Example 2. Given $x \in(3 \pi ; 3.5 \pi)$, find the angle $x$ in the following expressions:
(1) $\operatorname{tg} x=4$,
(2) $\operatorname{ctg} x=\sqrt{5}$.
(1) $\because \operatorname{arctg} 4$ is an angle in the I quadrant, and $x$ is an angle in the III quadrant, so the terminal sides of angle $x$ and $\operatorname{arctg} 4$ are symmetric with respect to the origin. Thus,
$$
\begin{array}{c}
x-\operatorname{arctg} 4=3 \pi, \\
\therefore \quad x=3 \pi+\operatorname{arctg} 4 .
\end{array}
$$
(2) Similarly, $x=3 \pi+\operatorname{arcctg} \sqrt{5}$.

In general, if the terminal sides of angle $x$ and $\alpha$ are symmetric with respect to the x-axis, then $x+\alpha=(2 k+1) \pi$; if they are symmetric with respect to the y-axis, then $x+\alpha=2 k \pi$; if they are symmetric with respect to the origin, then $x-\alpha=(2 k+1) \pi$; if they coincide, then $x-\alpha=2 k \pi (k \in Z)$.

The above two examples are handled using the $R M I$ and symmetry principles. It is also not difficult to study using the general solutions of simple trigonometric equations, which is not detailed here.
(Author's unit: Tianjin Nationalities High School)