Given sin x=3/5, pie /2<x<pi what is sin (x/2)=? is sin (x/2)=3/10?

Expert Answers
sciencesolve eNotes educator| Certified Educator

You need to use the following trigonometric identity that helps you to evaluate the sine of half angle, such that:

`sin (x/2) = sqrt((1 - cos x)/2)`

You need first to evaluate `cos x` , hence, you should use the basic formula of trigonometry, such that:

`sin^2 x + cos^2 x = 1 => cos^2 x = 1 - sin^2 x`

`cos x = +-sqrt(1 - sin^2 x)`

Since the problem provides the information that `x in (pi/2,pi)` , then you need to keep only the negative value of cos x because `cos x <0` in quadrant 2.

`cos x = -sqrt(1 - (3/5)^2) => cos x = -sqrt(16/25) => cos x = -4/5`

Replacing `-4/5` for `cos x` in `sin (x/2) = sqrt((1 - cos x)/2)` yields:

`sin (x/2) = sqrt((1 + 4/5)/2) => sin (x/2) = sqrt(9/10)`

`sin (x/2) = 3/sqrt10 => sin (x/2) = (3sqrt10)/10`

Hence, evaluating the value of `sin(x/2)` , under the given condition, yields `sin (x/2) = (3sqrt10)/10` (not `3/10` , as you have suggested).