Given angle x, where 0 <= x < 360 (degrees), cos(x) is equal to a unique value.

Determine the value of to the nearest degree. Justify your answer.

To determine the value of x where 0 ≤ x < 360 degrees such that cos(x) equals a unique value, we need to find the value of x that corresponds to that particular cosine value.

The cosine function is a periodic function with a period of 360 degrees. This means that after every 360 degrees, the cosine function repeats its values. To find the values of x for which cos(x) is equal to a unique value, we first need to identify the range of possible unique cosine values.

The cosine function ranges between -1 and 1. Therefore, any value outside this range will not correspond to a unique cosine value. For example, if we consider cos(x) = 2, there is no x value within the given range (0 ≤ x < 360) that satisfies this equation.

Now, let's consider the unique cosine values within the range -1 ≤ cos(x) ≤ 1. These unique values occur at specific angles, which are known as the "special angles" in trigonometry. The special angles are: 0°, 30°, 45°, 60°, 90°, 120°, 135°, 150°, 180°, 210°, 225°, 240°, 270°, 300°, 315°, and 330°.

To determine the value of x to the nearest degree, we need to find the closest special angle to the given unique cosine value. By matching the unique cosine value to its corresponding special angle, we can determine the value of x.

For example, if the unique cosine value is 0.5, the closest special angle is 60°. Therefore, the value of x to the nearest degree for cos(x) = 0.5 is 60°.

In summary, to find the value of x to the nearest degree where cos(x) equals a unique value, we need to match the unique cosine value with the closest special angle within the range 0 ≤ x < 360 degrees.