Precalculus II
Purpose of Course showclose
Precalculus II continues the indepth study of functions addressed in Precalculus I by adding the trigonometric functions to your function toolkit. In this course, you will cover families of trigonometric functions, as well as their inverses, properties, graphs, and applications. Additionally, you will study trigonometric equations and identities, the laws of sines and cosines, polar coordinates and graphs, parametric equations and elementary vector operations.
You might be curious how the study of trigonometry, or “trig,” as it is more often referred to, came about and why it is important to your studies still. Trigonometry, from the Greek for “triangle measure,” studies the relationships between the angles of a triangle and its sides and defines the trigonometric functions used to describe those relationships. Trigonometric functions are particularly useful when describing cyclical phenomena and have applications in numerous fields, including astronomy, navigation, music theory, physics, chemistry, and – perhaps most importantly, to the mathematics student – calculus.
In this course, you will begin by establishing the definitions of the basic trig functions and exploring their properties and then proceed to use the basic definitions of the functions to study the properties of their graphs, including domain and range, and to define the inverses of these functions and establish the properties of these. Through the language of transformation, you will explore the ideas of period and amplitude and learn how these graphical differences relate to algebraic changes in the function formulas. You will also learn to solve equations, prove identities using the trig functions, and study several applications of these functions.
Course Information showclose
Course Designers: Alessandra Bianchini
Primary Resources: This course is comprised of a range of different free, online materials. However, the course makes primary use of the following materials:
Requirements for Completion: In order to complete this course, you will need to work through each unit and all of its assigned materials. Most units will require you to complete checkpoint exercises that review the textbook information as well as selfreflective short answer questions that ask you to apply the information. In addition to these, you will also need to complete the Final Exam.
Note that you will only receive an official grade on your final exam. However, in order to adequately prepare for this exam, you will need to have a strong command of all the material covered in the course. The most efficient and effective way for you to learn this material is to simply work through all the units and complete all the activities as the instructor and course designer have presented them.
In order to pass this course, you will need to earn a 70% or higher on the Final Exam, which is administered electronically through the Saylor.org Moodle system. Your score on the exam will be tabulated as soon as you complete it. If you do not pass the exam, you may take it again.
Time Commitment: This course should take you approximately 72 hours to complete. Each unit includes a time advisory that lists the amount of time you should expect to spend on each subunit. These are only approximate times meant to help you plan your time accordingly, and could vary considerably for you. Please particularly note that the time commitment for each unit varies significantly; for example, Unit 2 should take you less than 15 hours, while Unit 4 will likely take you 25 hours.
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It is a good idea to use the time estimates to help you plan in advance when you will find time to complete each unit. It may be useful to take a look at these time advisories and to determine how much time you have over the next few weeks to complete each unit, and then to set goals for yourself.
For example, Unit 1 should take you 17 hours. Perhaps you can sit down with your calendar and decide to complete subunits 1.1 and 1.2 (a total of 6 hours) on Monday night; subunit 1.3 (a total of 3 hours) on Tuesday night; etc.
Tips/Suggestions: The skills and concepts that you learn in this course are designed to be applied to all the other courses you will take during your college experience. As you complete each unit, look for ways to immediately apply that unit’s main ideas to other work you are doing.
This course has been developed through a partnership with the Washington State Board for Community and Technical Colleges. Unless otherwise noted, all materials are licensed under a Creative Commons Attribution 3.0 Unported License. The Saylor Foundation has modified some materials created by the Washington State Board for Community and Technical Colleges in order to best serve our users.

This course features a number of Khan Academy™ videos. Khan Academy™ has a library of over 3,000 videos covering a range of topics (math, physics, chemistry, finance, history and more), plus over 300 practice exercises. All Khan Academy™ materials are available for free at www.khanacademy.org.

