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MA003: Precalculus II

Unit 3: Trigonometric Equations and Identities   The motion of a mass on a spring is a typical example of simple harmonic motion and can be described using sine waves and trig functions. The amplitude, period, and midline of the functions that describe the motion all give insight into the behavior of the moving object and allow for predictions about its behavior.

This unit builds on the equation-solving techniques we developed in the previous unit by expanding our repertoire of useful identities and equation solving techniques. You will continue to study the properties of amplitude, period, and midline, and you will apply these to harmonic motion and mathematical modeling. 

Unit 3 Time Advisory
This unit should take you approximately 16 hours to complete.

☐    Subunit 3.1: 3.5 hours

☐    Subunit 3.2: 4 hours

☐    Readings: 2.5 hours

☐    Assignment: 1.5 hours

☐    Subunit 3.3: 3.5 hours

☐    Subunit 3.4: 2.5 hours

☐    Unit 3 Assignment (Review): 1.5 hours

☐    Unit 3 Assessment: 1 hour

Unit3 Learning Outcomes
Upon successful completion of this unit, the student will be able to:

  • solve applied problems involving harmonic motion including problems with changing amplitude;
  • apply the addition and subtraction formulas, double angle formulas, half-angle formulas, and algebraic techniques to solve equations, simplify expressions, and prove identities; and
  • solve trigonometric equations for all solutions, providing exact solutions when appropriate, and using technology to find approximate solutions when appropriate.

3.1 Solving Trigonometric Equations with Identities   - Reading: Lippman and Rasmussen’s Precalculus: An Investigation of Functions: “Chapter 7: Trigonometric Equations and Identities” Link: Lippman and Rasmussen’s Precalculus: An Investigation of Functions: “Chapter 7: Trigonometric Equations and Identities (PDF)
 
Instructions: Read pages 409–415 to learn some additional techniques for solving trig equations. This reading covers the topics outlined in subunits 3.1.1 and 3.1.2 
 
Terms of Use: The article above is released under a Creative Commons Attribution-Share-Alike License 3.0 (HTML). It is attributed to Lippman & Rasmussen.

3.1.1 Review of Pythagorean, Negative Angle, and Reciprocal Identities   Note: This topic is covered by the reading below subunit 3.1. Focus on page 410 and work all relevant examples and ‘Try It Now” problems to review some basic identities. Solutions to the Try It Now problems appear at the end of each selected reading.

3.1.2 Solving Equations by Factoring, Using the Quadratic Formula, and Using Identities to Simplify   Note: This topic is covered by the reading below subunit 3.1. We can apply some of our algebraic equation solving techniques to trigonometric equations in conjunction with our identities to solve more complicated problems. Focus on pages 410–415 of the Chapter 7 reading for examples of this, working through the examples and ‘Try It Now” problems. Solutions to the Try It Now problems are located at the end of each selected reading.

  • Web Media: YouTube: MathTV’s “Trig: Solving Equations 1” Link: YouTube: MathTV’s “Trig: Solving Equations 1 (YouTube)
     
    Instructions: Carefully watch the video. Be sure to work through any examples. Use what you learned to redo any problems from the readings that were difficult to complete.

    Watching the 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: patrickJMT’s “Solving a Trigonometric Equation by Factoring, Example 1” Link: YouTube: patrickJMT’s “Solving a Trigonometric Equation by Factoring, Example 1 (YouTube)
     
    Instructions: Carefully watch the video. Be sure to work through any examples. Use what you learned to redo any problems from the readings that were difficult to complete.

    Watching the 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: ExamSolutions’ “Trigonometry: Solving Equations Using Identities (Example 2)” Link: YouTube: ExamSolutions’ “Trigonometry: Solving Equations Using Identities (Example 2)” (YouTube)
     
    Instructions: Carefully watch the video. Be sure to work through any examples. Use what you learned to redo any problems from the readings that were difficult to complete.

    Watching this video should take approximately 15 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 7.1 Graded Homework” Link: Washington State Board for Community and Technical Colleges’ MyOpenMath: “Section 7.1 Graded Homework (HTML)

    Instructions: Complete this assessment after you have worked through all the readings, videos, and problems for subunit 3.1. 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 7 Quiz 8” Link: Washington State Board for Community and Technical Colleges’  “Chapter 7 Quiz 8 (PDF)
     
    Instructions: Complete this quiz after you have completed the videos and readings for subunit 3.1. 

