Sines and cosines (part 1)

A friend asked me the following question. What x-values satisfy the equation


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.

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