A neat problem from AoPS’s Precalculus book showing the connection between complex numbers and trig functions

Last week my younger son was working through a really neat problem from Art of Problem Solving Precalculus book:

We actually did a project on it at the end of last week, but the video for the last part of that project got messed up in the camera memory, unfortunately. I’ll put the first two videos of the original project at the bottom of this post – the calculations that start this project are what was in the video that didn’t record properly.

So, to start today’s project by showing how to solve the original problem by doing calculations with complex numbers:

Next I had my older son look at the same approach on a pretty famous problem that also boils down to calculating the sum of arctan values. I had planned to look at the geometric interpretation of this problem in the next video, but by happy coincidence my older son saw the geometric idea right away.

Since my older son saw the geometric connection in the last problem much more quickly than I thought he would, I had my younger son talk through the geometry here. At the end I asked the boys if they thought the geometric solution or the complex number solution was more illuminating:

Finally, off camera I asked the kids to come up with two similar problems that they thought would be fun to try to solve. They came up with two interesting ones that we played around with for about 20 min and then discussed what we found in the video below:

Definitely a fun project. The connection between trig functions and complex numbers is something that I think many kids would find fascinating. I love that that Art of Problem Solving took the time to illustrate this amazing connection!

Below are the two videos from the project with my younger son working through the original problem. I’m sad that them video with the final calculations didn’t record properly, but at least that part was mostly mechanical and easy enough to repeat.

Part 1:

Part 2: