First of all, I'm not a mathematician but a programmer interested in shaders (and thus, applied vector, trigonometric and graph maths) so bear with me.
Recently, trying to create a interpolation of a linear input (from 0 to 1) I found out that a sine of a sine (sin(sin(x)) returns a sort of "cubic" sine.
For example, in a graph, if we input a sine wave function we have a cyclic function that, to me, has a sort of exponential characteristic to it (even though I'm not able to justify it mathematically, it looks like sin(x) is x^2 but cyclic.
Now, I'm aware of the relationship of sine waves and the circle, and in fact, what I was looking for was the wave function of a rounded square, rather than a perfect circle, and I don't understand why sin(sin(x)) gives back that (apparent) wave.
What I am asking is, if sin(x) looks like this:
why sin(sin(x)) looks like this:
Or sin(sin(sin(sin(x)))) looks like this:
Thank you in advance.
Best Answer
The $n$-th iterate of sine (denoted $\sin^{(n)}$) has been studied in great detail by N.G. de Bruijn, Asymptotic Methods in Analysis, pages 157–166. He finds, for $0<x<\pi$, that $$\sin^{(n)}x=\sqrt{\frac{3}{n}}\biggl(1-\tfrac{3}{10}n^{-1}\ln n+{\cal O}(n^{-1})\biggr),$$ so independent of $x$ to leading order in $n$.
This confirms the finding by Michael Hardy, that the iterated sign normalized by its maximum converges to unity.