Final answer:
To find cos x and x, we compare the provided cosine values to those of standard angles in trigonometry, identifying the angles as 60 degrees, 45 degrees, 30 degrees, and 75 degrees for the respective options.
Step-by-step explanation:
To find cos x and then x, we use known values of the cosine function and its relationship to angles in the unit circle. Given cos x = 0.5, we know from standard trigonometry tables or the unit circle that the angle whose cosine value is 0.5 is 60 degrees. This corresponds to option A. For option B, cos x = 0.7071 roughly matches the cosine of 45 degrees. In option C, the value cos x = 0.866 is the cosine of 30 degrees. Lastly, option D presents us with cos x = 0.2588, which does not correspond to a common angle in trigonometry tables, but it's close to the cosine of 75 degrees.
These answers are found by referencing a trigonometry table or the unit circle, which provides the cosine values for specific standard angles. As cosines are periodic, there could be multiple correct angles for each cosine value that are 360 degrees apart (for example, 60 degrees and 420 degrees). However, in the context of a trigonometry class, it is common to work within the first round of the unit circle, which is between 0 and 360 degrees.