One way would be to use the output arguments from rlocus to get the closed loop pole locations and the associated gain.
Here's the root locus plot.
s=tf('s');
GH=(s+8)/((s+6)*(s+3)*(s+10));
rlocus(GH),sgrid
Get closed loop pole locations and associated gains.
[r,k] = rlocus(GH);
Compute the angles from the -x-axis of the pole locations.
angles = 180/pi*atan2(imag(r),-real(r));
At this point, I'll leave the rest to you to figure out the value of the gain k that corresponds to the desired angle. A couple of things to keep in mind. Some problems might have no value of k that satisfies the criteria and others might have more than one, so you'll have to consider all possbilities. The other is that rlocus does not gurantee (at least I don't think it does) that each column of r is a continous branch of the root locus (though I do think rlocus tries very hard to make that so). So there's a possbility that two or more columns of r can jump from one branch to another, and if that happens at the critical point where the angle is what you're looking for there will be complications.
