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. 

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.
