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

Step 1 ：First, we need to understand the problem. We are asked to find the solutions for $x$ in the interval $0\le x\le 2\pi$ 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 $0\le x\le 2\pi$ 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 $0\le x\le 2\pi$ are $\boxed{{\displaystyle x=\frac{\pi }{3},x=\frac{5\pi }{3}}}$.