Learning Outcomes showclose
 measure angles in degrees and radians, and relate them to arc length;
 solve problems involving right triangles and unit circles using the definitions of the trigonometric functions;
 solve problems involving nonright triangles;
 relate the equation of a trigonometric function to its graph;
 solve trigonometric equations using inverse trig functions;
 prove trigonometric identities;
 solve trig equations involving identities;
 relate coordinates and equations in Polar form to coordinates and equations in Cartesian form;
 perform operations with vectors and use them to solve problems;
 relate equations and graphs in Parametric form to equations and graphs in Cartesian form;
 link graphical, numeric, and symbolic approaches when interpreting situations and analyzing problems;
 write clear, correct, and complete solutions to mathematical problems using proper mathematical notation and appropriate language; and
 communicate the difference between an exact and an approximate solution and determine which is more appropriate for a given problem.
Course Requirements showclose
√ have access to a computer;
√ have continuous broadband Internet access;
√ have the ability/permission to install plugins or software (e.g., Adobe Reader or Flash);
√ have the ability to download and save files and documents to a computer;
√ have the ability to open Microsoft files and documents (.doc, .ppt, .xls, etc.);
√ be competent in the English language; and
√ have read the Saylor Student Handbook.
Unit Outline show close

Unit 1: Trigonometric Functions of Angles
Imagine standing some distance from a building and trying to guess at its height. With some simple measurements and the tools you learn in this unit, you will be able to find the height of the building precisely. This is one of the many application problems you will be able to solve after your study of trigonometry.
Unit 1 Time Advisory show close
In this unit, you will explore the properties of circles and use those properties to investigate angles within the circle. In particular, you will begin with a new definition of angle measure related to arc length in a circle. A review of the equation for a circle with radius r leads to a definition of the sine and cosine functions, and you will use these to define the remaining trigonometric functions and explore their basic properties and identities. These definitions will be used to derive similar definitions for right triangle trigonometry, which is precisely the tool needed to solve problems like the one mentioned above.
Unit 1 Learning Outcomes show close

1.1 Circles
 Reading: Lippman and Rasmussen’s Precalculus: An Investigation of Functions: “Chapter 5: Trigonometric Angles of Functions”
Link: Lippman and Rasmussen’s Precalculus: An Investigation of Functions: “Chapter 5: Trigonometric Angles of Functions” (PDF)
Instructions: Read pages 297301 of Chapter 5 to learn about circles in trigonometry. Note that this reading covers the material in subunits 1.1.1 through 1.1.4.
Terms of Use: The article above is released under a Creative Commons AttributionShareAlike License 3.0 (HTML). It is attributed to Lippman & Rasmussen.
 Reading: Lippman and Rasmussen’s Precalculus: An Investigation of Functions: “Chapter 5: Trigonometric Angles of Functions”

1.1.1 The Pythagorean Theorem
Note: This topic is covered in the reading under subunit 1.1. To learn about the Pythagorean Theorem focus on the material on page 297.

1.1.2 The Distance Formula
Note: This topic is covered in the reading under subunit 1.1. To learn about the Distance Formula, focus on the material on pages 297 and 298. Pay particular attention to the examples in the text, and work the “Try It Now” problems; solutions are available at the end of each section of reading.
 Web Media: YouTube: Math Warehouse’s “Distance Formula Tutorial, Practice Problems, and Other Goodies”
Link: YouTube: Math Warehouse’s “Distance Formula Tutorial, Practice Problems, and Other Goodies” (YouTube)
Instructions: Watch this video for a comprehensive overview of the distance formula.
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: Math Warehouse’s “Distance Formula Tutorial, Practice Problems, and Other Goodies”

1.1.3 Equation of a Circle
Note: This topic is covered by the reading under subunit 1.1. Focus on the material on pages 298 and 299 to learn about the equation of a circle. Pay particular attention to the examples in the text, and work the “Try It Now” problems; solutions are available at the end of each section of reading.
 Web Media: YouTube: Brightstorm’s “Equation of a Circle”
Link: YouTube: Brightstorm’s “Equation of a Circle” (YouTube)
Instructions: Watch this video for a better understanding of writing the equation of a circle.
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: Brightstorm’s “Equation of a Circle”

