A Generalization of the Angle Doubling Formulas for Trigonometric Functions

Abstract

SummaryThe angle doubling formula sin 2θ = 2 sin θ cos θ for the sine function is well known. By replacing the cosine in this formula with sin (π/2 - θ), we see that sin 2θ can be written as the product of two sine functions where the second sine function is obtained from the basic sine function by only using a phase shift of the angle θ and a reflection about the horizontal axis. In this paper, we will show that, for any natural number n, sin nθ can be written as the product of n sine functions involving only phase shifts of the angle θ and a possible reflection about the horizontal axis. Similar formulas will be derived for the cosine and tangent functions.