3 sin ^2 x = 8 sin x -5
Move the terms to the left hand side
3 sin ^2 x - 8 sin x -5 =0
Let m = sin x
3 m^2 -8m -5 =0
Factor
(3m-5)(m-1) =0
Using the zero product property
3m-5 =0 m-1 =0
3m =5 m=1
m = 5/3 m=1
Now replace m with sin x
sin x = 5/3 sin x =1
Taking the inverse sin of each side
sin ^-1 ( sin x) = sin ^-1( 5/3) sin ^-1(sin x) = sin ^-1 (1)
x = no solution x = pi/2 + 2pi n where n is an integer
We limit our solution to the interval [0,2pi)
x = pi/2