1.1.4 Points on a Circle
Note: Points on a circle is one of the topics included in the reading listed under subunit 1.1. Pay particular attention to pages 300 and 301 and to the examples in the text. Work through the “Try It Now” problems; solutions are available at the end of each section of reading.
 Web Media: YouTube: Mathew Mattakad’s “Finding Points of Intersection between a Line and a Circle”
Link: YouTube: Mathew Mattakad’s “Finding Points of Intersection between a Line and a Circle” (YouTube)
Instructions: Watch this video and work through the examples discussed within. Then use what you have learned to redo 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 Mathew Mattakad, and the original version can be found here.  Assessment: Washington State Board for Community and Technical Colleges’ MyOpenMath: “Section 5.1: Circles Graded Homework”
Link: Washington State Board for Community and Technical Colleges’ MyOpenMath: “Section 5.1: Circles Graded Homework” (HTML)
Instructions: Once you have completed the readings and viewed the material for all of subunit 1.1, 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.
 Web Media: YouTube: Mathew Mattakad’s “Finding Points of Intersection between a Line and a Circle”

1.2 Angles
 Reading: Lippman and Rasmussen’s Precalculus: An Investigation of Functions: “Chapter 5: Trigonometric Angles of Functions”
Link: Lippman and Rasmussen’s Precalculus: An Investigation of Functions: “Chapter 5: Trigonometric Angles of Functions” (PDF)
Instructions: Read pages 307317 of Chapter 5 to learn about angles in trigonometry. Pay particular attention to the new form of angle measure, the radian. A complete grasp of this concept will serve you well through the remainder of the course. Also note that this reading covers the material in subunits 1.2.1 through 1.2.5.
Terms of Use: The article above is released under a Creative Commons AttributionShareAlike License 3.0 (HTML). It is attributed to Lippman & Rasmussen.
 Reading: Lippman and Rasmussen’s Precalculus: An Investigation of Functions: “Chapter 5: Trigonometric Angles of Functions”

1.2.1 Measuring Angles
Note: This topic is covered by the reading listed under subunit 1.2. Focus on page 307 to master measuring angles.

1.2.2 Working with Angles in Degrees
Note: This topic is covered by the reading listed under subunit 1.2. Read pages 308 and 309, and be sure to work through the examples for finding coterminal angles before you move on. Pay particular attention to the examples in the text, and work the “Try It Now” problems; solutions are available at the end of each section of reading.
 Web Media: YouTube: patrickJMT’s “Coterminal Angles  Example 3”
Link: YouTube: patrickJMT’s “Coterminal Angles  Example 3” (YouTube)
Instructions: This video provides further examples of finding coterminal angles. Watch it and rework problems from the reading that you found difficult.
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: clindelof’s “How to Find Coterminal Angles (Positive and Negative) Degrees: Precalculus”
Link: YouTube: clindelof’s “How to Find Coterminal Angles (Positive and Negative) Degrees: Precalculus” (YouTube)
Instructions: This video provides further examples of finding coterminal angles. Watch it and rework problems from the reading that you found difficult.
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 “Coterminal Angles  Example 3”

1.2.3 Angles in Radians
Note: This topic is covered by the reading listed under subunit 1.2. Pages 309313 focus on angles in radians. A thorough understanding of the definition of a radian as well as the conversion between radians and degrees is important to your success in this course. Pay particular attention to the examples in the text, and work the “Try It Now” problems; solutions are available at the end of each section of reading.
 Web Media: YouTube: eHow’s “Mathematics: Geometry & Trigonometry: How to Convert Degrees to Radians”
Link: YouTube: eHow’s “Mathematics: Geometry & Trigonometry: How to Convert Degrees to Radians” (YouTube)
Instructions: This video reviews the concept of radian/degree conversion. Watch the brief video clip and rework any problems in the chapter that you had difficulty with.
Watching this video should take approximately 1 minute.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.  Web Media: YouTube: AcademicLeadersEd’s “Conversion of Radians to Degrees and Viceversa (Easy!)”
Link: YouTube: AcademicLeadersEd’s “Conversion of Radians to Degrees and Viceversa (Easy!)” (YouTube)
Instructions: This brief video reviews the concept of radian/degree conversion. Watch the video clip and rework any problems in the chapter that you had difficulties with.
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: American Public University System’s “Radian Measure of Coterminal Angles”
Link: YouTube: American Public University System’s “Radian Measure of Coterminal Angles” (YouTube)
Instructions: This video reviews the concept of radian/degree conversion and revisits the earlier topic of coterminal angles. Watch the video and rework any problems in the chapter that you had difficulty with.
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: eHow’s “Mathematics: Geometry & Trigonometry: How to Convert Degrees to Radians”

