Mathematical Culture and Thought

Mathematical Culture and Thought

A Generalization of the Angle Doubling Formulas for Trigonometric Functions

Document Type : Translation

Authors
A. M. Kohestani
Y. Nejadhoseinian
Graduated in pure mathematics from University of Mazandaran, Iran
10.30504/mct.2025.1574.2109
Abstract
The 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.
Keywords
sine
double formula
trigonometry
cosine
Subjects

Brouwer, G., A Generalization of the angle doubling formulas for trigonometric functions, Math. Mag., 90 (1) (2017), 12-18.

[1]    Beyer, W. A., Louck, J. D., Zeilberger, D. A generalization of a curiosity that Feynman remembered all his life, Math. Mag., 69 (1996), 43–44.
[2]    Gauss, C. F., Disquisitiones Arithmeticae, Springer-Verlag, New York, 1986.
[3]    Gleick, J., Genius: The Life and Science of Richard Feynman, first ed., Pantheon Books, New York, 1992.
[4]    Moreno, S. G., García-Caballero, E. M. , A geometric proof of Morrie’s law, Amer. Math. Monthly, 122 (2016), 168.
[5]    Moreno, S. G., García-Caballero, E. M. , A geometric proof of a Morrie-type formula, Math. Mag., 90 (2016), 214–215.
[6]    Wantzel, M. L., Recherches sur les moyens de reconnaître si un problème de Géométrie peut se résoudre avec la règle et le compas, J. Math. Pures Appl., 2 (1837), 366–372.

  • Receive Date 03 October 2025
  • Revise Date 21 November 2025
  • Accept Date 22 November 2025
  • Publish Date 22 May 2026