You can solve this as a quadratic equation, let X = sin(x) to turn the equation into

5X^2 - 8X = -3

5X^2 - 8X + 3 = 0

5X^2 - 5X - 3X + 3 = 0

5X(X - 1) - 3(X - 1) = 0

(X - 1)(5X - 3) = 0

X - 1 = 0 or 5X - 3 = 0

X = 1 or X = 3/5

sin(x) = 1 or sin(x) = 3/5.

Both of these equations will have solutions, it's just you won't be able to get an EXACT answer for sin(x) = 3/5...

For sin(x) = 1, the only acceptable solution is pi/2.