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MA003: Precalculus II

Unit 2: Periodic Functions   Populations that oscillate over seasons or years, temperatures that oscillate over days or hours, and tidal depths are all natural phenomena that can be determined using trigonometric functions. Understanding the graphs of the trigonometric functions and how slight changes in the function formulas affect their graphs gives a more precise way of modeling such phenomena, gathering data, and allowing for predictions to be made. 

This unit continues the study of the trigonometric functions with an in-depth study of their graphical properties and transformations and how they relate to the function formulas. You will also study the appropriate domain restrictions necessary to create the inverse trig functions and explore their properties. You will use the identities established in the previous unit, as well as some new identities, and the inverse trig functions to simplify trigonometric expressions and solve trig equations. You will also begin to use the trig functions to model natural phenomena like those discussed above.

Unit 2 Time Advisory
This unit should take you approximately 14 hours to complete.

☐    Subunit 2.1: 3 hours

☐    Subunit 2.2: 2 hours

☐    Subunit 2.3: 2 hours

☐    Subunit 2.4: 2 hours

☐    Subunit 2.5: 2 hours

☐    Unit 2 Assignment (Review): 1 hour

☐    Unit 2 Assessment: 2 hours

Unit2 Learning Outcomes
Upon successful completion of this unit, the student will be able to:

  • sketch the graphs of trigonometric functions and their transformations and state the domain and range of these functions;
  • identify the amplitude, period, midline (vertical shift), and phase shift (or horizontal shift) from a graph, formula, table, or verbal description;
  • given the graph of trigonometric functions or their transformations determine a formula for the graph;
  • use trigonometric functions to model periodic behavior described numerically or verbally (such as fitting a graph to a table of values, a formula for the height of a seat on a Ferris wheel, modeling daylight hours or tide charts, or other periodic situations);
  • evaluate inverse trigonometric functions involving the basic angles by applying the domain and range of these functions without the use of a calculator;
  • convert trigonometric expressions such as cos(arctan x) into algebraic expressions;
  • identify the graphical relationships between sine, cosine, and tangent and their inverses; and
  • associate/link/relate the domain and range of the inverse trigonometric functions with the domain and range of trigonometric functions.  

2.1 Sinusoidal Graphs   - Reading: Lippman and Rasmussen’s Precalculus: An Investigation of Functions: “Chapter 6: Periodic Functions” Link: Lippman and Rasmussen’s Precalculus: An Investigation of Functions: ** “Chapter 6: Periodic Functions (PDF)
 
Instructions: The graphs of the sinusoidal functions have some important features that help us construct them, and make them useful for modeling. Read pages 353-365 to gain an understanding of the properties of these graphs. This reading also covers the topics outlined in subunits 2.1.1 through 2.1.5.

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2.1.1 Periodic Functions   Note: Periodic Functions are covered in the reading listed under subunit 2.1; pay attention in particular to pages 353-355, and be sure to work through any examples or “Try It Now” problems. Solutions to these problems are located at the end of the selected reading.

2.1.2 Domain and Range of the Sine and Cosine Functions   Note: This topic is covered by the reading under subunit 2.1. Focus on page 355 of the reading to see how the graphs of sine and cosine are derived from the unit circle and what their domain and ranges are.

2.1.3 Period of Sine and Cosine   Note: Periodicity is an important property of the tr. Read pages 355 and 356 of the reading listed under subunit 2.1 to begin to develop this concept. 

2.1.4 Negative Angle Identities   Note: Read pages 356 and 357 of the reading listed under subunit 2.1 to see how the symmetry of the graphs of the trig functions yields a new group of identities.  

2.1.5 Transforming Sine and Cosine Functions   Note: Understanding the relationship between algebraic changes in the formulas for the trig functions and the resulting changes in their graphs will yield a deeper understanding of how to apply the functions to real world applications. Pages 357 and 358 of the reading listed under subunit 2.1 begin to delve into this topic. 

  • Web Media: YouTube: patrickJMT’s “Graphing Trigonometric Functions - Graph Transformations P1” Link: YouTube: patrickJMT’s “Graphing Trigonometric Functions - Graph Transformations P1 (YouTube)
     
    Instructions: Watch this video to deepen your understanding of the relationship between changes in algebraic formulas and the graphs of the trig functions.

    Watching this video should take approximately 10 minutes.
     
    Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

2.1.5.1 Midline   Notes: Read pages 358-360 of the reading listed under subunit 2.1 and work through the relevant examples and “Try It Now” problems to gain an understanding of midline shifts in the trig functions. Solutions to these problems are found at the end of the selected reading.

