Question

sin((3pi/2)+x)

Answer 1

There's a sine rule that states: sin(a+b) = sin(a).cos(b) + cos(a).sin(b).

Here a = 3/2*PI and b = x. The known values for sin(3/2*PI) = -1 and cos(3/2*PI) = 0.

This reduces your problem to the following:
sin( 3*PI/2 + x) = sin(3*PI/2)*cos(x) + cos(3*PI/2)*sin(x) =
-1 * cos(x) + 0 * sin(x) = -cos(x)

Therefore sin ( 3*PI/2 + x) = - cos(x)