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MA003: Precalculus II

Unit 4: Further Applications of Trigonometry   Imagine a boat that leaves port, travels some distance, turns some measurable amount, and then continues on. How would you measure the distance between the boat and the port? The techniques you have developed in previous units for such calculations require the existence of a right angle to work. This unit introduces several new tools for modeling and analyzing the world around us, including an extension of right triangle trigonometry to non-right triangles, which is what you will need to solve the problem above.

In addition, you will learn about using the polar coordinate system and parametric equations as alternatives to the Cartesian coordinate system, vectors, and complex numbers.  

Unit 4 Time Advisory
This unit should take you approximately 25 hours to complete.

☐    Subunit 4.1: 5 hours

☐    Readings and Web Media: 2.5 hours

☐    Assignments: 2 hours

☐    Assessment: 0.5 hours

☐    Subunit 4.2: 4.5 hours ☐    Readings and Web Media: 2 hours

☐    Assignment: 2 hours

☐    Assessment: 0.5 hours

☐    Subunit 4.3: 4 hours

☐    Readings and Web Media: 2.5 hours

☐    Assignment: 1.5 hours

☐    Subunit 4.4: 3.5 hours

☐    Subunit 4.5: 3.5 hours 

☐    Unit 4 Assignment (Review): 2 hours

☐    Unit 4 Assignment 2 (Review Worksheet): 1.5 hours

☐    Unit 4 Assessment: 1 hour

Unit4 Learning Outcomes
Upon successful completion of this unit, the student will be able to: 

  • apply the law of sines and the law of cosines to solve for unknown sides and angles of triangles and solve applied problems associated with these laws;
  • convert polar coordinates into rectangular coordinates and vice versa;
  • convert polar equations into rectangular equations and vice versa;
  • sketch the graphs of polar equations by hand and by calculator;
  • evaluate powers and roots of complex numbers using the polar form of complex numbers (DeMoivre’s Theorem or Euler’s formula);
  • perform basic operations with vectors graphically and component-wise;
  • solve vector application problems such as navigation and resultant forces;
  • given a set of parametric equations, convert them into a Cartesian equation and determine a parameterization of a Cartesian equation including identifying the bounds; and
  • sketch the graphs of parametric equations by hand and by calculator.

4.1 Non-right Triangles: Law of Sines and Cosines   - Reading: Lippman and Rasmussen’s Precalculus: An Investigation of Functions: “Chapter 8: Further Applications of Trigonometry” Link: Lippman and Rasmussen’s Precalculus: An Investigation of Functions: “Chapter 8: Further Applications of Trigonometry (PDF)
 
Instructions: Pages 451–466 introduce the idea of using trigonometric functions in triangles other than right triangles. Read this selection carefully. This selection also covers the topics outlined in subunits 4.1.1 and 4.1.2.
 
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.

4.1.1 The Law of Sines   Note: This topic is covered by the reading assigned below subunit 4.1. Focus on pages 451–455 of the selection to learn about the Law of Sines. Take special note of the ambiguous case and be sure to understand when you can and cannot apply the law of sines. Read carefully, working through the examples and the “Try It Now” problems.  

  • Web Media: YouTube: Mindbites.com’s “Pre-Calculus: The Law of Sines” Link: YouTube: Mindbites.com’s “Pre-Calculus: The Law of Sines (YouTube)

    Instructions: Watch the video and try to rework any problems that you had difficulty with 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: rlp85hotmail’s “The Law of Sines, Part 1” Link: YouTube: rlp85hotmail’s “The Law of Sines, Part 1 (YouTube)
     
    Instructions: Watch the video and try to rework any problems that you had difficulty with in the reading.

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

4.1.2 The Law of Cosines   Note: This topic is covered by the reading assigned below subunit 4.1. Focus on pages 455–461 of the selection to develop your understanding of the law of cosines, noting when you use it instead of the law of sines. Work through the examples and the “Try It Now” problems. 

  • Web Media: YouTube: AlRichards314’s “The Cosine Law” Link: YouTube: AlRichards314’s “The Cosine Law (YouTube)
     
    Instructions: Watch the video, paying close attention to any example problems, and rework any of the problems from the reading that were difficult for you.

