Answer
The exact value of $y$ in this case is $$y=\frac{\pi}{6}$$
Work Step by Step
$\DeclareMathOperator{\as}{arcsec}$
$$y=\as \frac{2\sqrt 3}{3}$$
First, we see that the domain of inverse secant function is $(-\infty,\infty)$. Therefore, in fact when we deal with inverse secant function, we do not need to do this checking step.
The range of inverse secant function is $[0,\frac{\pi}{2})\hspace{0.2cm}U\hspace{0.2cm}(\frac{\pi}{2},\pi]$. In other words, $y\in[0,\frac{\pi}{2})\hspace{0.2cm}U\hspace{0.2cm}(\frac{\pi}{2},\pi]$.
We can rewrite $y=\as\frac{2\sqrt 3}{3}$ into $\sec y=\frac{2\sqrt 3}{3}$
We know that $$\sec\frac{\pi}{6}=\frac{1}{\cos\frac{\pi}{6}}=\frac{1}{\frac{\sqrt 3}{2}}=\frac{2}{\sqrt3}=\frac{2\sqrt 3}{3}$$
And $\frac{\pi}{6}$ belongs to the range $[0,\frac{\pi}{2})\hspace{0.2cm}U\hspace{0.2cm}(\frac{\pi}{2},\pi]$.
Therefore, the exact value of $y$ here is $$y=\frac{\pi}{6}$$
