A rectangle $ABCD$ is inscribed in a circle, with side $AB$ equal to $a$. From the end $K$ of the diameter $KP$, which is parallel to side $AB$, side $BC$ is seen at an angle $\beta$. Find the radius of the circle.
Let $R$ be the desired radius. Consider the case when point $K$ lies on the arc $A D$, not containing point $B$. Then, in the right triangle $A B C$, it is known that

$$
\angle B A C=\angle B K C=\beta, 2 R=A C=\frac{A B}{\cos \angle B A C}=\frac{\Delta}{\cos \beta}
$$

Therefore, $R=\frac{\square}{2 \cos \beta}$.

If point $K$ lies on the arc $A B D$, then similarly we find that

$$
R=-\frac{\Delta}{2 \cos \beta}
$$

## Answer

$\frac{a}{2|\cos \beta|} \cdot$