1.2.4 Arclength and Area of a Sector
Note: This section is covered in the reading under subunit 1.2. Read the material on pages 298 and 299 to learn about arclength and area of a sector, and work through the exercises. Complete the “Try It Now” exercises; solutions are available at the end of each section of reading.
 Web Media: YouTube: Mindbites.com’s “Trig: Radians, Arc Length, Solve 1st Deg Equations”
Link: YouTube: Mindbites.com’s “Trig: Radians, Arc Length, Solve 1st Deg Equations” (YouTube)
Instructions: This video provides further instruction for computing arc lengths. Watch the video and rework any problems you had difficulty with in the readings.
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: Mr. Pilarski’s “Arc Length Theorem: How to Find the Length of an Arc”
Link: YouTube: Mr. Pilarski’s “Arc Length Theorem: How to Find the Length of an Arc” (YouTube)
Instructions: This video provides further instruction for computing arc lengths. Watch the video and rework any problems you had difficulty with in the readings.
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: Tucson Math Doc’s “How to Find Arc Length? (Shortcut!)”
Link: YouTube: Tucson Math Doc’s “How to Find Arc Length? (Shortcut!)” (YouTube)
Instructions: This video provides further instruction for computing arc lengths. Watch the video and rework any problems you had difficulty with in the readings.
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: BabelMath’s “Area of a Sector”
Link: YouTube: BabelMath’s “Area of a Sector” (YouTube)
Instructions: This video provides further instruction for computing the area of a sector. Watch the video and rework any problems you had difficulty with in the readings.
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: Mindbites.com’s “Trig: Radians, Arc Length, Solve 1st Deg Equations”

1.2.5 Linear and Angular Velocity
Note: Linear and angular velocity is covered in the reading for subunit 1.2. Pay careful attention to pages 315317, and be sure to work through all the examples. Make sure you attempt the “Try It Now” problems; solutions for these are found at the end of the section of reading.
 Web Media: YouTube: 123MrBee’s “Angular Velocity Speed Word Problem: Trigonometry Math Help”
Link: YouTube: 123MrBee’s “Angular Velocity Speed Word Problem: Trigonometry Math Help” (YouTube)
Instructions: Watch this video for a better understanding of angular velocity.
Watching this video should take approximately 6 minutes.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.  Activity: Washington State Board for Community and Technical Colleges’ “Music Listening Technology” Worksheet
Link: Washington State Board for Community and Technical Colleges’ “Music Listening Technology” Worksheet (PDF)
Instructions: Changes in the technology for reproducing music provide an interesting context for studying circular speeds. Complete the worksheet by reading the text and answering all 6 questions to study the differences between linear and angular velocities for CD’s and records.  Assessment: Washington State Board for Community and Technical Colleges’ MyOpenMath: “Section 5.2: Angles Graded Homework”
Link: Washington State Board for Community and Technical Colleges’ MyOpenMath: “Section 5.2: Angles Graded Homework” (HTML)
Instructions: Once you have completed the readings and viewed the material for all of subunit 1.2, 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 5 Quiz 1”
Link: Washington State Board for Community and Technical Colleges’ “Chapter 5 Quiz 1” (PDF)
Instructions: This assessment covers subunits 1.1.and 1.2. Complete this quiz after you have worked through the readings, web media, and assignments for subunits 1.1 and 1.2.
 Web Media: YouTube: 123MrBee’s “Angular Velocity Speed Word Problem: Trigonometry Math Help”

