Can you deduce the familiar properties of the sine and cosine functions starting from these three different mathematical representations?
Look at the advanced way of viewing sin and cos through their power series.
Get further into power series using the fascinating Bessel's equation.
Looking at small values of functions. Motivating the existence of
the Taylor expansion.
Build series for the sine and cosine functions by adding one term
at a time, alternately making the approximation too big then too
small but getting ever closer.