3.2 Addition and Subtraction Identities   - Reading: Lippman and Rasmussen’s Precalculus: An Investigation of Functions: “Chapter 7: Trigonometric Equations and Identities” Link: Lippman and Rasmussen’s Precalculus: An Investigation of Functions: “Chapter 7: Trigonometric Equations and Identities (PDF)
 
Instructions: Add some additional identities to your problem solving arsenal by reading pages 417-430. This selection also covers the topics outlined in subunits 3.2.1–3.2.3.
 
Terms of Use: The article above is released under a Creative Commons Attribution-Share-Alike License 3.0 (HTML). It is attributed to Lippman & Rasmussen.

3.2.1 The Sum and Difference Identities   Note: The first new set of identities is covered on pages 417–419 of the Chapter 7 reading, listed below subunit 3.2. Read through these, solving the examples and ‘Try It Now” problems. Solutions to the Try It Now exercises are located at the end of each selected reading.

  • Web Media: YouTube: patrickJMT’s “Using the Sum and Difference Identities for Sine, Cosine, and Tangent, Ex 1” Link: YouTube: patrickJMT’s “Using the Sum and Difference Identities for Sine, Cosine, and Tangent, Ex 1 (YouTube)
     
    Instructions: Watch this entire video to learn about how to use sum and difference identities for sine, cosine, and tangent.

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

3.2.2 Combining Waves with Equal Period   Note: This topic is covered by the reading below subunit 3.2. In particular, pay attention to the selection on pages 419–423, which introduce a set of rules for combining and simplifying trig functions. Work through the examples and “Try It Now” problems. Solutions to the Try It Now exercises may be found at the end of each selected reading.

3.2.3 Product to Sum and Sum to Product Identities   Note: This topic is covered by the reading assigned below subunit 3.2. Converting from products of trig functions to sums, and vice versa, can be useful in solving some equations. The reading on pages 423–427 covers how to do this. Carefully read the examples and complete the ‘Try It Now” exercises. Solutions to the Try It Now exercises may be found at the end of each selected reading.

  • Web Media: YouTube: bullcleo’s “Sum to Product and Product to Sum Identities” Link: YouTube: bullcleo’s “Sum to Product and Product to Sum Identities (YouTube)

    Instructions: Carefully watch the video. Be sure to attempt any examples on your own, and use what you learn to re-do any problems that you found difficult in the reading.

    Watching the 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 7.2 Graded Homework” Link: Washington State Board for Community and Technical Colleges’ MyOpenMath: “Section 7.2 Graded Homework (HTML)

    Instructions: Complete this assignment when you have worked through all the readings, videos, and problems for subunit 3.2. 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 7 Quiz 9” Link: Washington State Board for Community and Technical Colleges’ “Chapter 7 Quiz 9 (PDF)
     
    Instructions: Complete this quiz after working through the material in subunit 3.2.

3.3 Double Angle Identities   - Reading: Lippman and Rasmussen’s Precalculus: An Investigation of Functions: “Chapter 7: Trigonometric Equations and Identities” Link: Lippman and Rasmussen’s Precalculus: An Investigation of Functions: “Chapter 7: Trigonometric Equations and Identities (PDF)
 
Instructions: Read pages 431–441 to learn about simplifying trig expressions and solving trig equations involving double angles. This reading also covers the topics outlined in subunits 3.3.1 and 3.3.2.
 
Terms of Use: The article above is released under a Creative Commons Attribution-Share-Alike License 3.0 (HTML). It is attributed to Lippman & Rasmussen.

3.3.1 Using the Double Angle Identities   Note: This topic is covered by the reading assigned below subunit 3.3. Focus on pages 431-436 and work through all problems in the text to begin to grasp the double angle identities.  

  • Web Media: YouTube: davidmpeabody’s “RHS Pre Calculus-Double Angle Identities” Link: YouTube: davidmpeabody’s “RHS Pre Calculus - Double Angle Identities (YouTube)
     
    Instructions: Carefully watch the video. Be sure to attempt any examples on your own, and use what you learn to redo any problems in the reading that you found difficult.

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

3.3.2 Power Reduction and Half Angle Identities   Note: This topic is covered by the reading assigned below subunit 3.3. Focus on pages 436-441 to learn to solve equations involving half angles and powers of trig functions. As usual, work through the examples carefully.