1.3 Points on Circles Using Sine and Cosine
 Reading: Lippman and Rasmussen’s Precalculus: An Investigation of Functions: “Chapter 5: Trigonometric Angles of Functions”
Link: Lippman and Rasmussen’s Precalculus: An Investigation of Functions: “Chapter 5: Trigonometric Angles of Functions” (PDF)
Instructions: Read pages 321330 of Chapter 5 to learn points on circles using sine and cosine. The unit circle is one of the key concepts in trigonometry, and a complete understanding how the coordinates from the equation of the circle are used to create the trig functions is fundamental to understanding the derivations of the graphs of the functions and all the useful identities we will study in later sections. Committing the unit circle to memory is a useful skill. This reading covers the material in subunits 1.3.1 through 1.3.4.
Terms of Use: The article above is released under a Creative Commons AttributionShareAlike License 3.0 (HTML). It is attributed to Lippman & Rasmussen.
 Reading: Lippman and Rasmussen’s Precalculus: An Investigation of Functions: “Chapter 5: Trigonometric Angles of Functions”

1.3.1 The Sine and Cosine Functions
Note: This section is covered by the reading listed under subunit 1.3. Pay attention to the readings on pages 321323 and in particular the relationship between the terminal points of an angle on the unit circle and its sine and cosine.
 Web Media: YouTube: sgordo3398’s “Section 5.3 #1”
Link: YouTube: sgordo3398’s “Section 5.3 #1” (YouTube)
Instructions: This video is optional for the student, but may provide particular insight into some of the assigned problems for subunit 1.3.
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: sgordo3398’s “Section 5.3 #1”

1.3.2 Pythagorean Identity
Note: Note that this topic is covered by the reading listed under subunit 1.3. Pay attention to the information on page 323 to understand how the Pythagorean Identity follows naturally from the equation of the unit circle. As always, work through examples and “Try It Now” problems; solutions to the Try It Now problems are available at the end of the reading section.

1.3.3 Unit Circle Values
Note: Note that this topic is covered by the reading listed under subunit 1.3. There are several important values for the sine and cosine functions on the unit circle. Read pages 324327 and begin to commit these values to memory. Be sure to work the “Try It Now” problems and go over the example problems. Solutions to the Try It Now problems appear at the end of the reading selection.
 Web Media: YouTube: sgordo3398’s “Section 5.3 #2”
Link: YouTube: sgordo3398’s “Section 5.3 #2” (YouTube)
Instructions: This optional video may prove useful in solving the exercises in the assignment for subunit 1.3.
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: 123MrBee’s “How to Find a Point on the Unit Circle Pt1: Trigonometry Math Help”
Link: YouTube: 123MrBee’s “How to Find a Point on the Unit Circle Pt1: Trigonometry Math Help” (YouTube)
Instructions: Watch this video to clarify your understanding of the unit circle, and rework the exercises or “Try It Now” problems 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.
 Web Media: YouTube: sgordo3398’s “Section 5.3 #2”

