Question

if a vertical line is dropped from the x - axis to the point (12, - 9) in the diagram below, what is the value of sec θ?

Explanation:

Step1: Recall secant definition

$\sec\theta=\frac{r}{x}$, where $r$ is the distance from the origin to the point $(x,y)$ and $x$ is the x - coordinate of the point.

Step2: Calculate $r$ using the distance formula

$r = \sqrt{x^{2}+y^{2}}$, with $x = 12$ and $y=-9$. So $r=\sqrt{12^{2}+(-9)^{2}}=\sqrt{144 + 81}=\sqrt{225}=15$.

Step3: Find $\sec\theta$

Since $\sec\theta=\frac{r}{x}$ and $r = 15$, $x = 12$, then $\sec\theta=\frac{15}{12}=\frac{5}{4}$.

Answer:

$\frac{5}{4}$