Worksheet: The Graphs of Reciprocal Trigonometric Functions

In this worksheet, we will practice graphing cosecant, secant, and cotangent functions by understanding that they are related to the graphs of the sine, cosine, and tangent functions.

Q1:

Which of the following is the graph of the function 𝑓(𝑥)=1(𝑥)sin?

  • A
  • B
  • C
  • D

Q2:

Consider the graph of 𝑦=𝑥sin, as shown in the figure. We are going to use the graph to help us identify some key features of the graph of 𝑦=𝑥cosec.

Use the fact that cosecsin𝑥=1𝑥 to determine whether each of the points 𝐴, 𝐵, 𝐶, and 𝐷 will represent the position of an asymptote, a local maximum, or a local minimum on the graph of 𝑦=𝑥cosec.

  • AAsymptote: 𝐴 and 𝐶, local maximum: 𝐵, local minimum: 𝐷
  • BAsymptote: 𝐴 and 𝐵, local maximum: 𝐶, local minimum: 𝐷
  • CAsymptote: 𝐵 and 𝐷, local maximum: 𝐴, local minimum: 𝐶
  • DAsymptote: 𝐴 and 𝐷, local maximum: 𝐵, local minimum: 𝐶
  • EAsymptote: 𝐴 and 𝐶, local maximum: 𝐷, local minimum: 𝐵

Identify which of the following is the graph of 𝑦=𝑥cosec.

  • A
  • B
  • C
  • D
  • E

Q3:

Identify the graph of 𝑦=𝑥sec.

  • A
  • B
  • C
  • D
  • E

Q4:

Identify the graph of 𝑦=𝑥cot.

  • A
  • B
  • C
  • D
  • E

Q5:

The three graphs shown represent the functions 𝑓(𝑥), 𝑔(𝑥), and (𝑥).

The function 𝑓 is represented by the graph C. Which of the following is the equation of 𝑓?

  • A𝑓(𝑥)=𝑥cos
  • B𝑓(𝑥)=𝑥tan
  • C𝑓(𝑥)=𝑥csc
  • D𝑓(𝑥)=𝑥sin
  • E𝑓(𝑥)=𝑥sec

The function 𝑔 is represented by the graph A. Which of the following is the equation of 𝑔?

  • A𝑔(𝑥)=3𝑥sec
  • B𝑔(𝑥)=𝑥sec
  • C𝑔(𝑥)=𝑥csc
  • D𝑔(𝑥)=3𝑥csc
  • E𝑓(𝑥)=𝑥tan

The function is represented by the graph B. Which of the following is the equation of ?

  • A(𝑥)=𝑥sec
  • B𝑓(𝑥)=𝑥tan
  • C(𝑥)=4𝑥sec
  • D(𝑥)=𝑥csc
  • E(𝑥)=4𝑥csc

Q6:

Consider the graph of 𝑦=𝑥cos as shown in the figure. We are going to use the graph to help us identify some key features of the graph of 𝑦=𝑥sec.

Use the fact that seccos𝑥=1𝑥 to determine which points from 𝐴, 𝐵, 𝐶, and 𝐷 can be used to identify the position of an asymptote, a local maximum, or a local minimum on the graph of 𝑦=𝑥sec.

  • AAsymptote: 𝐵, local maximum: 𝐴 and 𝐶, local minimum: 𝐷
  • BAsymptote: 𝐴, local maximum: 𝐵 and 𝐶, local minimum: 𝐷
  • CAsymptote: 𝐷, local maximum: 𝐴 and 𝐶, local minimum: 𝐵
  • DAsymptote: 𝐴 and 𝐵, local maximum: 𝐶 and 𝐷
  • EAsymptote: 𝐵, local maximum: 𝐷, local minimum: 𝐴 and 𝐶

Identify which of the following is the graph of 𝑦=𝑥sec.

  • A
  • B
  • C
  • D
  • E

Q7:

Consider the graph of 𝑦=𝑥tan as shown in the figure. We are going to use the graph to help us identify some key features of the graph of 𝑦=𝑥cot.

Use the fact that cottan𝑥=1𝑥 to determine which of the points 𝐴, 𝐵, 𝐶, and 𝐷 can be used to identify the position of an asymptote, a zero, a positive, or a negative on the graph of 𝑦=𝑥cot.

  • AAsymptote: 𝐴, zero: 𝐵, positive: 𝐶, negative: 𝐷
  • BAsymptote: 𝐷, zero: 𝐶, positive: 𝐵, negative: 𝐴
  • CAsymptote: 𝐴, zero: 𝐶, positive: 𝐵, negative: 𝐷
  • DAsymptote: 𝐶, zero: 𝐵, positive: 𝐴, negative: 𝐷
  • EAsymptote: 𝐵, zero: 𝐴, positive: 𝐶, negative: 𝐷

Identify which of the following is the graph of 𝑦=𝑥cot.

  • A
  • B
  • C
  • D
  • E

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