Correspondence between geometrical and differential definitions of the sine and cosine functions and connection with kinematics

TL;DR

This paper demonstrates the equivalence of geometrical and differential definitions of sine and cosine functions using elementary geometry and calculus, providing insights into circular motion and promoting abstract thinking in physics.

Contribution

It offers a novel, accessible proof of the equivalence between geometrical and differential definitions of sine and cosine without prior assumptions, linking mathematics and physics concepts.

Findings

Proves the equivalence using elementary methods

Connects geometric and differential approaches to harmonic functions

Enhances understanding of circular and harmonic motion

Abstract

In classical physics, the familiar sine and cosine functions appear in two forms: (1) geometrical, in the treatment of vectors such as forces and velocities, and (2) differential, as solutions of oscillation and wave equations. These two forms correspond to two different definitions of trigonometric functions, one geometrical using right triangles and unit circles, and the other employing differential equations. Although the two definitions must be equivalent, this equivalence is not demonstrated in textbooks. In this manuscript, the equivalence between the geometrical and the differential definition is presented assuming no a priori knowledge of the properties of sine and cosine functions. We start with the usual length projections on the unit circle and use elementary geometry and elementary calculus to arrive to harmonic differential equations. This more general and abstract treatment not only reveals the equivalence of the two definitions but also provides an instructive perspective on circular and harmonic motion as studied in kinematics. This exercise can help develop an appreciation of abstract thinking in physics.