A fun conversation about complex numbers from a problem that didn’t look so interesting at first

Last night my younger son was working in Art of Problem Solving’s Precalculus book and came across this problem:

Following the fun conversation we had last night, I thought it would be good to revisit the problem tonight for a project. We started by talking through the problem and reviewing the first solution that came to his mind:

Next we moved to a second solution – this one involving geometric series. Here we also talked through a solution that didn’t come up last night.

Finally, we talked through a solution involving trig. As I mentioned in the twitter thread, this was a fun one for me because on a stand alone basis, calculating Cos(36) – Cos(72) is a pretty difficult problem:

When I first saw the problem here, I have to admit that I thought it looked pretty dull. Turned out to be a terrific problem to illustrate a variety of different ways to talk about complex numbers! Definitely a fun surprise.