8,9 The relationship between the angle size and the length of the arc and chord

From vertex $A$ of square $A B C D$ with side length 1, two rays are drawn intersecting the square such that vertex $C$ lies between the rays. The angle between the rays is $\beta$. Perpendiculars are drawn from vertices $B$ and $D$ to the rays.

Find the area of the quadrilateral with vertices at the bases of these perpendiculars.
The desired area is the difference between the areas of two triangles with angle $\beta$ at vertex $A$. The sides of these triangles can be easily expressed through the angles $\beta_{1}$ and $\beta_{2}$ formed by the rays with the sides of the square (see figure). Therefore, $S=1 / 2 \cos \beta_{1} \cos \beta_{2} \sin \beta-1 / 2 \sin \beta_{1} \sin \beta_{2} \sin \beta=1 / 2 \cos \left(\beta_{1}+\beta_{2}\right) \sin \beta=1 / 2 \sin ^{2} \beta$.

![](https://cdn.mathpix.com/cropped/2024_05_06_eab8c6bbe58aee0b55b3g-02.jpg?height=520&width=531&top_left_y=1944&top_left_x=795)

## Answer

$1 / 2 \sin ^{2} \beta$.