Since π ≤ θ ≤ 2π (meaning θ is an angle that terminates in either the 3rd or 4th quadrant), we know that sin(θ) < 0.
Recall the Pythagorean identity,
sin²(θ) + cos²(θ) = 1
so that
sin(θ) = -√(1 - cos²(θ)) = -√(1 - (3/4)²) = -√7/4
By definition of tangent,
tan(θ) = sin(θ)/cos(θ) = (-√7/4)/(3/4) = -√7/3
Recall the double angle identity for sine:
sin(2θ) = 2 sin(θ) cos(θ) = 2 (-√7/4) (3/4) = -3√7/8