5.3.3 * Let the complex number $z$ satisfy that the absolute value of an argument of $\frac{z+1}{z+2}$ is $\frac{\pi}{6}$. Find the locus of the point $Z$ corresponding to the complex number $z$ and the range of the principal value of the argument of the complex number $z$.
Let $-2, -1$ correspond to points $A, B$ respectively. According to the problem, let $\angle A Z B = \frac{\pi}{6}$. As shown in the figure, the trajectory of point $Z$ is two segments of arcs (excluding the endpoints). Their centers are $O_{1}, O_{2}$, and the corresponding complex numbers are
$$
-\frac{3}{2} + \frac{\sqrt{3}}{2} \mathrm{i}, -\frac{3}{2} - \frac{\sqrt{3}}{2} \mathrm{i},
$$

with the radius of each circle being 1.
As shown in the figure, $\tan \alpha = \frac{\sqrt{3}}{3}, \alpha = \frac{\pi}{6}, \beta = \arcsin \frac{\sqrt{3}}{3}$, so

the principal value range of the argument of the complex number $z$ is
$$
\left[\frac{5 \pi}{6} - \arcsin \frac{\sqrt{3}}{3}, \pi\right) \cup \left(\pi, \frac{7 \pi}{6} + \arcsin \frac{\sqrt{3}}{3}\right] .
$$