Final answer:
After calculating sin(θ) using the Pythagorean identity due to θ being in the second quadrant where sine is positive, and applying the double angle formula, the exact value of sin(2θ) is found to be -24/25, which does not match any of the given options.
Step-by-step explanation:
To find the exact value of sin(2θ) where cos(θ) = -4/5, and θ is in quadrant 2, we first need to establish the value of sin(θ).
In the second quadrant, sine is positive and cosine is negative. Since we know cos(θ) and the Pythagorean identity sin²(θ) + cos²(θ) = 1, we can solve for sin(θ) as follows:
sin²(θ) = 1 - cos²(θ) = 1 - (-4/5)² = 1 - 16/25 = 9/25
sin(θ) = ±√(9/25), but since we are in the second quadrant, sin(θ) = √(9/25) = 3/5.
Now use the double angle formula for sine: sin(2θ) = 2 * sin(θ) * cos(θ).
sin(2θ) = 2 * (3/5) * (-4/5) = -24/25
None of the options provided match the calculated value. Therefore, there might be a typo or error in the question or the provided options.