TITLE: Why aren't the graphs of $\sin(\arcsin x)$ and $\arcsin(\sin x)$ the same?
QUESTION [14 upvotes]: (source for above graph)

(source for above graph)
Both functions simplify to x, but why aren't the graphs the same?

REPLY [3 votes]: Actually, my comment above is a bit imprecise. I shall make it more precise here.
The motivation is clear. We want to invert the function $\sin:\Bbb R\to \Bbb R$. But this function isn't injective, nor is it surjective, and that is a problem, since a function is invertible if and only if it is bijective.
So if we want to invert $\sin$, we first have to restrict its domain and codomain appropriately to make it bijective. The restriction $\sin:[-\frac{\pi}{2},\frac{\pi}{2}]\to[-1,1]$ is a bijective function, so it has an inverse. (Stricly speaking, this is a new function, but we usually use the same name for it, which might add to the confusion.) And it is the inverse of this function that is called $\arcsin:[-1,1]\to [-\frac{\pi}{2},\frac{\pi}{2}]$. That's why $\arcsin$ is defined only on the interval $[-1,1]$ and the functions $\sin:[-\frac{\pi}{2},\frac{\pi}{2}]\to[-1,1]$ and $\arcsin:[-1,1]\to [-\frac{\pi}{2},\frac{\pi}{2}]$ are inverse to each other.
Now, we can still look at what happens if we take the original function $\sin:\Bbb R\to\Bbb R$ and compose it with $\arcsin:[-1,1]\to [-\frac{\pi}{2},\frac{\pi}{2}]$, but then the functions are not inverse anymore and we get the behaviour you describe above.