3. $y=\sqrt{-\operatorname{tg} x-1}+\frac{\sqrt{16-x^{2}}}{1-\log \frac{\sqrt{3}}{2} \sin x}$ The domain of the function is
$\begin{array}{l}{\left[-4,-\frac{5 \pi}{4}\right) \cup\left(\frac{\pi}{2}\right.} \\ \left.\frac{2 \pi}{3}\right) \cup\left(\frac{2 \pi}{3}, \frac{3 \pi}{4}\right)\end{array}$

The translation is:

$\begin{array}{l}{\left[-4,-\frac{5 \pi}{4}\right) \cup\left(\frac{\pi}{2}\right.} \\ \left.\frac{2 \pi}{3}\right) \cup\left(\frac{2 \pi}{3}, \frac{3 \pi}{4}\right)\end{array}$

Note: The text is already in a mathematical notation that is the same in both Chinese and English, so the translation is identical to the original text.