What does an equation like $\sin(\sin(x))$ mean? I know it can be seen as a composite function $f(f(x))$, where $f(x)=\sin(x)$. Is there a way to simplify functions like this, and where will this be used? Thanks in advance.
P.S. I have looked at the graph of $f(x)=\sin(\sin(x))$ and compared it to the graph of $g(x)=\sin(x)$. They look pretty similar, but $f(x)=\sin(\sin(x))$ has a smaller amplitude
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$\begingroup$$\sin(\sin x))$ is not an equation.
There is no way to simplify it.
It will be used to test whether you have learned the Chain Rule, when you get to Calculus.
$\endgroup$ 1 $\begingroup$Besides hydrodynamics and wave equations, it also shows up in frequency modulation applications, which is electrical engineering. It expands exactly using Bessel functions of the first kind
$$ \sin(\sin(x)) = J_1(1) \sin(x) + J_3(1) \sin(3x) + J_5(1) \sin(5x) + … $$
Go to the Wikipedia entry for Frequency Modulation for more info
$\endgroup$ $\begingroup$In line with the link given in @Mathster's comment, we can look at the Fourier series of $\sin(\sin(x)$. If that is an unfamiliar term, what that means is basically the following equation holds:
$$\sin(\sin(x)) \approx 0.8801 \sin(x)+ 0.0391 \sin(3x) + 0.0005 \sin(5x).$$
(See the plot of the difference of the two functions here.) The numbers in the expression given are rounded to four decimal places and we could add more terms of the form $\sin((2n+1)x)$, but their coefficients will get smaller and smaller. If you look at the graph given you'll see the error is less than $0.00003$.
$\endgroup$ $\begingroup$It can be useful a lot when dealing with extensions of iterative functions to real or complex plane such as in case of collatz function:
$C\left(x\right)=\frac{x}{2}\cos^2\left(\frac{\pi x}{2}\right)+\frac{\left(3x+1\right)}{2}\cos^2\left(\frac{\pi\left(x-1\right)}{2}\right)$
and then $C(C(C(C...(x)))...)$
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