4. A plane intersects the faces and edge of a dihedral angle, and the projection of the edge onto this plane coincides with the bisector of the angle formed by the intersection of the plane with the dihedral angle. The edge forms an angle of magnitude $\alpha$ with the considered plane, and the magnitude of the angle at the intersection is $\beta$. Find the magnitude of the given dihedral angle.
4. In Figure 82, we have: $l$ - the edge of the dihedral angle, $[O A)$ and $\left[O A^{\prime}\right)$ - the intersections of its faces with a given plane, $[O P)$ - the bisector of angle $A O A^{\prime}$. From point $P$, which lies on the bisector $[O P)$, we draw a perpendicular $[P M]$ to the edge $l$. The plane $O P M$ is the plane of symmetry of the lines $O A$ and $O A^{\prime}$, and $(A A^{\prime}) \perp (O P)$. Let $|O P|=p$; then we get: $|P M|=p \sin \alpha, |A P| = p \operatorname{tg} \frac{\beta}{2}$. Let $\gamma$ be the desired value of the linear angle $A M A^{\prime}$, we get:

$$
\operatorname{tg} \frac{\gamma}{2}=\frac{|A P|}{|P M|}=\frac{\operatorname{tg} \frac{\beta}{2}}{\sin \alpha}
$$