arcsin(3/5) is some theta between -pi/2 and pi/2 with sintheta = 3/5.
Furthermore, with -pi/2 <= theta <= pi/2 and sin theta a positive number, we conclude that theta is between 0 and pi/2.
We want to find sin2 theta and we already know sin theta, so if we find cos theta, then we can ue the double angle formula for sine.
You've probably done this kind of problem many times by now. theta is in the first quadrant and sin theta = 3/5, find cos theta.
Use your favorite method -- draw a triangle, or a unit circle, or an angle in standard position, or skip the picture and use cos theta = +-sqrt(1-sin^2 theta) (recall that our theta is in Quadrant 1, so its cosine is positive.)
All of the above is really explanation of our thought process.
All we really need to write is something like:
Let theta = arcsin(3/5), then sin theta = 3/5 and
cos theta = 4/5
And sin(2 theta) = 2 sin theta cos theta.
So, putting it all together we get:
sin(2arcsin(3/5)) = 2(3/5)(4/5) = 24/25
