Given that $\cos 2 x=\frac{2}{3}$, with $\frac{\pi}{2}< x< \pi$, determine the exact value $\cos x$
\[
\cos x=
\]
(Write answer in radical form. Rationalize the denominator.)

Step 1 ：We are given that \(\cos 2x = \frac{2}{3}\) and \(\frac{\pi}{2}

Step 2 ：We know that \(\cos 2x = 2\cos^2 x - 1\). We can use this identity to solve for \(\cos x\).

Step 3 ：Setting \(\cos 2x = 2\cos^2 x - 1\) equal to \(\frac{2}{3}\), we get \(2\cos^2 x - 1 = \frac{2}{3}\).

Step 4 ：Solving for \(\cos^2 x\), we get \(\cos^2 x = \frac{5}{6}\).

Step 5 ：Since \(\cos x\) is negative in the second quadrant, we take the negative square root when solving for \(\cos x\), giving us \(\cos x = -\sqrt{\frac{5}{6}}\).

Step 6 ：Final Answer: The exact value of \(\cos x\) is \(\boxed{-\sqrt{\frac{5}{6}}}\).