  • Web Media: YouTube: MrGibsonrhs’s “Writing an Equation of a Sin/Cos Function When Given the Graph” Link: YouTube: MrGibsonrhs’s “Writing an Equation of a Sin/Cos Function When Given the Graph (YouTube)
     
    Instructions: Watch the video to get a better understanding of midline shifts and re-work any examples that gave you trouble.
     
    Watching this video should take approximately 5 minutes.

    Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.   

2.1.5.2 Transformations of Sine and Cosine   Note: This topic is covered by the reading beneath subunit 2.1. In particular, pages 361-363 deal specifically with transformations of sine and cosine. Read these pages and work through the examples and ‘Try It Now” problems. Solutions to these problems are available at the end of the selected reading.

  • Web Media: YouTube: Brightstorm’s “Precalculus: Transforming the Graphs of Sine and Cosine” Link: YouTube: Brightstorm’s “Precalculus: Transforming the Graphs of Sine and Cosine (YouTube)
     
    Instructions: Watch the video to deepen your understanding of transformations of the graphs of sine and cosine.

    Watching this video should take approximately 5 minutes.
     
    Terms of Use: Please respect the copyright and terms of use displayed on the webpage above

  • Web Media: YouTube: Khan Academy’s “Determining the Equation of a Trigonometric Function” Link: YouTube: Khan Academy’s “Determining the Equation of a Trigonometric Function (YouTube)

    Instructions: Watch this video to understand how to derive the equation of a trig function from the transformations of its graph.

    Watching this video should take approximately 10 minutes.

    Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License. It is attributed to the Khan Academy.

  • Web Media: YouTube: Brightstorm’s “Precalculus: Finding an Equation for the Sine or Cosine Wave” Link: YouTube: Brightstorm’s “Precalculus: Finding an Equation for the Sine or Cosine Wave (YouTube)
     
    Instructions: Watch this video to understand how to derive the equation of a trig function from the transformations of its graph.

    Watching this video should take approximately 3 minutes.
     
    Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

  • Web Media: YouTube: MathTV’s “Trigonometry: Finding an Equation from a Graph” Link: YouTube: MathTV’s “Trigonometry: Finding an Equation from a Graph (YouTube)
     
    Instructions: Watch this video to understand how to derive the equation of a trig function from the transformations of its graph.

    Watching this video should take approximately 6 minutes. 
     
    Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.   

2.1.5.3 Horizontal Shifts of Sine and Cosine   Note: This topic is covered by the reading under subunit 2.1. In particular, please read pages 363-365 to learn about horizontal shifts of sine and cosine. Work through the examples and “Try It Now” problems. Solutions to these problems are available at the end of the selected reading.

  • Web Media: YouTube: Brightstorm’s “Precalculus: More Transformations of Sine and Cosine” Link: YouTube: Brightstorm’s “Precalculus: More Transformations of Sine and Cosine” (YouTube)
     
    Instructions: After watching this video use what you have learned to work any problems that gave you trouble in the reading.

    Watching this video should take approximately 3 minutes.
     
    Terms of Use: Please respect the copyright and terms of use displayed on the webpage above. 

  • Web Media: YouTube: AIRichards314’s “Using Transformations to Sketch the Graphs of Sinusoidal Functions” Link: YouTube: AIRichards314’s “Using Transformations to Sketch the Graphs of Sinusoidal Functions” (YouTube)
     
    Instructions: After watching this video use what you have learned to work any problems that gave you trouble in the reading.

    Watching this video should take approximately 10 minutes.
     
    Terms of Use: Please respect the copyright and terms of use displayed on the webpage above. 

  • Assessment: Washington State Board for Community and Technical Colleges’ MyOpenMath: “Section 6.1 Graded Homework” Link: Washington State Board for Community and Technical Colleges’ MyOpenMath: “Section 6.1 Graded Homework (HTML)

    Instructions: Once you have completed the readings and viewed the material for all of subunit 2.1, complete this assessment. You will need to create a free account in order to access the material. If you have already created an account, log in to access the assessment. Your answers will be scored upon submission. 

  • Assessment: Washington State Board for Community and Technical Colleges’ “Chapter 6 Quiz 5” Link: Washington State Board for Community and Technical Colleges’ “Chapter 6 Quiz 5 (PDF)
     
    Instructions: This quiz uses the information from subunit 2.1 as well as in the previous sections. Complete it after you have worked through the material in this subunit. 

