Q: A real-world application: The angle of elevation $$x (in $$radians) to the top of a building is measured from a point $$d$$ meters away. The height $$h of $$the building is given by $$h = d \tan x. If$$ $$d = 50$$ meters and $$x = \frac{\pi}{6}$$, what is the height of the building?

$$h = 50 \times \sqrt{3} \approx 86.60$$ meters

Question 1

Which of the following best describes the domain of the function $$y = 	an x$$?

Accepted Answer

All real numbers except odd multiples of $$\frac{\pi}{2}$$

Question 2

What is the period of the function $$y = \cot x$$?

Accepted Answer

$$\pi$$

Question 3

Which transformation occurs when graphing $$y = 3 	an 2x$$ compared to $$y = 	an x$$?

Accepted Answer

The graph is vertically stretched by a factor of 3 and horizontally compressed by a factor of 2.

Question 4

For the function $$y = 	an x$$, what happens to the value of $$y as$$ $$x$$ approaches $$\frac{\pi}{2}$$ from the left?

Accepted Answer

$$y$$ approaches $$+\infty$$

Question 5

Which of the following statements is true about the graph of $$y = \csc x$$?

Accepted Answer

It has vertical asymptotes at integral multiples of $$\pi.$$

Question 6

Given $$y = 2 \sec 2x$$, what is the period of the function?

Accepted Answer

$$\pi$$

Question 7

Which of the following best describes the range of $$y = \sec x$$?

Accepted Answer

$$(-\infty, -1] \cup [1, \infty)$$

Question 8

If $$y = \cot x$$, where do the vertical asymptotes occur?

Accepted Answer

At integral multiples of $$\pi$$

Question 9

Which of the following is true about the symmetry of $$y = 	an x$$?

Accepted Answer

It is an odd function with origin symmetry.

Question 10

A real-world application: The angle of elevation $$x (in $$radians) to the top of a building is measured from a point $$d$$ meters away. The height $$h of $$the building is given by $$h = d 	an x. If$$ $$d = 50$$ meters and $$x = \frac{\pi}{6}$$, what is the height of the building?

Accepted Answer

$$h = 50 	imes \sqrt{3} \approx 86.60$$ meters

Question 11

Which of the following is the correct reciprocal identity for $$\csc x$$?

Accepted Answer

$$\csc x = \frac{1}{\sin x}$$

Question 12

For the function $$y = \cot x$$, what is the y-coordinate of the point that is $$\frac{1}{4} of $$the way between two consecutive asymptotes?

Accepted Answer

$$y = A$$

Question 13

Which of the following describes the range of $$y = \csc x$$?

Accepted Answer

$$(-\infty, -1] \cup [1, \infty)$$

Question 14

If $$y = 	an(Bx - C)$$, how do you find the equations of two consecutive vertical asymptotes?

Accepted Answer

$$Bx - C = -\frac{\pi}{2}$$ and $$Bx - C = \frac{\pi}{2}$$

Question 15

Which of the following is true about the function $$y = \sec x$$?

Accepted Answer

It is an even function with y-axis symmetry.

Question 16

Suppose you are graphing $$y = \cot x$$ for $$x in$$ $$[0, 2\pi]. $$How many vertical asymptotes will appear in this interval?

Accepted Answer

2

Question 17

Which of the following is the correct process for graphing $$y = 2 \sec 2x$$?

Accepted Answer

Graph $$y = 2 \cos 2x$$, draw vertical asymptotes at its x-intercepts, and sketch the reciprocal curve.

Question 18

Which of the following is NOT a characteristic of the graph of $$y = 	an x$$?

Accepted Answer

It has vertical asymptotes at integral multiples of $$\pi.$$

Question 19

A medical imaging device uses the function $$y = 	an x to $$model the angle of a rotating sensor. If the sensor cannot operate when $$y is $$undefined, at which angles (in radians) should the device avoid operating?

Accepted Answer

At $$x = \frac{\pi}{2} + n\pi$$, where $$n is $$any integer

Graphs of Other Trigonometric Functions: Tangent, Cotangent, Secant, and Cosecant

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