My younger son is working through Martin Weissman’s An Illustrated Theory Numbers and came across this exercise last week:

Today we finally got to doing a project on the problem. We worked through the first 4 parts and will save the last part for another project.

Here’s the first part of the problem which is mostly a discussion of how you can think about points on the unit circle using complex numbers:

The next part of the problem asked to show that if x is a 5th root of unity then . I forgot to zoom out after we zoomed in on the problem, but I do finally remember to zoom out around 1:30 – sorry about that:

Part c was the part that gave my younger son a lot of trouble, but luckily my older son was able to help out with the ideas about sums and products of roots require to get through this step:

Finally, we find the roots of the quadratic polynomial from the last part and find the exact value for cos(72). What a fun project!