We can use the formula for the sine of an addition of angles to answer this question:
Recall that:
sin(A + B) = sin(A) cos(B) + sIn(B) cos(A)
which in our case gives:
sin (x + 3pi/2) = sin(x) cos(3pi/2) + sin(3pi/2) cos(x)
now, we recall that 3pi/2 is one of the "special angles"of trigonometry, where the sin gives a value of -1, and the cosine, a value of "0".
Then we end up with the following:
sin(x) * (0) + (-1) cos(x) = - cos(x)
Therefore, the simplified answer is:
- cos(x) (negative cosine of x)