cos²x + sin²x = 1
tan x = sin x / cos x
So, if you know that sin (A) = -1/2, you get
cos²A + (-1/2)² = 1 <=> cos²A = 1 - 1/4 <=> cos²A = 3/4
cos A = ± sqrt(3)/2
Whether cos A is positive or negative depends on the quadrant that the angle A is in.
Once you know that sin A = -1/2 and cos A = ± sqrt(3)/2 , you can calculate tan A by applying the second formula:
tan A = (-1/2)/(± sqrt(3)/2) so tan A = ±(1/sqrt(3)) , again depending on the quadrant.
This is my first post on these forums, but I hope I was able to help.