Final answer:
The solutions for cos(4x)=1/2 in the interval [0,2π) are x = π/12, x = 5π/12, x = 13π/12, and x = 17π/12. The options provided in the question do not match these correct answers.
Step-by-step explanation:
To find all solutions for cos(4x)=1/2 in the interval [0,2π), we need to consider the general solution for the cosine function where it equals 1/2. The cosine of an angle is 1/2 at α = π/3 and α = 5π/3 in the standard position on the unit circle. Given the periodic nature of the cosine function, the solutions in the interval [0,2π) for 4x can also include 4x = π/3 + 2kπ and 4x = 5π/3 + 2kπ, where k is an integer.
Since 4x is within the range [0,8π), we divide this range by 4 to get the range for x, which is [0,2π). We find that x can equal π/12, 5π/12, 13π/12, and 17π/12 in the interval [0,2π) when reduced to their simplest form. Therefore, we can express these solutions as x = π/12, x = 5π/12, x = 13π/12, and x = 17π/12.
None of the provided options (A through D) match the correct answers; thus, they might have been misprinted or are incorrect. To help the student understand the concept, it is important to guide them through the unit circle and the properties of the cosine function.