Last week we used an old AMC problem to explore Cos(72):

My older son is studying trig now and today we went over the formulas for trig sums -> 2*Cos(x)*Cos(y) = Cos(x – y) + Cos(x + y). Reminded me of one of the contest problems that really made me curious about math. #30 from the 1975 AHSME: pic.twitter.com/0jfNqXkskv

Today we built a decagon with our Zometool set to see if we could approach the problem a different way:

I started by having the kids explore the decagon and having my older son explain where cos(72) and cos(36) were (roughly) on the shape:

Next we used a T-square to try to get good approximations for both Cos(72) and Cos(36). The T-square + Zometool combination was a little harder for the kids than I was expecting, but we got there.

Finally, I wrapped up with a challenge question for my older son. If we know that Cos(36) – Cos(72) = 1/2, find the value of Cos(36). He did a nice job working through this problem:

I’ve enjoyed playing around with properties of angles that arent usually part of the trig curriculum. We might have one more project on 72 degrees this weekend – I’m thinking of playing with the idea that Tan(72) is close to 3, but haven’t quite figured out that project yet.