Determine the solution( $(s)$ for the following where $0 \pi \leq x \leq 2 \pi$ : $\cos (x)=\frac{1}{2}$

Step 1 ：First, we need to understand the problem. We are asked to find the solutions for \(x\) in the interval \(0 \leq x \leq 2 \pi\) such that \(\cos(x) = \frac{1}{2}\).

Step 2 ：We know that the cosine function has a period of \(2 \pi\), and it equals \(\frac{1}{2}\) at two points within one period: \(x = \frac{\pi}{3}\) and \(x = \frac{5\pi}{3}\).

Step 3 ：Therefore, the solutions to the equation \(\cos(x) = \frac{1}{2}\) in the interval \(0 \leq x \leq 2 \pi\) are \(x = \frac{\pi}{3}\) and \(x = \frac{5\pi}{3}\).

Step 4 ：We can check our solutions by substituting them back into the original equation. For \(x = \frac{\pi}{3}\), we have \(\cos(\frac{\pi}{3}) = \frac{1}{2}\), which is true. For \(x = \frac{5\pi}{3}\), we have \(\cos(\frac{5\pi}{3}) = \frac{1}{2}\), which is also true.

Step 5 ：Therefore, the solutions to the equation \(\cos(x) = \frac{1}{2}\) in the interval \(0 \leq x \leq 2 \pi\) are \(\boxed{x = \frac{\pi}{3}, x = \frac{5\pi}{3}}\).