I assume 3π/2 < x < 2π, so that x is in the 4th quadrant. For angles x terminating in Q4, we have sin(x) < 0 and cos(x) > 0, so tan(x) = sin(x)/cos(x) < 0.
This means that, using the Pythagorean identity,
cos(x) = + √(1 - sin²(x)) = 4/5
tan(x) = (-3/5) / (4/5) = -3/4
Recall the double angle identities:
sin(2x) = 2 sin(x) cos(x) = 2 (-3/5) (4/5) = -24/25
cos(2x) = 2 cos²(x) - 1 = 2 (4/5)² - 1 = 7/25
tan(2x) = 2 tan(x) / (1 - tan²(x)) = 2 (-3/4) / (1 - (-3/4)²) = -24/7
