If cos x = 5/13 and sin x < 0, find sin (x/2) .

Answer 1

If cos(x) is positive and sin(x) is negative, then angle x must terminate in Quadrant IV.

Since we are looking for an angle that is half of "x", then that angle must terminate in Quadrant II.

The half angle identity for sine looks like this:

Since our answer is in Q II, and sine is positive in Q II, we can choose the "+" from the "±" in the identity.

So sin(x/2) = √[(1 - 5/13)/2] = √[(8/13)/2] = √(4/13) = 2/√13 = (2√13)/13