17. Three-sided roof. The tower in King Arthur's castle is topped with a roof that is a triangular pyramid, where all the plane angles at the apex are right angles. The three roof slopes are painted in different colors. The red roof slope is inclined to the horizontal at an angle $\alpha$, and the blue slope at an angle $\beta$. Find the probability that a raindrop, falling vertically onto the roof at a random point, lands on the green slope.
Solution. We will assume that the drop falls on the roof randomly in the sense that the probability of it hitting a certain area of the roof is proportional to the area of the projection of this area onto a horizontal surface. Then we should be interested in the areas of the projections of the slopes onto the triangular base

![](https://cdn.mathpix.com/cropped/2024_05_06_61d9b6af49c57885674cg-20.jpg?height=560&width=491&top_left_y=317&top_left_x=1482)
of the roof.

Let the area of the green slope be $S'$. Then the area of the projection of this slope onto the plane of the triangle $R B G$, lying at the base of the roof, is $S_{O R B}=S_{g} \cos \gamma$, where $\gamma$ is the unknown angle of inclination of the blue slope.

Let $S$ be the area of the triangle $R G B$: $S_{g}=S \cos \gamma$. Therefore,

$$
S_{O R B}=S \cos ^{2} \gamma
$$

![](https://cdn.mathpix.com/cropped/2024_05_06_61d9b6af49c57885674cg-20.jpg?height=534&width=657&top_left_y=1024&top_left_x=1316)

The desired probability is

$$
\frac{S_{O R B}}{S}=\frac{S \cos ^{2} \gamma}{S}=\cos ^{2} \gamma
$$

It remains to find $\cos ^{2} \gamma$. Introducing the notations $S_{r}$ and $S_{b}$ for the areas of the red and blue slopes and performing similar calculations, we find:

$$
S_{O B G}=S \cos ^{2} \alpha ; S_{O R G}=S \cos ^{2} \beta
$$

Then

$$
S=S_{O B G}+S_{O R G}+S_{O R B}=S\left(\cos ^{2} \alpha+\cos ^{2} \beta+\cos ^{2} \gamma\right)
$$

from which $\cos ^{2} \alpha+\cos ^{2} \beta+\cos ^{2} \gamma=1$. Therefore, $\cos ^{2} \gamma=1-\cos ^{2} \beta-\cos ^{2} \alpha$.

Answer: $\cos ^{2} \gamma=1-\cos ^{2} \beta-\cos ^{2} \alpha$.[^2]