3. In $\angle A$, there is a fixed point $P$, draw a line through $P$ intersecting the two sides at $B$ and $C$. When does $\frac{1}{P B}+\frac{1}{P C}$ reach its maximum value?
$$
\begin{array}{l}
\frac{1}{P B}+\frac{1}{P C}=\frac{1}{\rho_{B}}+\frac{1}{\rho_{C}} \\
=\frac{\sin (\theta+\alpha)}{\rho_{0} \sin \alpha} \\
+\frac{\sin (\theta-\beta)}{\rho_{0} \sin \beta} \\
=\frac{\sin (\theta+\alpha) \sin \beta+\sin (\theta-\alpha) \sin \alpha}{\rho_{0} \sin \alpha \cdot \sin \beta} \\
=\frac{\sin \theta \cdot \sin (\alpha+\beta)}{\rho_{0} \sin \alpha \cdot \sin \beta}. \\
\end{array}
$$

Obviously, when $\theta=90^{\circ}$, i.e., $A P \perp B C$, $\frac{1}{P B}+\frac{1}{P C}$ reaches its maximum value.