SOLUTION
Let us make a simple diagram of the position of the angle using the points (5, -9).
Now, from the right angled triagle in the diagram, let us find the hypotenues, using the Pythagoras theorem.
From the Pythagoras theorem
![\begin{gathered} h^2=5^2+9^2 \\ h^2=25+81 \\ h=\sqrt[]{106} \end{gathered}](https://img.qammunity.org/2023/formulas/mathematics/college/rhkxdxrnt9g33mudbud72jnuc5xgvl3kmd.png)
Now,
![\begin{gathered} \cos \theta=\frac{adjacent}{\text{hypotenuse }}=\frac{5}{\sqrt[]{106}} \\ \cos \theta=\frac{5}{\sqrt[]{106}} \\ \text{Also, in this quadrant, cos}\theta\text{ is positive } \end{gathered}](https://img.qammunity.org/2023/formulas/mathematics/college/2kxg8eeov3e8fyggcal9wqvagb2gw7bzzl.png)
Then
![\begin{gathered} \sec \theta=(1)/(\cos \theta) \\ \sec \theta=\frac{\sqrt[]{106}}{5} \end{gathered}](https://img.qammunity.org/2023/formulas/mathematics/college/wwjht8rcem48a1hz1pwqd6050zqi2bfg4o.png)
Hence, the answer is
![\sec \theta=\frac{\sqrt[]{106}}{5}](https://img.qammunity.org/2023/formulas/mathematics/college/z6qz9lll40savikhp8zf8zfuei3dwlvz1c.png)