mathmatical recreation: What is Sin(Sin(x)) ?



Hello All,

<disclaimer> Please forgive my imprecise use of terminology as I am not
a mathematician but only an humble engineer.
</disclaimer>

I am trying to determine what sort of function you get when you compose
two standard trig functions? Say, for example, sin(sin(x)) or
sin(cos(x)). I can't seem to find any analytic solution that would
let me restate these as sums or products of elementary functions (trig
or otherwise).

If you plot these, they are obviously periodic waveforms, but no longer
pure sinusoids. Though they look a little like sine waves, they're too
"fat" to be sinusoids. If you plot the error function (the difference
between sin(sin(x)) and an appropriately scaled sine wave) you get a
waveform that looks like a mix of the fundamental plus the third
harmonic. I tried to approximate the error function and hence
sin(sin(x)) by adding in a small amount of the fundamental and third
harmonic by trial and error. The following formula approximates the
target function within 1% over an entire cycle:

sin(sin(x)) = sin(1)*sin(x) + (sin(x)+sin(3x))/25.5 (approximately)

Alas, there's no value of the fudge factor (the "25.5") that produces a
perfect fit so I'm probably missing some terms in my approximation.

So, does anyone know of a closed-form or series expression for
sin(sin(x)), sin(cos(x)), etc -- something like a Taylor series or a
Fourier series. I was hoping that I could find the expression/solution
through experimentation or by searching for a web article on the topic.
But I can't seem to crack this on my own so I'm going to "rely on the
kindness of strangers" and ask for an explanation. Thanks in advance.


-Bob Miller

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