# Sines and cosines (part 1)

A friend asked me the following question. What $x$-values satisfy the equation $\sin(\sin(\sin(\sin(x))))=\cos(\cos(\cos(\cos(x))))$?

It occurred to me to ask a more general question. Let $s(x)=\sin(x)$ and $c(x)=\cos(x)$. Let $s^n=s\circ\cdots\circ s$ be the composition of $s$ with itself $n$ times. Similarly, let $c^m$ be the composition of $c$ with itself $m$ times.

What are the solutions to $s^n(x)=c^m(x)$?

Is it possible answer this question? Always? Sometimes? Never?

I’ll post my thoughts on this in a follow-up post.

[Update: I forgot to add that when I was looking up this problem online I found that it was a problem in the 1994 Russian Math Olympiad.]

1. samjshah says:

This was a great question. I had fun working on it. I’m a little sad that you already posted your solution – I wanted to have a crack at working on it and posting how far I got. I was going in a similar direction as you, and I even was using the same Grapher program on the Mac to generate my curves!

I too found that as m and n increase, we tend to get something that approximates a line. I like the cobweb plots. I don’t know why I didn’t think of them immediately, since this is precisely what they’re for!

I might make this into a math club investigation for next year.

2. Sorry to post the solution too quickly. I posted the solution separately so readers could read the question and the solution separately, but maybe a time lag would have been better—remove all temptation!

3. Nathan says:

A numerical solution is 0.756887 + 0.610155 i, but I don’t know a good way to find or express it analytically.