Solving an equation for a space dependent variable

Clear["Global`*"]

c = 1/2;
b = 11/10;

{x1, x2} = {-10, 10};

eqn = f[x] == c Cos[2 f[x]] + b x;

f[x_] = f[x] /. 
   DSolve[{D[eqn, x], f[0] == (f0 /.
         Solve[(eqn /. x -> 0 /. f[0] -> f0), f0,
           Reals][[1]])}, f[x], {x, x1, x2}][[1]] //
  FullSimplify

(* InverseFunction[-(1/2) Cos[2 #1] + #1 &][(11 x)/10] *)

Plot[f[x], {x, x1, x2}]

Plot[Sin[f[x]], {x, x1, x2}]

This kind of problem can be solved with FixedPoint iteration. I've used 500 iterations max:

c = 1/2.;
b = 1.1;
g[x_, y_] := c Cos[2 y] + b*x
fp[x_] := FixedPoint[g[x, #] &, x, 500]
ListLinePlot[Table[{x, fp[x]}, {x, -10, 10, .25}], 
 AxesLabel -> {"x", "f[x]"}]
ListLinePlot[Table[{x, Sin@fp[x]}, {x, -10, 10, .25}], 
 AxesLabel -> {"x", "Sin[f[x]]"}]

You can verify that fp is good by checking the error at a given $x$ for example when $x=5$ we have the error g[5, fp[5]] - fp[5] which is a very small number: 1.40502*10^-10. Increase the iterations if you want it smaller.