Q: Which of the following is the correct domain for the tangent function $$\tan(x)$$?

$$x \neq \dfrac{\pi}{2} + n\pi,\ n \in \mathbb{Z}$$

Question 1

Which of the following is the correct domain for the tangent function $$	an(x)$$?

Accepted Answer

$$x 
eq \dfrac{\pi}{2} + n\pi,\ n \in \mathbb{Z}$$

Question 2

What is the period of the cotangent function $$\cot(x)$$?

Accepted Answer

$$\pi$$

Question 3

Which equation represents the vertical asymptotes of the cosecant function $$\csc(x)$$?

Accepted Answer

$$x = n\pi,\ n \in \mathbb{Z}$$

Question 4

Given $$y = \cot 6\left(x - \dfrac{\pi}{6}\right)$$, what is the period of this function?

Accepted Answer

$$\dfrac{\pi}{6}$$

Question 5

Which of the following best describes the range of the cosecant function $$\csc(x)$$?

Accepted Answer

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

Question 6

A real-world application: In engineering, the tangent function is used to model the slope of a ramp. If the angle of elevation is $$45^\circ$$, what is the slope of the ramp?

Accepted Answer

$$1$$

Question 7

Which transformation occurs in the graph of $$y = 	an(3x + \dfrac{\pi}{2})$$ compared to $$y = 	an(x)$$?

Accepted Answer

Horizontal compression by a factor of $$\dfrac{1}{3}$$ and left shift by $$\dfrac{\pi}{6}$$

Question 8

If $$	an(x) is $$undefined at $$x = \dfrac{\pi}{2}$$, what is the behavior of the graph at this point?

Accepted Answer

There is a vertical asymptote at $$x = \dfrac{\pi}{2}$$

Question 9

Which of the following is true about the cotangent function $$\cot(x)$$?

Accepted Answer

$$\cot(x) is $$undefined at $$x = n\pi,\ n \in \mathbb{Z}$$

Question 10

Which of the following best describes the symmetry of the tangent function $$	an(x)$$?

Accepted Answer

$$	an(x) is an $$odd function

Question 11

A medical imaging device uses the cosecant function to model wave intensity. If $$\csc(x) is $$undefined at $$x = n\pi$$, what does this mean for the device's operation?

Accepted Answer

The device cannot operate at angles that are integer multiples of $$\pi$$

Question 12

Which of the following is the correct formula for the cotangent function?

Accepted Answer

$$\cot(x) = \dfrac{\cos(x)}{\sin(x)}$$

Question 13

If $$y = -3\cot(x - \dfrac{\pi}{3})$$, what is the amplitude of the function?

Accepted Answer

The function does not have an amplitude

Question 14

Which of the following statements is true about the range of the tangent function $$	an(x)$$?

Accepted Answer

$$(-\infty, \infty)$$

Question 15

A sound engineer uses the cotangent function to model phase shifts. If the function is $$y = \cot(2x)$$, what is the period?

Accepted Answer

$$\dfrac{\pi}{2}$$

Question 16

Which of the following describes the effect of the transformation $$y = 	an(x + \dfrac{\pi}{4})$$?

Accepted Answer

The graph shifts left by $$\dfrac{\pi}{4}$$

Question 17

Which of the following is NOT a feature of the graph of $$y = \csc(x)$$?

Accepted Answer

Horizontal asymptotes at $$y = 0$$

Question 18

If $$y = 2	an(x)$$, what is the period of the function?

Accepted Answer

$$\pi$$

Question 19

Which of the following best describes the vertical asymptotes of $$y = 	an(3x)$$?

Accepted Answer

Graphs of the Other Trigonometric Functions

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