Isn't the "CAST Rule" just an awareness of the sign of the basic trigonometric functions?
Positive Sine: Quadrants I and II
Positive Cosine: Quadrants I and IV
Positive Tangent: Quadrants I and III
From a Reference Angle point of view, |sin(x)| = 1/2 produces x = 30º(Quadrant I), 150º (Quadrant II), 210º (Qudarant III), or 330º (Quadrant IV).
Knowing that, in this case, we have sin(x) = -1/2, discard those in Quadrants I and II, since we know the sine to be positive there. This produces x = 210º or x = 330º.