the function y = (a sin x + b cos x + c)/(d sin x + e cos x + f) (where d^2 + e^2 < f^2) is changed to the form 
A sin (x) + B cos (x) + C = 0.
We know that, the equation has solution when and only when A^2 + B^2 >= C^2. 
restart;
a := 1;
b := 4;
c := 0;
d := 1;
e := -2;
f := 3;
solve((-d*y+a)^2+(-e*y+b)^2 >= (-f*y+c)^2 , y)

@Kitonum I usedWith the function (sin(x)+4*cos(x))/(sin(x)-2*cos(x)+3), 
The order of results Min and Max are not correct.

PS. With the function, (sin(x)-3*cos(x))/(sin(x)+cos(x)+2), the results too bad. 

@acer I got the results. Thank you very much.

I like the function TeXForm[HoldForm]] of Mathematica. It is wonderfull. 



 

@Kitonum If I have a list of some different cubes, how can I use your code?

@Kitonum Thank you very much. I did.

@Kitonum I got Error, unable to match delimiters