Question 107: Let the complex number $z$ satisfy $\operatorname{Arg}\left(\frac{z+1}{z+2}\right)= \pm \frac{\pi}{6}$, find the range of $\arg z$.

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Question 107, Answer: Let the points corresponding to $-2$ and $-1$ be $A$ and $B$, respectively, and the point corresponding to the complex number $z$ be $Z$. According to the given conditions, $\angle \mathrm{AZB}=\frac{\pi}{6}$.

As shown in the figure, the trajectory of point $\mathrm{Z}$ consists of two segments of arcs (excluding the endpoints). Let the centers of the two arcs be $\mathrm{O}_{1}$ and $\mathrm{O}_{2}$, respectively. Then the corresponding complex numbers are $-\frac{3}{2}+\frac{\sqrt{3}}{2} \mathrm{i}$ and $-\frac{3}{2}-\frac{\sqrt{3}}{2} \mathrm{i}$, and the radii of the two arcs are both 1.

Draw tangents from 0 to the two arcs, with the points of tangency being $\mathrm{Z}_{1}$ and $\mathrm{Z}_{2}$. Let $\angle A O O_{1}=\alpha$ and $\angle O_{1} O Z_{1}=\beta$, then $\alpha=\frac{\pi}{6}$ and $\beta=\arcsin \frac{\sqrt{3}}{3}$. Therefore, $\arg z \in\left[\frac{5 \pi}{6}-\arcsin \frac{\sqrt{3}}{3}, \pi\right) \cup\left(\pi, \frac{7 \pi}{6}+\arcsin \frac{\sqrt{3}}{3}\right]$.