Q: Which of the following is the reciprocal identity for cotangent?

$$\cot(x) = \dfrac{1}{\tan(x)}$$

Q: If $$y = 1 - \tan(x - \dfrac{\pi}{2})$$, what is the $$y-$$intercept of the graph?

$$y = 1 - \tan(-\dfrac{\pi}{2}) ($$undefined)

Q: Which of the following statements is true about the graph of $$y = \tan(x)$$ and $$y = \cot(x)$$?

$$y = \tan(x)$$ has vertical asymptotes at $$x = \dfrac{\pi}{2} + n\pi$$, while $$y = \cot(x)$$ has vertical asymptotes at $$x = n\pi.$$

Question 1

Which of the following is the period of the function $$y = 	an(bx)$$?

Accepted Answer

$$\dfrac{\pi}{b}$$

Question 2

For the function $$y = 	an(x)$$, at which $$x-$$values do vertical asymptotes occur?

Accepted Answer

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

Question 3

Given the function $$y = -2\cot\left(\dfrac{x}{2}\right)$$, what is its period?

Accepted Answer

$$4\pi$$

Question 4

Which transformation is applied to the graph of $$y = 	an(x) to $$obtain $$y = 	an(x - \dfrac{\pi}{2})$$?

Accepted Answer

A horizontal shift to the right by $$\dfrac{\pi}{2}$$ units

Question 5

If $$y = 	an(3x)$$, what is the distance between consecutive vertical asymptotes?

Accepted Answer

$$\dfrac{\pi}{3}$$

Question 6

Which of the following is the reciprocal identity for cotangent?

Accepted Answer

$$\cot(x) = \dfrac{1}{	an(x)}$$

Question 7

A medical device uses the function $$y = 	an(x) to $$model the angle of a rotating arm. If the device is programmed to operate only between $$x = -\dfrac{\pi}{2}$$ and $$x = \dfrac{\pi}{2}$$, what is the range of $$y in $$this interval?

Accepted Answer

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

Question 8

For the function $$y = \cot(x)$$, where do its vertical asymptotes occur?

Accepted Answer

$$x = n\pi$$, where $$n is $$any integer

Question 9

A sound engineer is analyzing a signal modeled by $$y = 	an(x - \dfrac{\pi}{4}). $$What is the phase shift of the signal?

Accepted Answer

$$\dfrac{\pi}{4}$$ units to the right

Question 10

Which of the following best describes the graph of $$y = 	an(x)$$?

Accepted Answer

It is an odd function with vertical asymptotes at $$x = \dfrac{\pi}{2} + n\pi$$ and repeats every $$\pi$$ units.

Question 11

If $$y = 1 - 	an(x - \dfrac{\pi}{2})$$, what is the $$y-$$intercept of the graph?

Accepted Answer

$$y = 1 - 	an(-\dfrac{\pi}{2}) ($$undefined)

Question 12

A physicist models a pendulum's displacement using $$y = \cot(2x). $$What is the period of this function?

Accepted Answer

Question 13

Which of the following statements is true about the graph of $$y = 	an(x)$$ and $$y = \cot(x)$$?

Accepted Answer

$$y = 	an(x)$$ has vertical asymptotes at $$x = \dfrac{\pi}{2} + n\pi$$, while $$y = \cot(x)$$ has vertical asymptotes at $$x = n\pi.$$

Question 14

If $$y = 	an(2x)$$, what is the period and where are the vertical asymptotes located?

Accepted Answer

Question 15

A student graphs $$y = 	an(x)$$ and notices the function is undefined at $$x = \dfrac{\pi}{2}. $$What is the reason for this?

Accepted Answer

Because $$\cos(x) = 0 at$$ $$x = \dfrac{\pi}{2}$$, making $$	an(x)$$ undefined.

Question 16

Which of the following describes the effect of the coefficient $$a in$$ $$y = a	an(x)$$?

Accepted Answer

It stretches or compresses the graph vertically by a factor of $$|a|.$$

Question 17

If $$y = 	an(x)$$ models the angle of a ramp, and the ramp is rotated so the function becomes $$y = 	an(x + \dfrac{\pi}{3})$$, what transformation has occurred?

Accepted Answer

A phase shift of $$\dfrac{\pi}{3}$$ units to the left

Question 18

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

Accepted Answer

$$y = 	an(x) is an $$odd function, so $$	an(-x) = -	an(x)$$

Graphs of Tangent and Cotangent Functions

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