Final answer:
Given sinθ1 = -√3/2 in the third quadrant, we use the Pythagorean identity to find cosθ1. After simplification, we get cosθ1 = -1/2, as cosine is negative in the third quadrant.
Step-by-step explanation:
The student is asking about the cosine of an angle given the sine, where the angle is in the third quadrant. In the third quadrant, both sine and cosine are negative. Given that the sine of angle θ1 is √3/2, we can use the Pythagorean identity sin¸(θ1) + cos¸(θ1) = 1 to find the cosine. Since sin(θ1) = -√3/2, we get (-√3/2)² + cos¸(θ1) = 1, which simplifies to 3/4 + cos¸(θ1) = 1. Solving for cos¸(θ1), we get cos¸(θ1) = 1 - 3/4 = 1/4, and thus cos(θ1) = ±√(1/4). Because θ1 is in the third quadrant where cosine is negative, cos(θ1) = -1/2.