Elementary trigonometry based on a first order differential equation

TL;DR

This paper demonstrates that functions satisfying a specific first-order differential equation can exhibit all properties of sine functions, providing elementary derivations and proofs for periodicity and identities.

Contribution

It introduces a novel approach to elementary trigonometry based on a first-order differential equation, linking sine functions to differential equations with boundary conditions.

Findings

Functions satisfying f'(x) = f(x+a) behave like sine functions

Periodic properties are derived from the differential equation

Trigonometric identities are proved using this approach

Abstract

It is shown that with appropriate boundary conditions, a real function satisfying the differential equation $f'(x) = f(x+a)$ has all known properties of the sine function. A number of elementary derivations are presented including proofs for periodicity and trigonometric identities.