If theta is an angle in standard position and its terminal side passes through the...

1. Identify the coordinates: The problem states that the terminal side of angle $\theta$ passes through the point $(-4, 9)$. So, we have $x = -4$ and $y = 9$.
2. Calculate the distance $r$: The distance $r$ from the origin to the point $(x, y)$ is calculated using the Pythagorean theorem: $r = \sqrt{x^2 + y^2}$.
Substituting the given values: $r = \sqrt{(-4)^2 + 9^2} = \sqrt{16 + 81} = \sqrt{97}$.
3. Recall the definition of secant: The secant of an angle $\theta$ in standard position is defined as the ratio of the distance $r$ to the x-coordinate of the point on the terminal side: $\sec \theta = \frac{r}{x}$.
4. Substitute values and simplify: Substitute the calculated value of $r$ and the given x-coordinate into the secant formula: $\sec \theta = \frac{\sqrt{97}}{-4}$.
5. Express in simplest radical form: The expression is already in simplest radical form. However, it is customary to write the negative sign in front of the fraction. Therefore, $\sec \theta = -\frac{\sqrt{97}}{4}$.