Find the exact value of sinx/2 if sinx=1/4 and x is such that pi/2

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
]