Final answer:
Given cosθ = 2√5 is not within the valid range for cosine values, we cannot accurately solve for sinθ. If cosθ were a possible value within the range of [-1,1], we could use the Pythagorean identity to find sinθ for a negative result, but the given value is incorrect to perform such a calculation.
Step-by-step explanation:
To find the value of sinθ given that cosθ = 2√5 and sinθ < 0, we use the Pythagorean identity sin²θ + cos²θ = 1. First, we need to observe that the given cosθ value is not possible since the cosine function has a range of [-1,1]. Assuming there's a typo and that cosθ should be within its valid range, we would solve for sinθ as follows:
sin²θ = 1 - cos²θ. If cosθ were normalized to a possible value, like √(2/5), then we would have:
sin²θ = 1 - (√(2/5))² = 1 - 2/5 = 3/5,
Therefore, sinθ = ±√(3/5), and given that sinθ < 0, the value would be negative.
However, without a correct value for cosθ, we cannot accurately determine the value of sinθ. Thus, we are unable to select A, B, C, or D as the correct answer and should clarify the given value of cosθ.