Final answer:
To determine cos(θ) for sin(θ) = -1/2 and π < θ < 3π/2, we note that the angle θ is in the third quadrant where cos(θ) is negative. Using the reference angle of 30° or π/6, we can conclude that cos(θ) = -√3/2. The result is also supported by the Pythagorean identity applied in the context of the unit circle.
Step-by-step explanation:
To find cos(θ) when sin(θ) = -1/2 and π < θ < 3π/2, we need to consider the unit circle. In this range of θ, we are in the third quadrant where both sine and cosine are negative. The value of sine being -1/2 suggests that θ corresponds to an angle whose reference angle is 30° or π/6 radians. Since cosine is the x-coordinate of the unit circle, in the third quadrant where both sine and cosine are negative, and knowing the reference angle, we can assert that cos(θ) = -√3/2.
The relationship between sin and cos, given by the Pythagorean identity sin²(θ) + cos²(θ) = 1, can also be used to find the cosine value. Considering that sin(θ) = -1/2 and the angles' range, we square the sine value to get 1/4, and subtract this from 1 to find cos²(θ). Solving for cos(θ) yields the negative square root because we are in the third quadrant, cos(θ) = -√(1-sin²(θ)) = -√(1-1/4) = -√3/2.