All solutions of the simultaneous equations 2 cos theta= square root 3 and sin theta=-1/2 are obtained by...?

This question amounts to finding when cos(x) = sqrt(3)/2 and sin(x) = -.5. I know your problem says theta, but that really doesn't matter. cos(x) = sqrt(3)/2 at pi/6. sin(x) = .5 also at pi/6. The problem then boils down to when is cos(x) positive and sin(x) negative. That occurs in the 4th quadrant. Since we always measure from the x-axis, then at -pi/6, cos(x) = sqrt(3)/2 and sin(x) = -.5. They are also equal for every multiple of 2pi, so you get 2npi - pi/6. Alternatively you could have had 2npi+11pi/6 or a myriad of other answers. Hope this helps