Final answer:
The value of cosθ1, given that sinθ1 = -√3/2 and θ1 is in the third quadrant, is -1/2. This conclusion is derived using the Pythagorean identity.
Step-by-step explanation:
The student asked about the value of cosθ1 given that angle θ1 is in the third quadrant and sinθ1 = -√3/2. In the third quadrant, both sine and cosine are negative, as per the trigonometric ASTC rule (All Students Take Calculus), which helps remember the sign of trigonometric functions in different quadrants. We can find the value of cosθ1 by using the Pythagorean identity: sin²θ + cos²θ = 1. Substituting the given sine value, we get (-√3/2)² + cos²θ1 = 1, which simplifies to 3/4 + cos²θ1 = 1. Solving for cosθ1 gives us cosθ1 = ±√(1 - 3/4), which yields cosθ1 = ±√(1/4) = ±1/2. Since the angle is in the third quadrant where cosine is negative, we conclude that cosθ1 = -1/2.