Maclaurin series

problem
Trig reps

Age

16 to 18

Challenge level

Can you deduce the familiar properties of the sine and cosine functions starting from these three different mathematical representations?
problem
Bessel's equation

Age

16 to 18

Challenge level

Get further into power series using the fascinating Bessel's equation.
problem
Taking trigonometry series-ly

Age

16 to 18

Challenge level

Look at the advanced way of viewing sin and cos through their power series.
problem
Building approximations for sin(x)

Age

16 to 18

Challenge level

Build up the concept of the Taylor series
problem
What do functions do for tiny x?

Age

16 to 18

Challenge level

Looking at small values of functions. Motivating the existence of the Taylor expansion.
problem
Towards Maclaurin

Age

16 to 18

Challenge level

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.