    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: TheMathClips’ “The Cosine Law (Finding an Unknown Angle)” Link: YouTube: TheMathClips’ “The Cosine Law (Finding an Unknown Angle)” (YouTube)
     
    Instructions: Watch the video, paying close attention to any example problems, and rework any of the problems from the reading that were difficult for you.

    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: calculatorbox’s “Law of Cosines” Link: YouTube: calculatorbox’s “Law of Cosines (YouTube)

    Instructions: Watch the video, paying close attention to any example problems, and rework any of the problems from the reading that were difficult for you.

    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: The Law of Cosines” Link: YouTube: MathTV’s “Trigonometry: The Law of Cosines (YouTube)
     
    Instructions: Watch the video, paying close attention to any example problems, and rework any of the problems from the reading that were difficult for you.

    Watching this video should take approximately 7 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 8.1 Non-Right Triangles: Law of Sines and Cosines Graded Homework” Link: Washington State Board for Community and Technical Colleges’ MyOpenMath: “Section 8.1 Non-Right Triangles: Law of Sines and Cosines Graded Homework (HTML)

    Instructions: Complete this assessment after you have worked through all the readings, videos, and problems for subunit 4.1. 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.

  • Assessment: Washington State Board for Community and Technical Colleges’ MyOpenMath: “Section 8.1 Applications Non-Right Triangles: Law of Sines and Cosines Graded Homework” Link: Washington State Board for Community and Technical Colleges’ MyOpenMath: “Section 8.1 Applications Non-Right Triangles: Law of Sines and Cosines Graded Homework (HTML)

    Instructions: Complete this assessment after you have worked through all the readings, videos, and problems for subunit 4.1. 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.

  • Assessment: Washington State Board for Community and Technical Colleges’ “Chapter 8 Quiz 11” Link: Washington State Board for Community and Technical Colleges’ “Chapter 8 Quiz 11 (PDF)
     
    Instructions: Take this unit quiz after you have completed all of the assigned material for subunit 4.1.

4.2 Polar Coordinates   - Reading: Lippman and Rasmussen’s Precalculus: An Investigation of Functions: “Chapter 8: Further Applications of Trigonometry” Link: Lippman and Rasmussen’s Precalculus: An Investigation of Functions: “Chapter 8: Further Applications of Trigonometry (PDF)
 
Instructions: Read the selection from pages 467–475. The selection defines a new system for graphing points and curves based on distances and angles rather than the horizontal and vertical distances used in the Cartesian Coordinate system. This reading covers the topics outlined in subunits 4.2.1 through 4.2.3.
 
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.

4.2.1 Defining Polar Coordinates   Note: This topic is covered by the reading assigned below subunit 4.2. Focus on pages 467-468 for the definition and introduction of the polar coordinate system. Read carefully, working through the examples and the ‘Try It Now” problems on your own.  

  • Web Media: patrickJMT’s “Polar Coordinates: Basic Graphing” Link: YouTube: patrickJMT’s “Polar Coordinates: Basic Graphing (YouTube)
     
    Instructions: Watch the entire video and rework any problems that you may have encountered in the reading section. 

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

4.2.2 Converting between Polar and Cartesian Coordinates   Note: This topic is covered by the reading assigned below subunit 4.2. This selection teaches the conversion back and forth between Cartesian and Polar Coordinate systems. Read pages 468 and 469 carefully, working through all problems.  

  • Web Media: YouTube: patrickJMT’s “Converting between Polar and Rectangular Cartesian) Coordinates, Ex 3” Link: YouTube: patrickJMT’s “Converting between Polar and Rectangular Cartesian Coordinates, Ex 3 (YouTube)
     
    Instructions: Watch the entire video. If you had any difficulty with any of the problems in the reading selection, retry them after viewing this video.

    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: TheIntegralCALC’s “Converting Polar Coordinates Example 1” Link: YouTube: TheIntegralCALC’s “Converting Polar Coordinates Example 1 (YouTube)
     
    Instructions: Watch the entire video. If you had any difficulty with the problems in the reading selection, retry them after viewing this video.

    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: S22105’s “Convert Rectangular Coordinates to Polar Form” Link: YouTube: S22105’s “Convert Rectangular Coordinates to Polar Form (YouTube)
     
    Instructions: Watch the entire video. If you had any difficulty with the problems in the reading selection, retry them after viewing this video.

    Watching this video should take approximately 7 minutes.

