  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.
