what part do you have trouble with? personally, i use the unit circle.
you know tanx = sinx / cosx = -sqrt(3)
since arctanx is negative, so you can narrow down the quadrant knowing either sinx is negative or cosx is negative.
quadrant 2: cosx is negative, sinx is positive
quadrant 4: cosx is positive, sinx is negative
looking at the unit circle, how do you get sinx / cosx = -sqrt(3)? with sinx = -sqrt(3)/2 and cosx = 1/2. which angle gives that? 5pi / 3, or -pi/3