Evaluate exactly sin12 Arc sin45 The answer should be an ex

Evaluate exactly sin(1/2 Arc sin4/5) . The answer should be an exact answer, not in decimal form.

Solution

sin ( 1/2 arc sin 4/5)

let y= 1/2arcsin(4/5)

sin2y = 4/5 ; cos2y = sqrt( 1-sin^2y) = 3/5

cos2y = 1- 2sin^2y

2sin^2y = sqrt( 1-cos2y) = 2/5

sin^2y = 1/5

siny = 1/sqrt5

y = arcsin(1/sqrt5)

So, sin ( 1/2 arc sin 4/5) = siny = sin(arcsin(1/sqrt5))

= 1/sqrt5