5. For which $a$ does the system of equations $\left\{\begin{array}{c}x \sin a-y \cos a=2 \sin a-\cos a \\ x-3 y+13=0\end{array}\right.$ have a solution $(x ; y)$ in the square $5 \leq x \leq 9,3 \leq y \leq 7 ?$
Solution. The line $L$, defined by the equation $x-3 y+13=0$, intersects the sides of the square at points $A(5 ; 6)$ and $B(8 ; 7)$.

![](https://cdn.mathpix.com/cropped/2024_05_06_6cabcf1712d23d5ceb2bg-03.jpg?height=362&width=528&top_left_y=2032&top_left_x=841)

The first equation of the system can be written as $(x-2) \sin a-(y-1) \cos a=0$.

A line with such an equation passes through the point $C(2 ; 1)$ for any $a$ and intersects the segment $[A ; B]$ when $a \in\left[\frac{\pi}{4}+\pi k, \operatorname{arctg} \frac{5}{3}+\pi k\right], k \in Z$.

Answer: $a \in\left[\frac{\pi}{4}+\pi k, \operatorname{arctg} \frac{5}{3}+\pi k\right], k \in Z$.