Question

the angle of elevation from the top of a house to a jet flying 8 miles above the house is x radians. if d represents the horizontal distance, in miles, of the jet from the house, express d in terms of a trigonometric function of x. then graph the function for 0 < x < π. express d in terms of a trigonometric function of x. d = □ then graph the function for 0 < x < π. choose the correct graph below. a. b. c. d.

Explanation:

Step1: Recall tangent function

We know that $\tan(x)=\frac{\text{opposite}}{\text{adjacent}}$ in a right - triangle. Here, the opposite side to the angle of elevation $x$ is 8 miles (height of the jet above the house) and the adjacent side is $d$ (horizontal distance). So, $\tan(x)=\frac{8}{d}$.

Step2: Solve for $d$

Cross - multiply the equation $\tan(x)=\frac{8}{d}$ to get $d\times\tan(x)=8$. Then, $d = \frac{8}{\tan(x)}=8\cot(x)$.

Step3: Analyze the graph of $y = 8\cot(x)$

The cotangent function $y=\cot(x)=\frac{\cos(x)}{\sin(x)}$ has vertical asymptotes at $x = 0$ and $x=\pi$ in the interval $(0,\pi)$. As $x$ approaches 0 from the right, $\cot(x)\to+\infty$, and as $x$ approaches $\pi$ from the left, $\cot(x)\to-\infty$. The function $y = 8\cot(x)$ is a vertical stretch of the basic cotangent function by a factor of 8.

Answer:

$d = 8\cot(x)$; The graph of $y = 8\cot(x)$ for $0