If $\sin(h(x))=f(x)$ and $\cos(h(x))=g(x)$ can we determine $h(x)$? 
If $\sin(h(x))=f(x)$ and $\cos(h(x))=g(x)$ for some fixed functions$f,g:\mathbb{R}\rightarrow [-1,1]$ can we determine $h(x)$?


To be honest, I was asked to solve a problem with functions of the form $f,g:\mathbb{R}\rightarrow [-1,1]$ involved. So I thought if we could substitute $f,g$ with sines and cosines without loss of generality,(the same way we substitute real numbers with trigonometric functions when solving classical inequalities problems) since we were given that $f^2(x)+g^2(x)=1 \quad \forall x \in \mathbb{R}$. I know that functions are not the same with real numbers, but this is actually the whole reason of my question. 
Therefore my thought is the following:
Since it is $f^2(x)+g^2(x)=1 \quad \forall x \in \mathbb{R}$ we can say that  $f(x)=\sin(h(x))$ and $g(x)=\cos(h(x))$ for some function $h(x)$ that its formula depends on $f(x)$ and $g(x)$. If the sine/cosine function was invertible in all its domain then we could determine $h(x)$ by applying the inverse function of sine in both sides of the equation $f(x)=\sin(h(x))$. But I'm stuck here...
 A: If you know that $f(x)^2+g(x)^2=1$ everywhere and you explicitly don't care about continuity, then it's easy to make a $h$ that works:
$$ h(x) = \begin{cases} \arccos(g(x)) & \text{if } f(x) \ge 0 \\
-\arccos(g(x)) & \text{otherwise} \end{cases} $$
This is then one of infinitely many $h$s that meet your specifications; the others arise by adding an arbitrary function $\mathbb R\to \{2\pi n\mid n\in\mathbb N\}$.
A: $[g(t), f(t)] = [\cos(h(t)), \sin(h(t))]$ is the point on the unit circle with counterclockwise angle $h(t)$ from the positive $x$ axis.  Obviously $h$ is determined only up to a multiple of $2\pi$.  One choice can be described using the two-argument form of $\arctan$.
A: If you know $\sin a$ and $\cos a$ then you can find $a$ up to congruence modulo $2\pi$.  That gives you $h(x)$ modulo $2\pi$, but it tells you nothing about the way in which $h$ depends on $x$ unless you know how $f(x)$ and $g(x)$ depend on $x$.
So if you've got a point wandering around on the circle as $x$ changes, then you know where it it is on the circle as expressed by radians or degrees or whatever your preferred unit is, but you don't know how many times the point wound completely around the circle while you weren't watching.
