Where Do the Terms of the Power Series Expansions of Sine and Cosine Functions Come from? Involutes!

TL;DR

This paper revisits Chaikovsky's elementary geometric proof of sine and cosine power series, emphasizing their geometric meanings and involving involutes, to aid teachers and students in understanding these expansions without advanced calculus.

Contribution

It reintroduces Chaikovsky's geometric approach to power series of sine and cosine, highlighting the role of involutes and providing computational tools for educational purposes.

Findings

Geometric interpretation of power series terms

Connection between involutes and series expansions

A Maxima procedure for involute computation

Abstract

In the 1930's, a Russian school teacher Y. S. Chaikovsky presented a proof of the power series expansion of the sine and cosine functions without using calculus. In doing so he also showed the geometrical meanings of the various terms in these power series expansions. Chaikovsky's ideas were first published by Leo S. Gurin as a Note in the American Mathematical Monthly in 1996. The proofs, though they use only elementary mathematics, require some combinatorial arguments which may be hard even for bright students. This paper is an attempt to bring the ideas of Chaikovsky once again to the attention of the mathematics teachers and students. The emphasis in this paper is on establishing the geometrical meanings of the various terms in the power series expansions of sine and cosine functions. This has been done taking recourse to methods of calculus. Since the concept of an involute is crucial in this investigation, we have also included a brief discussion on the concept of an involute of a curve and also a Maxima procedure for computing the parametric equations of the involute of a given curve.