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 $(-3, 4)$. So, we have $x = -3$ and $y = 4$.
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: $r = \sqrt{(-3)^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$.
3. Recall the definition of cosecant: The cosecant of an angle $\theta$ in standard position is defined as the ratio of the distance $r$ to the y-coordinate of the point on the terminal side: $\csc \theta = \frac{r}{y}$.
4. Substitute values and find the cosecant: Substitute the calculated value of $r$ and the given y-coordinate into the formula for cosecant.
$\csc \theta = \frac{5}{4}$.
Since the value is already in its simplest form and does not involve radicals, this is the final answer.