With school nearly over for the year I was looking for some ideas to explore with my younger son over the summer. I thought some introductory trig ideas might be fun since he saw some basic right triangle trig inn his math class at school this year.
The first thing that came to mind for me was a short exploration of what the functions sin(x) and cos(x) look like. It was fun to hear his ideas about these functions evolve over the course of our discussion this morning.
I started by asking him what he already knew:
After that introduction, I introduced the unit circle and asking him to make a guess as to what the graph of y = cos(x) would look like:
In the last video we looked only at the interval 0 to 90 degrees. Here we made a sketch of y = cos(x) and y = sin(x) from 0 to 360 degrees. It was fun to hear what he thought of these graphs as he was drawing them:
The discussion we had today was really fun and even had a few nice surprises. I’m excited to continue this discussion a bit more over the next couple of weeks.
Mike's Math Page 
The best thing I learned about trig in the last 30 years was this pictorial derivation of the addition formulae for sin and cos: https://i.stack.imgur.com/fvD9b.png
Start with the gray triangle with angle alpha, then put on the pink triangle with angle beta, label the new hypotenuse 1, enclose in the rectangle, and use the definitions to work out the lengths of everything else. Ta-da!
What a great set of videos to watch. The fact that he is starting with such solid foundation of understanding of concepts really makes such a difference. He knows to call an angle theta. He understands that cos and sin are ratios, and that when the hypotenuse is one that the legs of the triangle can be expressed at cos theta and sin theta. Then the unit circle becomes decipherable. Just so amazing to see how coming to trig with this kind of knowledge can demystify cos and sin.