Final answer:
The angle 2x when cos(x) = -1/6 and x is in quadrant III is 4π/3 because it's the only option in the range after doubling the interval for the third quadrant angle.
Step-by-step explanation:
To find the angle 2x when cos(x) = -1/6 and x is in quadrant III, we must first understand that the cosine function is negative in the third quadrant. In quadrant III, the angle x would be more than 180° (or π radians) but less than 270° (or 3π/2 radians). Knowing that, we can find the reference angle in quadrant I, where cosine is positive, and then reflect it to the third quadrant.
The reference angle for cos(x) with the same absolute value in the first quadrant would be cos⁻¹(1/6). This is not a typical value that we can find on the unit circle, so we cannot get an exact value for x. However, we know x is in the interval (π, 3π/2). To find 2x, we double this interval to get (2π, 3π). Of the given options, only choice d. 4π/3 falls within this range. Therefore, that is the correct answer.