Answer
$$y=\frac{5\pi}{6}$$
Work Step by Step
$$y=\sec^{-1}(-\frac{2\sqrt3}{3})$$
*Remember: $\sec^{-1}A=x$ means that $\sec x=A$
Therefore, to find $y$, we need to recall which value of $y$ would have $\sec y=-\frac{2\sqrt3}{3}$.
Now, if you find it hard to relate to secant function, use the following identity to switch it to cosine:
$$\cos y=\frac{1}{\sec y}=\frac{1}{-\frac{2\sqrt3}{3}}=-\frac{3}{2\sqrt3}=-\frac{\sqrt3}{2}$$
So an equivalent task is to find $y$ so that $\cos y=-\frac{\sqrt3}{2}$.
$\cos y\lt0$ shows that we can find a value of $y$ in either quadrant II or III of the unit circle. The choice is arbitrary. I would choose quadrant II here.
In quadrant II, $\frac{5\pi}{6}$ is the value of the angle with which $\cos(\frac{5\pi}{6})=-\frac{\sqrt3}{2}$, or similarly, $\sec(\frac{5\pi}{6})=-\frac{2\sqrt3}{3}$
Therefore, $$y=\frac{5\pi}{6}$$
