7.1. Two spheres are inscribed in a dihedral angle, touching each other. The radius of one sphere is twice that of the other, and the line connecting the centers of the spheres forms an angle of $45^{\circ}$ with the edge of the dihedral angle. Find the measure of the dihedral angle. Write the cosine of this angle in your answer, rounding it to two decimal places if necessary.
Answer: 0.56 (exact value: $\frac{5}{9}$).

Solution. Let the sought dihedral angle be denoted by $\alpha$. Let $R$ and $r$ be the radii, and $O_{R}$ and $O_{r}$ be the centers of the larger and smaller spheres, respectively. Let $A$ and $B$ be the feet of the perpendiculars dropped from $O_{R}$ and $O_{r}$ to the edge of the dihedral angle, and let $\beta$ be the angle between the edge of the dihedral angle and $O_{R} O_{r}, R=k r$. Then (see figure),

$O_{R} A=\frac{R}{\sin (\alpha / 2)}, O_{r} B=\frac{r}{\sin (\alpha / 2)} \quad \Rightarrow \quad C O_{R}=\frac{R-r}{\sin (\alpha / 2)}$.

From the right triangle $O_{R} O_{r} C$ we find

$$
\sin \beta=\frac{C O_{R}}{O_{R} O_{r}}=\frac{R-r}{(R+r) \sin (\alpha / 2)}=\frac{k-1}{k+1} \cdot \frac{1}{\sin (\alpha / 2)}
$$

![](https://cdn.mathpix.com/cropped/2024_05_06_a656db1b08002c401269g-5.jpg?height=309&width=783&top_left_y=2564&top_left_x=1139)

From this, $\alpha=2 \arcsin \left(\frac{k-1}{k+1} \cdot \frac{1}{\sin \beta}\right), \cos \alpha=1-2\left(\frac{k-1}{k+1} \cdot \frac{1}{\sin \beta}\right)^{2}$. For $k=2, \beta=45^{\circ}$, we get $\cos \alpha=\frac{5}{9}$.