$\sin^2$ notation and uses of the alternative. So I was taking my calculus class and I was shocked by the following:
Apparently its a convention for 
$\sin^2(\alpha)=(\sin(\alpha))^2$
As opposed to what I thought made more sense which was
$\sin^2(\alpha)=\sin^2(\alpha)=\sin(\sin(\alpha)))$.
This made more sense to me because for two reasons.
reason 1 : $\sin^{-1}(\alpha)\neq(\sin(\alpha))^{-1}$ 
in other words it denotes the inverse function of sine, not the multiplicative inverse of sine of a number.
reason 2: 
writing $(\sin(\alpha))^n$ is a lot faster than writing $\underbrace{\sin(\sin(\dots(\sin(}_{n\text{ of these}}\alpha\underbrace{))\dots))}_{n\text{ of these}}$
So I guess that the justification for this is that the composition of the sine function is not very useful.
My question is, are there any interesting things that actually do use composition of trigonometric functions? is there an agreed upon definition for the composition of trigonometric functions?
Regards.
 A: Please feel free to write (if you wish) $(\sin x)^2$ and $\arcsin x$ to avoid this confusion.  But learn to recognize the standard notations $\sin^2 x$ and $\sin^{-1} x$ for them, when others use these.  
You are correct that $\sin(\sin(x))$ is not often seen (because it is not often useful).  Do not write $\sin^2 x$ for it, unless you are writing a specialized paper, and include this convention at the beginning.  Also do not write $\sin^{(2)} x$ for it, as that usually means
$$
\frac{d^2}{dx^2} \sin x .
$$
Notations you can use for $(\sin \circ \;\sin)\, x$ are $\sin^{[2]} x$ and $\sin^{{\displaystyle \circ} 2} x$.  But even then, it would be best to include the explanation.
A: $\sin^2 x=(\sin x)^2=(\sin x)(\sin x)$ and that is not equal to $\sin(\sin x))$
Similarly: $(\sin x)^{-1}=\frac{1}{\sin x }$ but $\sin^{-1}x=\arcsin x$ Lastly, $\sin(x^2)=\sin(x*x)$.
A: Conventions are what somebody decided they'd be.  
For a parallel, consider that mathmeticians say i = sqrt(-1) while electrical engineers say j = sqrt(-1).  For EE's, i was already in use for current.
Note, however, that in naming unit vectors, i parallels the x axis, j parallels y, etc.  And since the complex plane LOOKS like an x-y coordinate system, it would seem more logical that sqrt(-1) be called j.
Just learn to live with it.  All the rest of did.
