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

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: 𝑦1 or 𝑦1, range: 𝑥,𝑥0,+180,180,+360,360,
  • BDomain: 𝑥,𝑥0,+180,180,+360,360,, range: 𝑦1 or 𝑦1
  • CDomain: 𝑥,𝑥0,+180,180,+360,360,, range: 𝑦0 or 𝑦1
  • DDomain: 𝑥,𝑥0,+180,180,+360,360,, range: 𝑦<1 or 𝑦>1
  • EDomain: 𝑥,𝑥0,+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, 𝑥𝑛𝜋, 𝑛, range: [0,)
  • BDomain: 𝑦, 𝑦𝑛𝜋, 𝑛, range: 𝑥
  • CDomain: 𝑦, range: 𝑥, 𝑥𝑛𝜋, 𝑛
  • DDomain: 𝑥, 𝑥𝑛𝜋, 𝑛, range: 𝑦
  • EDomain: 𝑥, 𝑥𝑛𝜋, 𝑛, range: (,0]

Q5:

Find the 𝑥-intercept and the 𝑦-intercept of the graph of 𝑦=𝑥sec for 0𝑥2𝜋.

  • 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 0𝑥2𝜋.

  • 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 0𝑥2𝜋.

  • A
  • B
  • C
  • D
  • E

Q6:

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

  • 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 0𝑥𝜋.

  • 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 0𝑥𝜋.

  • 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 0𝑥360.

Q8:

By sketching the graphs of 𝑦=𝑥sec, 𝑦=(𝜋𝑥)sec, and 𝑦=𝜋2𝑥csc, determine which two of the three functions are equal.

  • Aseccsc(𝜋𝑥)=(𝜋𝑥)
  • Bsecsec𝑥=(𝜋𝑥)
  • Cseccsc𝑥=𝜋2𝑥
  • Dseccsc(𝜋𝑥)=𝜋2𝑥

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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