sin(2asin(3/4))=?

sin(2asin(3/4))

 

let      $$\theta=asin(3/4)$$       You should recognise that this is an angle.

 

$$\\sin(2\theta)=2sin\theta cos\theta\\\\
sin\theta=\frac{3}{4}\\\\
opp=3,\; hyp=4\; adj=\sqrt{16-9}=\sqrt{7}\\\\
cos\theta=\frac{\sqrt7}{4}\\\\
2sin\theta cos\theta \\\\
=2\times\frac{3}{4}\times\frac{\sqrt7}{4}\\\\
=\frac{6\sqrt7}{16}\\\\\\
sin(2asin(3/4))=\frac{3\sqrt7}{8}\\\\\\$$