Build up the concept of the Taylor series
Get further into power series using the fascinating Bessel's equation.
Look at the advanced way of viewing sin and cos through their power series.
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.
Can you deduce the familiar properties of the sine and cosine functions starting from these three different mathematical representations?