What is the value of x in the diagram?

The two sides of the smaller triangle (on the right) opposite x are equal lengths.  (it is a 45-45-90 triangle)

Now for the larger , left triangle:

sin 30  = opp/hyp

1/2   =   opp/8         opp = 4

Now use the Pythagorean theorem to find x

x ^2  =  4^2  + 4^2        x= 5.66 units