sin x =1/4.
cos x = sqrt (1 - (sin x)^2)
=> -(sqrt 15) / 4
In the interval pi/2< x < pi
sin (x/2) = sqrt ((1 - cos x)/2)
=> sin (x/2) = sqrt[(4 - sqrt 15)/8]
The required value of sin (x/2) = sqrt[(4 - sqrt 15)/8]
We'll determine sin (x/2), using the half angle formula
sin (x/2) = +/- sqrt [ (1 - cos x) / 2 ]
We know, from enunciation, that:
Pi < x < Pi / 2
We'll divide by 2 the inequality:
Pi / 2 < x / 2 < Pi / 4
From the above inequality, the angle x/2 is in the 1st quadrant and the value of sin (x/2) is positive.
Since sin x = 1/4, we'll apply the trigonometric identity
(sin x)^2 + (cos x)^2 = 1 to determine cos x,
We'll recall that x is in 2nd quadrant where cos x is negative.
cos x = - sqrt(1 - sin 2x)
cos x = - sqrt(1 - 1/16)
cos x = - sqrt(15) / 4
We'll substitute cos x by its value in the formula for sin x/2.
sin x/2 = sqrt [ (1 - sqrt(15)/4) / 2 ]