Rational Sines, Cosines, and Tangents of Rational Multiples of 2π
Severino V. Gervacio
It is known that for 0 ≤ θ < 2π, where θ is a rational multiple of 2π, all of sin θ, cos θ, tan θ are irrational, except for exactly 16 values of θ which are rational multiples of 2π. In this paper, this same result is derived in a different way. Basically, this is done by expressing cos nθ as a polynomial in cos θ. Properties of this polynomial are then derived and used to prove the main results. Some interesting combinatorial identities using this polynomial are also obtained.