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

*From*: rwmm415@xxxxxxxxx*Date*: 26 Apr 2005 15:03:10 -0700

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

.

**Follow-Ups**:**Re: mathmatical recreation: What is Sin(Sin(x)) ?***From:*Robert Israel

**Re: mathmatical recreation: What is Sin(Sin(x)) ?***From:*flip

**Re: mathmatical recreation: What is Sin(Sin(x)) ?***From:*Phil Freedenberg

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