sin x = - 3/5. First, find cos x.
cos^2 x = 1 - sin^2 x = 1 - 9/25 = 16/25 --> cos x = +- 4/5
To Find sin (x/2), use trig identity:
2sin^2 x = 1 - cos 2x.
In this case: 2sin^2 (x/2) = 1 - cos x = 1 +- 4/5
a. 2sin^2 (x/2) = 1 -(- 4/5) = 9/5
sin^2 (x/2) = 9/10 --> sin (x/2) = +- 3/sqrt10 = +- (3sqrt10)/10
sin x < 0, cos x < 0, x is in Quadrant 3, x/2 is in quadrant 2, then,
sin (x/2) is positive.
sin (x/2) = (3sqrt10)/10
b. 2sin^2 (x/2) = 1 - 4/5 = 1/5
sin^2 (x/2) = 1/10 --> sin (x/2) = +- 1/sqrt10 = +- sqrt10/10.
x is in Quadrant 4, x/2 is also in quadrant 4, sin (x/2) is negative.
sin (x /2) = - sqrt10/10
Check by calculator.
sin x = - 3/5 --> x = - 36^@87 and x = 216^@87
a. x = - 36^@87 --> x/2 = - 18^@46 --> sin (x/2) = - 0.316 = = - sqrt10/10. OK
b. x = 216.87 --> x/2 = 108.46 --> sin (x/2) = 0.948 = (3sqrt10/10). OK