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

  • Web Media: YouTube: Brightstorm’s “Precalculus: Converting from Polar Coordinates to Rectangular” Link: YouTube: Brightstorm’s “Precalculus: Converting from Polar Coordinates to Rectangular (YouTube)
     
    Instructions: Watch the entire video. If you had any difficulty with the problems in the reading selection, retry them after viewing this video.

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

4.2.3 Polar Equations   Note: Now that you have developed an understanding of the polar coordinate system, you can begin to use it to graph and solve equations in polar forms. This topic is covered by the reading assigned below subunit 4.2. Focus on pages 470-475, which covers techniques for graphing and solving polar equations.  

  • Web Media: YouTube: patrickJMT’s “Converting between Polar and Rectangular Equations, Ex 1,” “Converting between Polar and Rectangular Equations, Ex 2,” and “Graphing Special Polar Equations; How Many Petals Will a Graph” Links: YouTube: patrickJMT’s “Converting between Polar and Rectangular Equations, Ex 1,” “Converting between Polar and Rectangular Equations, Ex 2,” and “Graphing Special Polar Equations; How Many Petals Will a Graph (YouTube)

    Instructions: Watch these videos. After viewing each resource, go back and rework any examples or ‘Try It Now” problems from the reading that were difficult.
     
    Terms of Use: Please respect the copyright and terms of use displayed on the webpage above. 

  • Web Media: YouTube: Khan Academy’s “Polar Coordinates 3” Link: YouTube: Khan Academy’s “Polar Coordinates 3 (YouTube)
     
    Instructions: Watch the entire video, being sure to go back and work any examples or ‘Try It Now” problems from the reading that were difficult. 

    Watching this video should take approximately 8 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.

  • Assessment: Washington State Board for Community and Technical Colleges’ MyOpenMath: “Section 8.2 Polar Coordinates Graded Homework” Link: Washington State Board for Community and Technical Colleges’ MyOpenMath: “Section 8.2 Polar Coordinates Graded Homework (HTML)

    Instructions: Complete this assessment after you have worked through all the readings, videos, and problems for subunit 4.2. 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.

  • Assessment: Washington State Board for Community and Technical Colleges’ “Chapter 8: Quiz 12” Link: Washington State Board for Community and Technical Colleges’ “Chapter 8: Quiz 12 (PDF)
     
    Instructions: Complete this assessment after you have worked through the readings, videos, and assignments for subunit 4.2.

4.3 Polar Form of Complex Numbers   - Reading: Lippman and Rasmussen’s Precalculus: An Investigation of Functions: “Chapter 8: Further Applications of Trigonometry” Link: Lippman and Rasmussen’s Precalculus: An Investigation of Functions: “Chapter 8: Further Applications of Trigonometry (PDF)
 
Instructions: Read pages 480–490 of Chapter 8 to learn how polar coordinates and complex numbers are related. This selection also covers the topics outlined in subunits 4.3.1 through 4.3.4.
 
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.

4.3.1 Imaginary Number i, Complex Numbers, and the Complex Plane   Note: This topic is covered by the reading assigned below subunit 4.3. Focus on pages 480 and 481 for a definition of complex and imaginary numbers. Work through the examples and the “Try It Now” problems. Solutions to the Try It Now exercises appear at the end of each selected reading.

  • Web Media: YouTube: derekownes’ “Complex Numbers, Part 3- The Complex Plane” Link: YouTube: derekownes’ “Complex Numbers, Part 3-The Complex Plane (YouTube)
     
    Instructions: Watch the video, and then review any difficult problems from the reading selection.

    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: videosbyjulieharland’s “Complex Numbers 1” Link: YouTube: videosbyjulieharland’s “Complex Numbers 1 (YouTube)
     
    Instructions: Watch the video, and then review any difficult problems from the reading selection.

    Watching this video should take approximately 11 minutes.
     
    Terms of Use: Please respect the copyright and terms of use displayed on the webpage above. 

  • Web Media: YouTube: videosbyjulieharland’s “Complex Numbers 2” Link: YouTube: videosbyjulieharland’s “Complex Numbers 2 (YouTube)
     
    Instructions: Watch the video, and then review any difficult problems from the reading selection.

    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: yourteachermathhelp’s “Imaginary Numbers: Algebra 2 Help” Link: YouTube: yourteachermathhelp’s “Imaginary Numbers: Algebra 2 Help (YouTube)
     
    Instructions: Watch this brief video, and then review any difficult problems from the reading selection.

