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=π‘₯sec.

  • A
  • B
  • C
  • D
  • E

Q2:

Identify the graph of 𝑦=π‘₯cot.

  • A
  • B
  • C
  • D
  • E

Q3:

Use the given graph of 𝑦=π‘₯csc to determine the domain and range of the cosecant function in degrees.

  • ADomain: π‘¦β‰€βˆ’1 or 𝑦β‰₯1, range: π‘₯βˆˆβ„,π‘₯β‰ 0,+180,βˆ’180,+360,βˆ’360,β€¦βˆ˜βˆ˜βˆ˜βˆ˜
  • BDomain: π‘₯βˆˆβ„,π‘₯β‰ 0,+180,βˆ’180,+360,βˆ’360,β€¦βˆ˜βˆ˜βˆ˜βˆ˜, range: π‘¦β‰€βˆ’1 or 𝑦β‰₯1
  • CDomain: π‘₯βˆˆβ„,π‘₯β‰ 0,+180,βˆ’180,+360,βˆ’360,β€¦βˆ˜βˆ˜βˆ˜βˆ˜, range: 𝑦≀0 or 𝑦β‰₯1
  • DDomain: π‘₯βˆˆβ„,π‘₯β‰ 0,+180,βˆ’180,+360,βˆ’360,β€¦βˆ˜βˆ˜βˆ˜βˆ˜, range: 𝑦<βˆ’1 or 𝑦>1
  • EDomain: π‘₯βˆˆβ„,π‘₯β‰ 0,+180,βˆ’180,+360,βˆ’360,β€¦βˆ˜βˆ˜βˆ˜βˆ˜, range: [βˆ’1,1]

Q4:

Using the given graph of 𝑦=π‘₯cot, determine the domain and range of the cotangent function in radians.

  • ADomain: π‘₯βˆˆβ„, π‘₯β‰ π‘›πœ‹, π‘›βˆˆβ„€, range: [0,∞)
  • BDomain: π‘¦βˆˆβ„, π‘¦β‰ π‘›πœ‹, π‘›βˆˆβ„€, range: π‘₯βˆˆβ„
  • CDomain: π‘¦βˆˆβ„, range: π‘₯βˆˆβ„, π‘₯β‰ π‘›πœ‹, π‘›βˆˆβ„€
  • DDomain: π‘₯βˆˆβ„, π‘₯β‰ π‘›πœ‹, π‘›βˆˆβ„€, range: π‘¦βˆˆβ„
  • EDomain: π‘₯βˆˆβ„, π‘₯β‰ π‘›πœ‹, π‘›βˆˆβ„€, range: (βˆ’βˆž,0]

Q5:

Find the π‘₯-intercept and the 𝑦-intercept of the graph of 𝑦=βˆ’π‘₯sec for 0≀π‘₯≀2πœ‹.

  • AThere is no π‘₯-intercept and 𝑦=1.
  • Bπ‘₯=βˆ’1 and there is no 𝑦-intercept.
  • CThere is no π‘₯-intercept and there is no 𝑦-intercept.
  • Dπ‘₯=1 and there is no 𝑦-intercept.
  • EThere is no π‘₯-intercept and 𝑦=βˆ’1.

Find the coordinates of the local maxima and minima of the graph of 𝑦=βˆ’π‘₯sec for 0≀π‘₯≀2πœ‹.

  • AMinimum: (0,βˆ’1),(2πœ‹,βˆ’1), maximum: (πœ‹,1)
  • BMaximum: (2πœ‹,βˆ’1), minimum: (πœ‹,1)
  • CMaximum: (0,βˆ’1),(2πœ‹,βˆ’1), minimum: (πœ‹,1)
  • DMaximum: (0,1),(2πœ‹,1), minimum: (πœ‹,βˆ’1)
  • EMaximum: (0,βˆ’1), minimum: (πœ‹,1)

Identify which of the following is the graph of 𝑦=βˆ’π‘₯sec for 0≀π‘₯≀2πœ‹.

  • A
  • B
  • C
  • D
  • E

Q6:

Find the π‘₯-intercept and the 𝑦-intercept of the graph of 𝑦=1βˆ’(2π‘₯)csc for 0≀π‘₯β‰€πœ‹.

  • Aπ‘₯=βˆ’3πœ‹4, and there is no 𝑦-intercept.
  • BThere is no π‘₯-intercept, and there is no 𝑦-intercept.
  • CThere is no π‘₯-intercept, and 𝑦=πœ‹4.
  • Dπ‘₯=πœ‹4, and there is no 𝑦-intercept.
  • EThere is no π‘₯-intercept, and 𝑦=βˆ’3πœ‹4.

Find the coordinates of the local maxima and minima of the graph of 𝑦=1βˆ’(2π‘₯)csc for 0≀π‘₯β‰€πœ‹.

  • AMaximum: ο€Όβˆ’3πœ‹4,0, minimum: ο€Ό3πœ‹4,2
  • BMaximum: ο€»πœ‹4,0, minimum: ο€Όβˆ’3πœ‹4,2
  • CMaximum: ο€Όβˆ’3πœ‹4,0, minimum: ο€»βˆ’πœ‹4,2
  • DMaximum: ο€»πœ‹4,0, minimum: ο€Ό3πœ‹4,2
  • EMaximum: ο€»βˆ’πœ‹4,0, minimum: ο€Όβˆ’3πœ‹4,2

Identify which of the following is the graph of 𝑦=1βˆ’(2π‘₯)csc for 0≀π‘₯β‰€πœ‹.

  • A
  • B
  • C
  • D
  • E

Q7:

By sketching the graphs 𝑦=2π‘₯cot and 𝑦=π‘₯cos on the same axes, find the number of solutions of the equation cotcos2π‘₯=π‘₯ for 0≀π‘₯≀360∘∘.

Q8:

By sketching the graphs of 𝑦=π‘₯sec, 𝑦=(πœ‹βˆ’π‘₯)sec, and 𝑦=ο€»πœ‹2βˆ’π‘₯csc, determine which two of the three functions are equal.

  • Aseccsc(πœ‹βˆ’π‘₯)=(πœ‹βˆ’π‘₯)
  • Bsecsecπ‘₯=(πœ‹βˆ’π‘₯)
  • Cseccscπ‘₯=ο€»πœ‹2βˆ’π‘₯
  • Dseccsc(πœ‹βˆ’π‘₯)=ο€»πœ‹2βˆ’π‘₯

Q9:

What is the period of 𝑦=2π‘₯sec? Give your answer in radians.

  • A2πœ‹
  • B3πœ‹4
  • Cπœ‹4
  • Dπœ‹2
  • Eπœ‹

Q10:

What is the period of 𝑦=π‘₯2cot? Give your answer in radians.

  • Aπœ‹2
  • B4πœ‹
  • C3πœ‹2
  • Dπœ‹
  • E2πœ‹

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