  • Web Media: YouTube: patrickJMT’s “Power Reducing Formulas for Sine and Cosine, Example 1” Link: YouTube: patrickJMT’s “Power Reducing Formulas for Sine and Cosine, Example 1 (YouTube)

    Instructions: Carefully watch the video. Be sure to attempt any examples on your own, and use what you learn to redo any problems that you found difficult in the reading.

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

  • Web Media: YouTube: patrickJMT’s “Half Angle Identities to Evaluate Trigonometric Expressions, Example 3” Link: YouTube: patrickJMT’s “Half Angle Identities to Evaluate Trigonometric Expressions, Example 3 (YouTube)
     
    Instructions: Carefully watch the video. Be sure to attempt any examples on your own, and use what you learn to redo any problems that you found difficult 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. 

  • Assessment: Washington State Board for Community and Technical Colleges’ MyOpenMath: “Section 7.3 Graded Homework” Link: Washington State Board for Community and Technical Colleges’ MyOpenMath: “Section 7.3 Graded Homework (HTML)

    Instructions: Complete this assignment after you have worked through all the readings, videos, and problems for subunit 3.3. 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.  

3.4 Modeling Changing Amplitude and Midline   - Reading: Lippman and Rasmussen’s Precalculus: An Investigation of Functions: “Chapter 7: Trigonometric Equations and Identities” Link: Lippman and Rasmussen’s Precalculus: An Investigation of Functions: “Chapter 7: Trigonometric Equations and Identities (PDF)
 
Instructions: Because real world phenomena are often modeled with trig functions, it is important to understand how changes to the functions affect the resulting graphs and the phenomena being modeled. To increase your understanding of this, read pages 442–448 of Chapter 7. This selection also covers the topics outlined in subunits 3.4.1 through 3.4.3. 
 
Terms of Use: The article above is released under a Creative Commons Attribution-Share-Alike License 3.0 (HTML). It is attributed to Lippman & Rasmussen.

3.4.1 Changing Midlines   Note: This topic is covered by the reading assigned below subunit 3.4. The selection on pages 442–444 covers changing the midline of a trig function. Work through the examples and “Try It Now” problems to master the techniques.  

3.4.2 Changing Amplitude   Note: This topic is covered by the reading assigned below subunit 3.4. Focus on page 445, which covers changing the amplitude of a trig function. Work through the examples and “Try It Now” problems to master the techniques.  

3.4.3 Damped Harmonic Motion   Note: This topic is covered by the reading assigned below subunit 3.4. Focus on pages 445–447, which covers trig functions with increasing or decreasing amplitudes. Work through the examples and “Try It Now” problems.  

  • Assessment: Washington State Board for Community and Technical Colleges’ MyOpenMath: “Section 7.4 Graded Homework” Link: Washington State Board for Community and Technical Colleges’ MyOpenMath: “Section 7.4 Graded Homework (HTML)

    Instructions: Complete this assignment after you have worked through all the readings, videos, and problems for subunit 3.4. 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 7 Quiz 10” Link: Washington State Board for Community and Technical Colleges’ “Chapter 7 Quiz 10 (PDF)
     
    Instructions: This assessment covers subunits 3.3 and 3.4; be sure to complete the material in both subunits before attempting this quiz.

Unit 3 Review   - Assessment: Washington State Board for Community and Technical Colleges’ MyOpenMath: “Chapter 7 Review” Link: Washington State Board for Community and Technical Colleges’ MyOpenMath: “Chapter 7 Review (HTML)

 Instructions: Complete this assessment after you have worked
through all the readings, videos, and problems for Unit 7. 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.
  • Assessment: Washington State Board for Community and Technical Colleges’ MyOpenMath: “Chapter 7 Review Identities” Link: Washington State Board for Community and Technical Colleges’ MyOpenMath: “Chapter 7 Review Identities (HTML)

    Instructions: Note that this review is optional. Complete this assessment after you have worked through all the readings, videos, and problems for unit 7. 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.

Unit 3 Assessment   - Assessment: Washington State Board for Community and Technical Colleges’ “Chapter 7 Exam B” Link: Washington State Board for Community and Technical Colleges’ “Chapter 7 Exam B (PDF)

 Instructions: This assessment covers all of Unit 3; be sure to
complete all of the material in each subunit before attempting this
practice exam.