Non 1-1 transformation of continuous random variable

[Kate2010]

Homework Statement

X is exponentially distributed with mean s.
Find P(Sin(X)> 1/2)

Homework Equations

f_X(x) = se^-sx, x$$\geq$$ 0
0, otherwise

F_X(x) = 1 - e^-sx, x$$\geq$$ 0
0 otherwise

The Attempt at a Solution

Let Y = sin X

F_Y (y) = P(Y$$\leq$$ y)
= P(sinX $$\leq$$ Y)
= P(X $$\leq$$ arcsin(y), X$$\geq$$ $$\pi$$ - arcsin(y)) {This is where I become slightly unsure}
=F_X(arcsin(y)) - F_X($$\pi$$ - arcsin(y))
=1-e^{-s(arcsin(y))} - (1-e^{-s($$\pi$$ - arcsin(y))})

From here I can differentiate to find the pdf and then use that to find sinX < 1/2.

 

[Kate2010]

Any ideas anyone?