Since the sine ratio is the side opposite theta over the hypotenuse when a right triangle is formed, AND since the hypotenuse of any right triangle is never negative, we are looking for where the side opposite the angle is negative. The sin is directly related to the rise on a graph, so we are looking for where y is negative in our x/y coordinate plane. Y is negative in QIII and QIV. In a right triangle, a 30-60-90 right triangle to be specific, the Pythagorean triple is, numerically speaking, [tex] (1,\sqrt{3},2) [/tex], with the 1 being across from the 30 degree angle, the square root of 3 being across from the 60 degree angle, and the hypotenuse across from the right angle. Since our y value is -sqrt(3) and we are in the third and fourth quadrants, we will use the reference angle in each quadrant for a 60 degree angle. The first quadrant is from 0 to 90 degrees, the second quadrant is from 90 to 180, the third is from 180 to 270, and the fourth is from 270 to 360. A 60 degree angle in the third quadrant has a reference angle of 180+60, which is 240 degrees. In the fourth quadrant, the 60 degree angle has a reference angle of 270+30 which is 300 degrees. Those are the angles you're looking for!
