
2, Article 3.
Available at: https://scholar.rose-hulman.edu/rhumj/vol22/iss2/3