Final answer:
Given cos θ = 3/4 in the range π ≤ θ ≤ 2π, tan θ is found to be -√7/3 and sin 2θ is -3√7/8, using trigonometric identities and considering the quadrant where θ lies.
Step-by-step explanation:
To solve for tan θ and sin 2θ given that cos θ = 3/4 and π ≤ θ ≤ 2π, we must consider the quadrant in which θ lies. In the specified interval, θ is in the third or fourth quadrant, where cosine is positive and sine and tangent are negative.
To find tan θ, we use the Pythagorean identity sin^2 θ + cos^2 θ = 1. Substituting cos θ = 3/4 and solving for sine, we get sin θ = -√(1 - (3/4)^2) = -√(1 - 9/16) = -√(7/16) = -√7/4. Therefore, tan θ = sin θ / cos θ = -√7/3.
For sin 2θ, we use the double-angle formula sin 2θ = 2 sin θ cos θ. Substituting the known values gives us sin 2θ = 2(-√7/4)(3/4), simplifying to sin 2θ = -3√7/8.