    Watching this video should take approximately 1 minute.
     
    Terms of Use: Please respect the copyright and terms of use displayed on the webpage above. 

4.3.2 Arithmetic on Complex Numbers   Note: This topic is covered by the reading assigned below subunit 4.3. Focus on pages 481–483, which cover addition, subtraction, multiplication, and division of the complex numbers. Make sure to work through any example problems.  

  • Web Media: YouTube: patrickJMT’s “Adding and Subtracting Complex (Imaginary) Numbers” Link: YouTube: patrickJMT’s “Adding and Subtracting Complex (Imaginary) Numbers (YouTube)
     
    Instructions: Watch the video, paying close attention to the examples, and then rework any problems that you had trouble with in the reading selection.

    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: FerranteMath’s “Imaginary Numbers: (Multiplying) (Example 1)” Link: YouTube: FerranteMath’s “Imaginary Numbers: (Multiplying) (Example 1)” (YouTube)
     
    Instructions: Watch the video, paying attention to the examples, and then rework any problems that you had trouble with in the reading selection.

    Watching this video should take approximately 2 minutes.
     
    Terms of Use: Please respect the copyright and terms of use displayed on the webpage above. 

  • Web Media: YouTube: patrickJMT’s “Complex Numbers: Dividing- Ex 2” and “Complex Numbers: Multiplying- Ex 1” Link: YouTube: patrickJMT’s “Complex Numbers: Dividing- Ex 2 and “Complex Numbers: Multiplying- Ex 1 (YouTube)
     
    Instructions: Watch the first video example on dividing complex numbers. Then, watch the video example on multiplying complex numbers. Pay attention to the examples provided in each video, and then rework any problems that you had trouble with in the reading selection.

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

4.3.3 Polar Form of Complex Numbers and Euler's Formula   Note: This topic is covered by the reading assigned below subunit 4.3. Learn how to write complex numbers in polar form by reading pages 483–486 carefully and working through the problems.  

  • Web Media: YouTube: waszel’s “The Polar Form of Complex Numbers” Link: YouTube: waszel’s “The Polar Form of Complex Numbers (YouTube)
     
    Instructions: After viewing and working through the examples in the video, try to rework any difficult problems from the reading selection. 

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

4.3.4 DeMoivre's Theorem   Note: This topic is covered by the reading assigned below subunit 4.3. Focus on pages 486–488, and work through the examples and “Try It Now” problems carefully. Solutions to the Try It Now exercises appear at the end of each reading selection.

  • Web Media: YouTube: MathDoctorBob’s “Example of De Moivre's Theorem” Link: YouTube: MathDoctorBob’s “Example of De Moivre's Theorem (YouTube)
     
    Instructions: Watch the video, paying attention to the examples, and then rework any problems that you had trouble with in the reading selection.

    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 8.3 Polar Form of Complex Numbers Graded Homework” Link: Washington State Board for Community and Technical Colleges’ MyOpenMath: “Section 8.3 Polar Form of Complex Numbers Graded Homework(HTML)

    Instructions: Complete this assignment after you have worked through all the readings, videos, and problems for subunit 4.3. 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.

  • Assessment: Washington State Board for Community and Technical Colleges’ “Chapter 8: Quiz 13” Link: Washington State Board for Community and Technical Colleges’ “Chapter 8: Quiz 13 (PDF)
     
    Instructions: Complete this quiz after you have finished the assignments, videos, and readings from subunit 4.3.

4.4 Vectors   - Reading: Lippman and Rasmussen’s Precalculus: An Investigation of Functions: “Chapter 8: Further Applications of Trigonometry” Link: Lippman and Rasmussen’s Precalculus: An Investigation of Functions: “Chapter 8: Further Applications of Trigonometry (PDF)
 
Instructions: Vectors are geometric objects with both distance and direction, and they have numerous applications. Read pages 491-502 from Chapter 8 carefully to understand these applications. This reading selection also covers the topics outlined in subunits 4.4.1 through 4.4.3. 
 
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.

4.4.1 A Geometric View of Vectors   Note: This topic is covered by the reading assigned below subunit 4.4. In particular, read page 491 for a geometric view of vectors, working through the examples as always.  

