How to find feasible set of $a_1 \cos x_1 + a_2 \cos x_2 + \cdots + a_n \cos x_n \ge 0$ I'm looking to describe the feasible set of $x_1, \ldots, x_n$ for the following inequality:
$$a_1\cos x_1 + a_2\cos x_2 + \cdots+ a_n\cos x_n \ge 0$$
For variables $x_1, \ldots, x_n$ restricted to domain $-k\pi \le x_i \le k\pi$ , $k \ge 1$ an integer, and $a_i$ are real numbers. 
Is this even possible to describe analytically?
EDIT - I should mention that numerical methods would be very welcome as well. For example, if the problem could somehow be split up into finding $x$ satisfying the intersection of a bunch of convex sets that would be fantastic. 
 A: Not an answer, but to get some intuition on what kind of regions you might get it is useful to take the simplest cases $n=2$ and $n=3$ and just plot the regions.
Below is a plot of the allowed region for two choices of $a = (a_1,a_2)$ for $n=2$ and two choices of $a=(a_1,a_2,a_3)$ for $n=3$. We see that depening on the vector $a$ the regions go from being a tilted chess-board (when $a$ is tilted $45^\circ$ wrt the coordinate axes) to stripes (in the limit of $a$ being aligned with one of the coordinate axes).
The Mathematica code to do this is for $n=2$:

a = {1, -1};
v = {Cos[x], Cos[y]};
f[x_, y_] = If[a.v < 0, -1, 1];
ContourPlot[f[x, y], {x, -10, 10}, {y, -10, 10}, Contours -> {0.0}]

and for $n=3$ use

ContourPlot3D[f[x, y, z], {x, -10, 10}, {y, -10, 10}, {z, -10, 10},
Contours -> {0.0}]

$$(a_1,a_2) = \{1,-1\}$$
$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~$
(source: folk.uio.no)
$$(a_1,a_2) = \{1,1/2\}$$
$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~$
(source: folk.uio.no)
$$(a_1,a_2,a_3) = \{1,1,-2\}$$
$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~$
$$(a_1,a_2,a_3) = \{2,1,1\}$$
$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~$
(source: folk.uio.no)
