3.39. The plane angle at the vertex of a regular hexagonal pyramid is equal to the angle between a lateral edge and the plane of the base. Find this angle.
3.39. By condition, $O A B C D E F$ is a regular hexagonal pyramid, $O O_{1} \perp(A B C), \angle F O E=\angle O F O_{1}$ (Fig. 3.41). Let $O F=l$ and $\angle F O E=\alpha$; then $F E=\sqrt{2 l^{2}-2 l^{2} \cos \alpha}=2 l \sin \frac{\alpha}{2}$. Considering that $O_{1} F=F E$, in $\triangle O O_{1} F$ we have

$$
\begin{gathered}
\cos \alpha=\frac{2 l \sin \frac{\alpha}{2}}{l} ;-2 \sin ^{2} \frac{\alpha}{2}+1=2 \sin \frac{\alpha}{2} ; 2 \sin ^{2} \frac{\alpha}{2}+2 \sin \frac{\alpha}{2}-1=0 \\
\sin \frac{\alpha}{2}=\frac{-1+\sqrt{3}}{2} ; \frac{\alpha}{2}=\arcsin \frac{\sqrt{3-1}}{2} ; \alpha=2 \arcsin \frac{\sqrt{3-1}}{2}
\end{gathered}
$$