Final answer:
To find cos2θ when sinθ is given and θ is in the second quadrant, we use the Pythagorean identity to find cosθ and then apply the double angle formula, which results in -41/49.
Step-by-step explanation:
If we know that sinθ = 2/7 and 90° < θ < 180°, this means θ is in the second quadrant where sine is positive and cosine is negative. From the Pythagorean identity, we have cos²θ = 1 - sin²θ which becomes cos²θ = 1 - (2/7)². Solving this, we find cos²θ = 1 - 4/49, so cos²θ = 45/49. Because we're in the second quadrant, the cosine value must be negative, therefore cosθ = -√(45/49). Now we use the double angle formula, cos2θ = 2cos²θ - 1, and substitute, yielding cos2θ = 2(-√(45/49))² - 1 = 2(45/49) - 1 = 90/49 - 49/49 = 41/49. Thus, the answer must be negative, because in the intervals 90° to 180°, cos2θ lies in either the second or third quadrant, both of which yield a negative result for the cosine of an angle. Therefore, the correct answer is C. -41/49.