The stated interval places the angle in the 3rd quadrant where sin is negative. Using arcsin (-5/13) and a calculator the reference angle is about 22.64 degrees from the x-axis, so the angle (x) is 180+22.64 = 202.64.
Cos(x/2) = cos(202.64/2) = cos (101.32) = appox. -.196.
Without a calculator (except to find the initial angle) you'll use the half angle formula . The half angle formula is + or - that square root, but I don't know how to add the "+ or -" into the code. You have to think about which quadrant holds the angle to determine whether the result is + or -. In this case the angle is in the third quadrant, half the angle is in the second quadrant, so the resulting cos is negative.
The initial angle was arcsin (-5/13) = 22.64. Add to 180 to get into the 3rd quadrant = 180+22.64=202.64.
202.64/2 = 101.32 and = + or - .196. Since it's in the 2nd quadrant it's -.196.
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