4.4.1.1 Adding Vectors Geometrically   Note: Page 492 of the reading from subunit 4.4 describes vector addition in a geometric context. Read carefully and work through the examples.  

4.4.1.2 Scaling Vectors Geometrically   Note: Page 493 of the reading from subunit 4.4 describes vector scaling in a geometric context. Read carefully and work through the examples.  

4.4.2 Component Form of Vectors   4.4.2.1 Magnitude and Direction of a Vector   Note: The geometric properties of vectors can be described and computed algebraically using component form. Read page 494 of the subunit 4.4 reading selection for an introduction to these ideas. 

  • Web Media: YouTube: Mindbites.com’s “Pre-Calculus: Finding Vector Magnitude & Direction” Link: YouTube: Mindbites.com’s “Pre-Calculus: Finding Vector Magnitude & Direction (YouTube)
     
    Instructions: Watch this video carefully, and use what you learn to rework any problems from the reading that you had difficulty solving.

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

4.4.2.2 Using Component Form   Note: The geometric properties of vectors can be described and computed algebraically using component form. Read page 494 of the subunit 4.4 reading selection for an introduction to these ideas. 

  • Web Media: YouTube: patrickJMT’s “Finding the Components of a Vector, Ex 1” Link: YouTube: patrickJMT’s “Finding the Components of a Vector, Ex 1 (YouTube)
     
    Instructions: Watch this video carefully, and use what you learn to rework any problems from the reading that you had difficulty solving.

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

4.4.2.3 Alternate Notation for Component Form   Note: Read pages 494-496 of the reading listed under subunit 4.4 to see an alternated notation for component form using unit vectors. 

4.4.3 Adding and Scaling Vectors in Component Forms   Note: Now that you have established the algebraic definitions for vectors, the selection on pages 496–499 of the reading under subunit 4.4 will teach you how to combine vectors using their components. Read carefully and work through the problems and examples.  

  • Web Media: YouTube: patrickJMT’s “Vector Basics: Drawing Vectors/Vector Addition” Link: YouTube: patrickJMT’s “Vector Basics: Drawing Vectors/Vector Addition (YouTube)
               
    Instructions: Watch the entire video. Work through any examples on your own, and be sure to rework the difficult exercises from the Chapter 8 reading. 

    Watching this video should take approximately 12 minutes.
     
    Terms of Use: Please respect the copyright and terms of use displayed on the webpage above. 

  • Web Media: YouTube: dlippmanmath’s “Equations of Planes: Parallel and Containing a Line” Link: YouTube: dlippmanmath’s “Equations of Planes: Parallel and Containing a Line” (YouTube)

    Instructions: Watch this video and work through the examples discussed within. Then use what you have learned to re-do any problems that you had difficulty with before.

    Watching this video should take approximately 15 minutes.

    Terms of Use: This resource is licensed under a Creative Commons Attribution 3.0 Unported License. It is attributed to dlippmanmath, and the original version can be found here.

  • Web Media: YouTube: dlippmanmath’s “Vector and Scalar Equations of a Plane” Link: YouTube: dlippmanmath’s “Vector and Scalar Equations of a Plane” (YouTube)

    Instructions: Watch this video and work through the examples discussed within. Then use what you have learned to re-do any problems that you had difficulty with before.

    Watching this video should take approximately 15 minutes.

    Terms of Use: This resource is licensed under a Creative Commons Attribution 3.0 Unported License. It is attributed to dlippmanmath, and the original version can be found here.

  • Web Media: YouTube: eHow’s “Math in Daily Life: Adding Vectors” Link: YouTube: eHow’s “Math in Daily Life: Adding Vectors (YouTube)
     
    Instructions: Watch the entire video. Work through any examples on your own, and be sure to rework the difficult exercises from the Chapter 8 reading.

    Watching this video should take approximately 2 minutes.
     
    Terms of Use: Please respect the copyright and terms of use displayed on the webpage above. 

  • Web Media: YouTube: GeometryMsK’s “Vectors Application Word Problem” Link: YouTube: GeometryMsK’s “Vectors Application Word Problem (YouTube)

    Instructions: Watch the video and work through any examples on your own, be sure to rework the difficult exercises from the Chapter 8 reading.

    Watching this video should take approximately 9 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 8.4 Vectors Graded Homework” Link: Washington State Board for Community and Technical Colleges’ MyOpenMath: “Section 8.4 Vectors Graded Homework (HTML)

    Instructions: Complete this assessment after you have worked through all the readings, videos, and problems for subunit 4.5. 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. 

