47. All edges of a regular hexagonal prism are equal to $a$. Find the area of the section made through the side of the base at an angle $\alpha$ to the base plane.
47. If $0<\alpha<\frac{\pi}{6}$, then $S=\frac{3 a^{2} \sqrt{3}}{2 \cos \alpha} ;$ if $\frac{\pi}{6} \leqslant \alpha<$ $<\operatorname{arctg} \frac{2}{\sqrt{3}}$, then $S=\frac{a^{2}}{6 \cos \alpha}\left(18 \operatorname{ctg} \alpha-3 \sqrt{3}-2 \sqrt{3} \operatorname{ctg}^{2} \alpha\right)$; if $\operatorname{arctg} \frac{2}{\sqrt{3}} \leqslant \alpha<\frac{\pi}{2}$, then $\quad S=\frac{a^{2}}{\sqrt{3} \sin \alpha}(\sqrt{3}+\operatorname{ctg} \alpha)$.