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.
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.
This problem begins to motivate the existence of the Maclaurin's series in which any function which is well-behaved at the origin can be written as an 'infinite polynomial' or 'power series'. This iterative method is a numerical way of finding the coefficients of the polynomial, although calculus can be used in cases where the derivatives of the function needing to be approximated are