Functions Defined By Differential Equations A Short Course I

Functions Defined by Differential Equations: A Short Course in Trigonometry
Author: Bushaw, D.
College Mathematics Journal, Vol. 2, no. 1, pp. 32-35, 1971.

In this article, the author uses differential equations to explain the study of trigonometry. He defines trigonometry as the study of the special initial value problem c'(t)=-s(t), s'(t)=c(t), c(0)=1, s(0)=0. He then goes on to prove several different theorems and corollaries about trigonometric functions using the system of differential equations. In particular, he proves the following:

  • c(u+v)=c(u)c(v), s(u+v)=c(u)s(v)+s(u)c(v), for every two real numbers u and v
  • c(2t)=c2(t)-s2(t), s(2t)=2c(t)s(t), for all t
  • c2(t)+s2(t)=1, for all t

The author also presents the fundamental theorem of trigonometry. (Summary by Anna Cho, Pomona College '13)

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