My older son is starting to learn trig and I wanted to show him the surprising connection between the exponential function and trig functions. Grant Sanderson’s video on Pythagorean triples from earlier this year came to mind:

After my son watched the video tonight I asked him to talk about some of the ideas. Here’s what he had to say – I was happy that many of the ideas that Sanderson presented sank in:

Now I showed him the incredible formula = Cos(x) + i*Sin(x). Although I just present the formula, the complex number ideas after that are exactly the same ideas that are in Sanderson’s video:

Now I showed him how this formula makes some of the standard trig identities really easy to both derive and remember. The ideas here were really the inspiration for the video because he asked me this morning if the trig functions were linear.

We specifically looked at formulas for Cos(2x) and Sin(2x) here.

Now we used the formulas in a couple of simple examples. First we looked at 45 degrees, then 30 degrees, and then we looked at the angles in a 3-4-5 triangle:

This project was a fun way to introduce an amazing idea in math and also hopefully a nice way to show my son that ideas in trig go way beyond triangles.