2.2 Graphs of the Other Trig Functions   - Reading: Lippman and Rasmussen’s Precalculus: An Investigation of Functions: “Chapter 6: Periodic Functions” Link: Lippman and Rasmussen’s Precalculus: An Investigation of Functions: ** “Chapter 6: Periodic Functions (PDF)
 
Instructions: Much like the sinusoidal functions, the remaining trig function graphs have some key features that are important to understand. Read pages 369- 374 to understand these. This reading selection covers the topics outlined in subunits 2.2.1 through 2.2.4.

Terms of Use: The article above is released under a [Creative
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2.2.1 The Graph of the Tangent Function   Note: To understand the graph of the tangent function, focus on pages 369-372 of the reading listed under subunit 2.2, working through the examples and “Try It Now” problems. Solutions to the Try It Now problems are available at the end of the selected reading.

2.2.2 The Graphs of the Secant, and Cosecant Functions   Note: To understand the graph of the secant and cosecant functions, focus on pages 372 and 373 of the reading listed under subunit 2.2, working through the examples and “Try It Now” problems. Solutions to the Try It Now problems are available at the end of the selected reading.

2.2.3 The Graphs of the Cotangent Function   Note: To understand the graph of the cotangent function, focus on pages 373 and 374 of the reading listed under subunit 2.2, working through the examples and “Try It Now” problems. Solutions to the Try It Now problems are available at the end of the selected reading.

2.2.4 Negative Angle Identities   Note: As for sine and cosine, we can use symmetry to establish a set of negative angel identities for the remaining trig functions. Focus on page 374 of the reading listed under subunit 2.2 to see these, working through the examples and “Try It Now” problems. Solutions to the Try It Now problems are available at the end of the selected reading.

  • Assessment: Washington State Board for Community and Technical Colleges’ MyOpenMath: “Section 6.2 Graded Homework” Link: Washington State Board for Community and Technical Colleges’ MyOpenMath: “Section 6.2 Graded Homework (HTML)

    Instructions: Once you have completed the readings and viewed the material for all of subunit 2.2, complete this assessment. You will need to create a free account in order to access the material. If you have already created an account, log in to access the assessment. Your answers will be scored upon submission.

  • Assessment: Washington State Board for Community and Technical Colleges’ “Chapter 6 Quiz 6” Link: Washington State Board for Community and Technical Colleges’ “Chapter 6 Quiz 6 (PDF)
     
    Instructions: Complete this assessment after completing the readings, videos, and assignment for subunit 2.2.

2.3 Inverse Trig Functions   - Reading: Lippman and Rasmussen’s Precalculus: An Investigation of Functions: “Chapter 6: Periodic Functions” Link: Lippman and Rasmussen’s Precalculus: An Investigation of Functions: ** “Chapter 6: Periodic Functions (PDF)
 
Instructions: The functions give us some powerful tools for equation solving. Read pages 379–384 to begin to understand them, their graphs, and their relationship to the trig functions. This reading covers the topics outlined in subunits 2.3.1 through 2.3.3.

Terms of Use: The article above is released under a [Creative
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attributed to Lippman & Rasmussen.

2.3.1 Arcsine, Arccosine, and Arctangent Functions and Their Graphs   Note: The inverse trig functions give us a set of tools for solving trig equations. Focus on pages 379 and 380 of the reading listed under subunit 2.3 to begin to understand these new functions. Be sure to work through the examples and attempt the “Try It Now” problems. Solutions to the Try It Now problems are available at the end of the selected reading.

  • Web Media: YouTube: Khan Academy’s “Inverse Trig Functions: Arcsin” and “Inverse Trig Functions: Arctan” Links: YouTube: Khan Academy’s “Inverse Trig Functions: Arcsin and “Inverse Trig Functions: Arctan (YouTube)
     
    Instructions: Watch the first video to gain a deeper understanding of the Arcsin function. Then, watch the second video to gain a deeper understanding of the Arctan function.

    Watching both videos should take approximately 25 minutes.

    Terms of Use: These resources are licensed under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License. They are attributed to the Khan Academy.

  • Web Media: YouTube: maths247’s “Graphs of Inverse Trig Functions Arccos, Arc Sin and Arc Tan Core 3 Mathematics A level.MP4” Link: YouTube: maths247’s “Graphs of Inverse Trig Functions Arccos, Arc Sin and Arc Tan Core 3 Mathematics A level.MP4 (YouTube)
     
    Instructions: Use this resource to develop an understanding of the graphs of the inverse trig functions. Be sure to reflect on how they are both similar to and different from the original trig functions.

    Watching this video should take approximately 15 minutes. 
     
    Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

2.3.2 Evaluating Inverses with the Unit Circle or a Calculator   Note: Work through Example 1 through Example 6 on pages 380-383 of the reading listed under subunit 2.3 to begin evaluating inverse functions for specific values. 

  • Web Media: YouTube: bwolky’s “Section 6.3 #3” and “Section 6.3 #4” Link: YouTube: bwolky’s “Section 6.3 #3 and “Section 6.3 #4 (YouTube)

    Instructions: These optional resources may be useful for solving some of the problems in the assignment at the end of this section.  

    Watching both videos should take approximately 10 minutes.
     
    Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.   

2.3.3 Simplifying and Evaluating Expressions Involving the Inverse Trig Functions   Note: This topic is covered by the reading below subunit 2.3. In particular, please focus on page 384 for a discussion of using the inverse trig function to simplify expressions and solve equations. Be sure to work the examples, as always. 

  • Assessment: Washington State Board for Community and Technical Colleges’ MyOpenMath: “Section 6.3 Graded Homework” Link: Washington State Board for Community and Technical Colleges’ MyOpenMath: “Section 6.3 Graded Homework" (HTML)

    Instructions: Once you have completed the readings and viewed the material for all of subunit 2.3, complete this assessment. You will need to create a free account in order to access the material. If you have already created an account, log in to access the assessment. Your answers will be scored upon submission. 

2.4 Solving Trig Equations   - Reading: Lippman and Rasmussen’s Precalculus: An Investigation of Functions: “Chapter 6: Periodic Functions” Link: Lippman and Rasmussen’s Precalculus: An Investigation of Functions: ** “Chapter 6: Periodic Functions (PDF)
 
Instructions: Now that you have an understanding of the inverse trig functions and the domains and ranges of both the trig and inverse trig functions, you can begin solving more complicated equations. Read pages 387-394 to understand how. This reading covers the topics outlined in subunits 2.4.1 and 2.4.2. 

 Terms of Use: The article above is released under a [Creative
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2.4.1 Methods for Solving Trig Equations   2.4.1.1 Solving Using Known Values   Note: This topic is covered in the reading below subunit 2.4. Focus on pages 387-389 for a discussion of using known values to solve trig equations. Be sure to work through the examples on your own and work and check the “Try It Now” problems. 

  • Web Media: YouTube: MathTV’s “Trig: Solving Equations 1” Link: YouTube: MathTV’s “Trig: Solving Equations 1 (YouTube)
     
    Instructions: Watch the video to review what you learned in the readings and use this to re-work any problems that you had difficulty with.

    Watching this video should take approximately 8 minutes.
     
    Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

  • Web Media: YouTube: dcollis2’s “Solving Trigonometric Equations—Easy” Link: YouTube: dcollis2’s “Solving Trigonometric Equations—Easy (YouTube)
     
    Instructions: Watch the video to review what you learned in the readings and use this to re-work any problems that you had difficulty with.

    Watching this video should take approximately 6 minutes. 
     
    Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

2.4.1.2 Solving Using the Inverse Trig Functions   Note: The inverse trig functions give us new tools for equation solving. Read pages 390–393 of the reading under subunit 2.4 to see how this works. Also, complete the examples and the “Try It Now” problems. Solutions to the Try It Now problems appear at the end of the selected reading.

  • Web Media: YouTube: bwolky’s “Section 6.3 #7” and “Section 6.3 #8” Link: YouTube: bwolky’s “Section 6.3 #7 and “Section 6.3 #8 (YouTube)
     
    Instructions: These optional resources may be useful for solving some of the problems in the assignment at the end of this section.

    Watching these videos should take approximately 10 minutes.
     
    Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

2.4.2 Finding Solutions in One Cycle or Period vs. Finding All Possible Solutions   Note: Because trig functions are periodic, equations involving them often have multiple solutions. Learn how to find the required ones by reading page 394 of Chapter 6, listed under subunit 2.4. Work through any examples or problems. 

  • Web Media: YouTube: bwolky’s “Section 6.3 #9” Link: YouTube: bwolky’s “Section 6.3 #9” (YouTube)
     
    Instructions: This optional resource may be useful for solving some of the problems in the assignment at the end of this section.  

    Watching this video should take approximately 5 minutes.
     
    Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

  • Assessment: Washington State Board for Community and Technical Colleges’ MyOpenMath: “Section 6.4 Graded Homework” Link: Washington State Board for Community and Technical Colleges’ MyOpenMath: “Section 6.4 Graded Homework (HTML)

    Instructions: Once you have completed the readings and viewed the material for all of subunit 2.4, complete this assessment. You will need to create a free account in order to access the material. If you have already created an account, log in to access the assessment. Your answers will be scored upon submission

2.5 Modeling with Trigonometric Equations   - Reading: Lippman and Rasmussen’s Precalculus: An Investigation of Functions: “Chapter 6: Periodic Functions” Link: Lippman and Rasmussen’s Precalculus: An Investigation of Functions: ** “Chapter 6: Periodic Functions (PDF)
 
Instructions: Trigonometry is very useful for modeling real world data. Read the selection on pages 397–403 to develop some modeling techniques. Note that this reading covers the topics outlined in subunits 2.5.1 and 2.5.2. 

 Terms of Use: The article above is released under a [Creative
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2.5.1 Solving Right Triangles for Angles   Note: Right triangles provide many practical applications of the trig functions. Learn to use these by reviewing pages 397-398 of the reading listed under subunit 2.5. Work through the examples and problems in the reading. 

  • Web Media: YouTube: patrickJMT’s “Right Triangles and Trigonometric Functions” Link: YouTube: patrickJMT’s “Right Triangles and Trigonometric Functions (YouTube)

    Instructions: Watch the video to review applications of right triangle trig, and rework any problems that you had difficulty with.

    Watching this video should take approximately 10 minutes. 
     
    Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

  • Web Media: YouTube: 100tunafish’s “Trig Review: Two Sides.MP4” Link: YouTube: 100tunafish’s “Trig Review: Two Sides.MP4 (YouTube)
     
    Instructions: Watch the video to review applications of right triangle trig, and rework any problems that you had difficulty with.

    Watching this video should take approximately 7 minutes. 
     
    Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

2.5.2 Modeling with Sinusoidal Functions   Note: Trig functions can be used to model real world phenomena that repeat in a predictable manner. Read pages 399–403 of the reading listed under subunit 2.5 to see how this works. Work the example problems and solve the “Try It Now” problems. Solutions to the Try It Now problems appear at the end of the selected reading.

  • Web Media: YouTube: Khan Academy’s “Ferris Wheel Trig Problem” Link: YouTube: Khan Academy’s “Ferris Wheel Trig Problem (YouTube)
     
    Instructions: Before completing this entire video, attempt to solve the problem stated at the beginning of the video. Then, watch the video to check your solution or get assistance finding one.

    Watching this video should take approximately 10 minutes.

    Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License. It is attributed to the Khan Academy.

  • Web Media: YouTube: aprancicus’s “Ferris Wheel Trig Word Problem Type2.mp4” Link: YouTube: aprancicus’s “Ferris Wheel Trig Word Problem Type2.mp4 (YouTube)

    Instructions: Before completing this entire video, attempt to solve the problem stated at the beginning of the video. Then, watch this video to check your solution.

    Watching this video should take approximately 5 minutes.
     
    Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

  • Assessment: Washington State Board for Community and Technical Colleges’ MyOpenMath: “Section 6.5 Graded Homework” Link: Washington State Board for Community and Technical Colleges’ MyOpenMath: “Section 6.5 Graded Homework (HTML)

    Instructions: Once you have completed the readings and viewed the material for all of subunit 2.5, complete this assessment. You will need to create a free account in order to access the material. If you have already created an account, log in to access the assesssment. Your answers will be scored upon submission.  

  • Assessment: Washington State Board of Community and Technical Colleges’ “Chapter 6 Quiz 7” Link: Washington State Board of Community and Technical Colleges’ “Chapter 6 Quiz 7 (PDF)
     
    Instructions: This assessment covers subunits 2.3–2.5. Be sure to complete all of those before completing the quiz.

Unit 2 Review   - Assessment: Washington State Board for Community and Technical Colleges’ MyOpenMath: “Chapter 6 Review” Link: Washington State Board for Community and Technical Colleges’ MyOpenMath: “Chapter 6 Review (HTML)

 Instructions: Once you have completed the readings and viewed the
material for all of unit 1, complete this assessment. You will need
to create a free account in order to access the material. If you
have already created an account, log in to access the assessment.
Your answers will be scored upon submission. This assessment will
provide a review for your unit assessment.

Unit 2 Assessment   - Assessment: Washington State Board for Community and Technical Colleges’ “Chapter 6 Exam” Link: Washington State Board for Community and Technical Colleges’ “Chapter 6 Exam (PDF)
                                            
Instructions: Complete this assessment after you have completed all of the readings and assignments for Unit 2, including the review assignment.