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

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