Range of a sum of sine waves Suppose I'm given a function
f(x) = sin(Ax + B) + sin(Cx + D)
is there a simple (or, perhaps, not-so-simple) way to compute the range of this function?  My goal is ultimately to construct a function g(x, S, T) that maps f to the range [S, T].  
My strategy is to first compute the range of f, then scale it to the range [0,1], then scale that to the range [S, T].  
Ideally I would like to be able to do this for an arbitrary number of waves, although to keep things simple I'm willing to be satisfied with 2 if it's the easiest route.
Numerical methods welcome, although an explicit solution would be preferable. 
 A: I'll assume that all variables and parameters range over the reals, with $A,C\neq0$. Let's see how we can get a certain combination of phases $\alpha$, $\gamma$:
$$Ax+B=2\pi m+\alpha\;,$$
$$Cx+D=2\pi n+\gamma\;.$$
Eliminating $x$ yields
$$2\pi(nA-mC)=AB-BC+\alpha C-\gamma A\;.$$
If $A$ and $C$ are incommensurate (i.e. their ratio is irrational), given $\alpha$ we can get arbitrarily close to any value of $\gamma$, so the range in this case is at least $(-2,2)$. If $AB-BC$ happens to be an integer linear combination of $2\pi A$ and $2\pi C$, then we can reach $2$, and the range is $(-2,2]$, whereas if $AB-BC$ happens to be a half-integer linear combination of $2\pi A$ and $2\pi C$ (i.e. and odd-integer linear combination of $\pi A$ and $\pi C$), then we can reach $-2$, and the range is $[-2,2)$. (These cannot both occur if $A$ and $C$ are incommensurate.)
On the other hand, if $A$ and $C$ are commensurate (i.e. their ratio is rational), you can transform $f$ to the form
$$f(u)=\sin mu+ \sin (nu+\phi)$$
by a suitable linear transformation of the variable, so $f$ is periodic. In this case, there are periodically recurring minima and maxima, and in general you'll need to use numerical methods to find them.
A: If ${A\over C}\in{\mathbb Q}$ we may assume $A$, $C\in{\mathbb Z}$. In this case $f$ is periodic with period $2\pi$, and the range of $f$ is found by evaluating $f$ at the zeros of $f'$. The latter have to be determined by solving a certain polynomial equation which one obtains by introducing the variable $z:=e^{ix}$.
If ${A\over C}\notin{\mathbb Q}$ then $f$ is almost periodic. In this case the range of $f$ is the open interval $\ ]{-2},2[\ $, because one can find a sequence $x_n\to\infty$ such that the $x_n$ are local maxima of $x\mapsto\sin(Ax+B)$ and at the same time "almost" local maxima of $x\mapsto\sin(Cx+D)$.
${\bf Edit}$ concerning the case ${A\over C}\notin{\mathbb Q}$: As noted in yoriki's answer the range might include one of $\pm2$  if $B$ and $D$ are such that "by coincidence" two local maxima or minima of $x\mapsto\sin(Ax+B)$ and $x\mapsto\sin(Cx+D)$ coincide.
