Lesson Worksheet: The Graphs of Reciprocal Trigonometric Functions Mathematics • 10th Grade

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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:

Identify the graph of 𝑦=𝑥sec.

  • A
  • B
  • C
  • D
  • E

Q2:

Identify the graph of 𝑦=𝑥cot.

  • A
  • B
  • C
  • D
  • E

Q3:

Use the given graph of 𝑦=𝑥csc to determine the domain and range of the cosecant function in degrees.

  • ADomain: 𝑦≤−1 or 𝑦≥1, range: 𝑥∈ℝ,𝑥≠0,+180,−180,+360,−360,…∘∘∘∘
  • BDomain: 𝑥∈ℝ,𝑥≠0,+180,−180,+360,−360,…∘∘∘∘, range: 𝑦≤−1 or 𝑦≥1
  • CDomain: 𝑥∈ℝ,𝑥≠0,+180,−180,+360,−360,…∘∘∘∘, range: 𝑦≤0 or 𝑦≥1
  • DDomain: 𝑥∈ℝ,𝑥≠0,+180,−180,+360,−360,…∘∘∘∘, range: 𝑦<−1 or 𝑦>1
  • EDomain: 𝑥∈ℝ,𝑥≠0,+180,−180,+360,−360,…∘∘∘∘, range: [−1,1]

Q4:

Using the given graph of 𝑦=𝑥cot, determine the domain and range of the cotangent function in radians.

  • ADomain: 𝑥∈ℝ, 𝑥≠𝑛𝜋, 𝑛∈ℤ, range: [0,∞)
  • BDomain: 𝑦∈ℝ, 𝑦≠𝑛𝜋, 𝑛∈ℤ, range: 𝑥∈ℝ
  • CDomain: 𝑦∈ℝ, range: 𝑥∈ℝ, 𝑥≠𝑛𝜋, 𝑛∈ℤ
  • DDomain: 𝑥∈ℝ, 𝑥≠𝑛𝜋, 𝑛∈ℤ, range: 𝑦∈ℝ
  • EDomain: 𝑥∈ℝ, 𝑥≠𝑛𝜋, 𝑛∈ℤ, range: (−∞,0]

Q5:

Find the 𝑥-intercept and the 𝑦-intercept of the graph of 𝑦=−𝑥sec for 0≤𝑥≤2𝜋.

  • AThere is no 𝑥-intercept and 𝑦=1.
  • B𝑥=−1 and there is no 𝑦-intercept.
  • CThere is no 𝑥-intercept and there is no 𝑦-intercept.
  • D𝑥=1 and there is no 𝑦-intercept.
  • EThere is no 𝑥-intercept and 𝑦=−1.

Find the coordinates of the local maxima and minima of the graph of 𝑦=−𝑥sec for 0≤𝑥≤2𝜋.

  • AMinimum: (0,−1),(2𝜋,−1), maximum: (𝜋,1)
  • BMaximum: (2𝜋,−1), minimum: (𝜋,1)
  • CMaximum: (0,−1),(2𝜋,−1), minimum: (𝜋,1)
  • DMaximum: (0,1),(2𝜋,1), minimum: (𝜋,−1)
  • EMaximum: (0,−1), minimum: (𝜋,1)

Identify which of the following is the graph of 𝑦=−𝑥sec for 0≤𝑥≤2𝜋.

  • A
  • B
  • C
  • D
  • E

Q6:

Find the 𝑥-intercept and the 𝑦-intercept of the graph of 𝑦=1−(2𝑥)csc for 0≤𝑥≤𝜋.

  • A𝑥=−3𝜋4, and there is no 𝑦-intercept.
  • BThere is no 𝑥-intercept, and there is no 𝑦-intercept.
  • CThere is no 𝑥-intercept, and 𝑦=𝜋4.
  • D𝑥=𝜋4, and there is no 𝑦-intercept.
  • EThere is no 𝑥-intercept, and 𝑦=−3𝜋4.

Find the coordinates of the local maxima and minima of the graph of 𝑦=1−(2𝑥)csc for 0≤𝑥≤𝜋.

  • AMaximum: −3𝜋4,0, minimum: 3𝜋4,2
  • BMaximum: 𝜋4,0, minimum: −3𝜋4,2
  • CMaximum: −3𝜋4,0, minimum: −𝜋4,2
  • DMaximum: 𝜋4,0, minimum: 3𝜋4,2
  • EMaximum: −𝜋4,0, minimum: −3𝜋4,2

Identify which of the following is the graph of 𝑦=1−(2𝑥)csc for 0≤𝑥≤𝜋.

  • A
  • B
  • C
  • D
  • E

Q7:

By sketching the graphs 𝑦=2𝑥cot and 𝑦=𝑥cos on the same axes, find the number of solutions of the equation cotcos2𝑥=𝑥 for 0≤𝑥≤360∘∘.

Q8:

By sketching the graphs of 𝑦=𝑥sec, 𝑦=(𝜋−𝑥)sec, and 𝑦=𝜋2−𝑥csc, determine which two of the three functions are equal.

  • Aseccsc(𝜋−𝑥)=(𝜋−𝑥)
  • Bsecsec𝑥=(𝜋−𝑥)
  • Cseccsc𝑥=𝜋2−𝑥
  • Dseccsc(𝜋−𝑥)=𝜋2−𝑥

Q9:

What is the period of 𝑦=2𝑥sec? Give your answer in radians.

  • A2𝜋
  • B3𝜋4
  • C𝜋4
  • D𝜋2
  • E𝜋

Q10:

What is the period of 𝑦=𝑥2cot? Give your answer in radians.

  • A𝜋2
  • B4𝜋
  • C3𝜋2
  • D𝜋
  • E2𝜋

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