If theta is an angle in standard position and its terminal side passes through the...
Check the final answer first, then review the worked steps.
Problem
If theta is an angle in standard position and its terminal side passes through the point (-4, 9), find the exact value of sec theta in simplest radical form.
Answer
\(-\frac{\sqrt{97}}{4}\)
Step-by-step solution
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, $\sec \theta = -\frac{\sqrt{97}}{4}$.