4.5 Parametric Equations   - Reading: Lippman and Rasmussen’s Precalculus: An Investigation of Functions: “Chapter 8: Further Applications of Trigonometry” Link: Lippman and Rasmussen’s Precalculus: An Investigation of Functions: “Chapter 8: Further Applications of Trigonometry (PDF)
 
Instructions: Up until this point in the course, we have been defining functions in terms of two variables: a dependent and an independent variable. Parametric equations give us a new way to define functions, determining the coordinates of a point based on functions of a third variable, often time. Read pages 504–512 to learn about these concepts. This reading also covers the topics outlined for subunits 4.5.1 through 4.5.3.
 
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.

4.5.1 Defining Parametric Equations   Note: This topic is covered by the reading assigned below subunit 4.5. This section is an introduction to the basic concepts of parametric equations. Read pages 504–507 carefully, working all examples and problems.   

  • Web Media: YouTube: sgordo3398’s “DSCF2359” Link: YouTube: sgordo3398’s “DSCF2359 (YouTube)
     
    Instructions: This optional video will assist you with some of the problems in the assignment at the end of subunit 4.5.
     
    Watching this video should take approximately 2 minutes.

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

4.5.2 Converting from Parametric to Cartesian Coordinates   Note: This topic is covered by the reading assigned below subunit 4.5. Now that you have an idea what parametric equations are, you can explore how to change between parametric and Cartesian coordinates. Focus on pages 507–509 of the reading selection, and work through the exercises and sample problems.  

  • Web Media: YouTube: math2bprof’s “Parametric Curves Part 1” Link: YouTube: math2bprof’s “Parametric Curves Part 1 (YouTube)
     
    Instructions: Watch the entire video. Be sure you understand the examples discussed, and use what you learn to re-work any examples in the reading that you found difficult. 

    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: maths247’s “(ii) Parametric Equations Core 4 Parametric to Cartesian Basic Examples” Link: YouTube: maths247’s “(ii) Parametric Equations Core 4 Parametric to Cartesian Basic Examples (YouTube)
     
    Instructions: Watch the video and make sure you understand the examples discussed; use what you learn to re-work any examples in the reading that you found difficult. 

    Watching this video should take approximately 4 minutes.
     
    Terms of Use: Please respect the copyright and terms of use displayed on the webpage above. 

4.5.3 Parameterizing Curves   Note: Now that you can go between parametric and Cartesian coordinates, you will learn how to change equations in a similar manner. The reading on pages 509–512 of the selection listed under subunit 4.5 covers this topic. 

  • Assessment: Washington State Board for Community and Technical Colleges’ MyOpenMath: “Section 8.5 Parametric Equations Graded Homework” Link: Washington State Board for Community and Technical Colleges’ MyOpenMath: “Section 8.5 Parametric Equations Graded Homework (HTML)

    Instructions: Complete this assessment after you have worked through all the readings, videos, and problems for subunit 4.5. 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 8: Quiz 14” Link: Washington State Board for Community and Technical Colleges’ “Chapter 8: Quiz 14 (PDF)
     
    Instructions: This quiz covers both subunits 4.4 and 4.5; be sure to complete the materials in both subunits before you attempt this quiz.

Unit 4 Review   - Assessment: Washington State Board for Community and Technical Colleges’ MyOpenMath: “Chapter Review Worksheet” Link: Washington State Board for Community and Technical Colleges’ MyOpenMath: “Chapter 4 Review (HTML)

 Instructions: Complete this review assessment after you have worked
through all the readings, videos, and problems for Unit 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’ MyOpenMath: “Chapter 4 Review” Link: Washington State Board for Community and Technical Colleges’ MyOpenMath: “Chapter Review Worksheet (HTML)

    Instructions: Complete this assessment worksheet after you have worked through all the readings, videos, and problems for Unit 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. 

Unit 4 Assessment   - Assessment: Washington State Board for Community and Technical Colleges’ “Chapter 4 Exam” Link: Washington State Board for Community and Technical Colleges’ “Chapter 4 Exam (PDF)

 Instructions: This assessment covers all of Unit 4; be sure to
complete the material for all subunits and the chapter review before
attempting this practice exam.