if y = -cos x, what x - value corresponds to a y - value of $\frac{1}{2}$ between $pi$ and $2pi$?

Explanation:

Step1: Set up the equation

Set $y =-\cos x=\frac{1}{2}$, then $\cos x =-\frac{1}{2}$.

Step2: Recall cosine values

We know that $\cos x =-\frac{1}{2}$ has solutions. The general solutions of $\cos x =-\frac{1}{2}$ are $x = 2k\pi\pm\frac{2\pi}{3},k\in\mathbb{Z}$.

Step3: Find the solution in the given interval

We want a solution between $\pi$ and $2\pi$. When $k = 1$ and we take the plus - sign in $x = 2k\pi\pm\frac{2\pi}{3}$, $x=2\pi-\frac{2\pi}{3}=\frac{4\pi}{3}$.

Answer:

$\frac{4}{3}$