3.42. A plane passing through a generatrix of a cylinder forms an acute angle $\alpha$ with the plane of the axial section containing the same generatrix. The diagonal of the rectangle obtained by the intersection of the cylinder with this plane is equal to $l$ and forms an angle $\beta$ with the plane of the base. Find the volume of the cylinder.
3.42. Given that $\angle B A C=\alpha, \angle B A B_{1}=\beta, A B_{1}=l$ (Fig. 3.44). From $\triangle B A B_{1}$ we find $A B=l \cos \beta, B B_{1}=l \sin \beta$. Since $O A=O B$ (radii of the base), $\triangle A O B$ is isosceles and $\alpha$ is the angle at its base $A B$. Therefore, $O B=\frac{1}{2} A B: \cos \alpha=\frac{l \cos \beta}{2 \cos \alpha}$. Thus,

$$
\begin{aligned}
& V=\pi \cdot O B^{2} \cdot B B_{1}=\pi \frac{l^{2} \cos ^{2} \beta}{4 \cos ^{2} \alpha} \cdot l \sin \beta= \\
& =\frac{\pi l^{3} \cos ^{2} \beta \sin \beta}{4 \cos ^{2} \alpha}=\frac{\pi l^{3} \sin 2 \beta \cos \beta}{8 \cos ^{2} \alpha}
\end{aligned}
$$