sin (2 arcsin (3/5))?

Question

what is sin(2arcsin(3/5))?

i know that without the 2 it would simply be 3/5 but i can't find any examples with a co-efficient in front of the arcsin

Answer 1

sin(2x) = 2sinx cosx

So compute 2 (3/5) cos(arcsin(3/5))

= 2 * (3/5) * (4/5) = 24/25

Answer 2

sin(2arcsin(3/5)) = 2 sin(arcsin(3/5)) cos(arcsin(3/5)) =
= 2*3/5 * +- sqrt(1 - (3/5)^2) = 6/5 * +- sqrt(1 -9/25) =
= 6/5 * +- sqrt(16/25) = +- 6/5 * 4/5 = +- 24/25

Let's determine the sine:
arc cos x = pi/2 + arcsin x

3/5 > 0, hence arcsin(3/5) is between 0 and pi/2
The cos of any number between 0 and pi/2 is
positive,
thus cos(arcsin(3/5)) is positive
and sin(2 arcsin(3/5)) = 24/25