1.3.4 Reference Angles
Note: This section gives a discussion on using reference angles to determine the sine or cosine of an angle of any measure. It is covered by the reading listed under subunit 1.3; pay attention to pages 327330, working through the examples and completing the “Try It Now” problems. Solutions to these problems appear at the end of the reading section.
 Web Media: Brightstorm’s “Evaluating Cosine”
Link: Brightstorm’s “Evaluating Cosine” (Flash Video)
Instructions: View this video and use the information in it to deepen your understanding of the reading.
Watching this video will take approximately 3 minutes.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.  Web Media: YouTube: speakerbill’s “Reference Angles Introduction”
Link: YouTube: speakerbill’s “Reference Angles Introduction” (YouTube)
Instructions: After watching this video be sure to rework the examples in the text that you previously had difficulty with.
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: Khan Academy’s “The Unit Circle Definition of Trigonometric Function”
Link: YouTube: Khan Academy’s “The Unit Circle Definition of Trigonometric Function” (YouTube)
Instructions: After watching this video be sure to rework the examples in the text that you previously had difficulty with.
Watching this video should take approximately 10 minutes.
Terms of Use: This resource is licensed under a Creative Commons AttributionNonCommercialNoDerivs 3.0 Unported License. It is attributed to the Khan Academy.  Web Media: YouTube: sgordo3398’s “Section 5.3 #7,” “Section 5.3 #9” and “Section 5.3 #10”
Links: YouTube: sgordo3398’s “Section 5.3 #7,” “Section 5.3 #9,” and “Section 5.3 #10” (YouTube)
Instructions: These videos relate to specific problems in the assignment for subunit 1.3 and will help you understand how to complete problems 7, 9, and 10 in this assignment.
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 5.3: Points on Circles using Sine and Cosine Graded Homework”
Link: Washington State Board for Community and Technical Colleges’ MyOpenMath: “Section 5.3: Points on Circles using Sine and Cosine Graded Homework” (HTML)
Instructions: Once you have completed the readings and viewed the material for all of subunit 1.3, complete this assessment. Your answers will be scored upon submission.  Assessment: Washington State Board for Community and Technical Colleges’ “Chapter 5 Quiz 2”
Link: Washington State Board for Community and Technical Colleges’ “Chapter 5 Quiz 2” (PDF)
Instructions: This quiz uses the information from subunit 1.3 as well as in the previous sections. Complete it after you have worked through the material in this subunit.
 Web Media: Brightstorm’s “Evaluating Cosine”

1.4 The Other Trigonometric Functions
 Reading: Lippman and Rasmussen’s Precalculus: An Investigation of Functions: “Chapter 5: Trigonometric Angles of Functions”
Link: Lippman and Rasmussen’s Precalculus: An Investigation of Functions: “Chapter 5: Trigonometric Angles of Functions” (PDF)
Instructions: Read pages 333338 of Chapter 5 to learn about the other trigonometric functions and some important identities, establishing some relationships between all six of the trigonometric functions. This reading covers the material in subunits 1.4.1 through 1.4.3.
Terms of Use: The article above is released under a Creative Commons AttributionShareAlike License 3.0 (HTML). It is attributed to Lippman & Rasmussen.
 Reading: Lippman and Rasmussen’s Precalculus: An Investigation of Functions: “Chapter 5: Trigonometric Angles of Functions”

1.4.1 Tangent, Secant, Cosecant, and Cotangent Functions
Note: The reading for this topic is contained in the reading for subunit 1.4. Pay attention to page 333 to develop an understanding of the four additional trig functions as ratios of the sine and cosine. As always, work through the examples on your own and complete the ‘Try It Now” problems, and check your solutions at the end of the selected reading.
 Web Media: Brightstorm’s “Reciprocal Trigonometric Functions”
Link: Brightstorm’s “Reciprocal Trigonometric Functions” (Flash Video)
Instructions: Watch this video to deepen your understanding of the tangent, cotangent, secant, and cosecant as reciprocals of sine and cosine.
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: sgordo3398’s “Section 5.4 #4”
Link: YouTube: sgordo3398’s “Section 5.4 #4” (YouTube)
Instructions: Watch this video; it will be particularly useful when you complete the assignment at the end of the subunit.
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: bullcleo1’s “Determining Trigonometric Function Values on a Calculator”
Link: YouTube: bullcleo1’s “Determining Trigonometric Function Values on a Calculator” (YouTube)
Instructions: Watch this video to learn how to accurately compute values of the trig functions using a calculator. There are numerous pitfalls in these calculations; the video can help you avoid them.
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: Brightstorm’s “Reciprocal Trigonometric Functions”

1.4.2 Some Important Trig Identities
Note: This topic is covered by the reading under subunit 1.4. Focus on the information on pages 333337 for some additional important trig functions. Be sure to work through the example problems yourself and attempt the “Try It Now” problems.
 Web Media: YouTube: MrEmrah57’s “11 2 Simplifying Trig Expressions”
Link: YouTube: MrEmrah57’s “11 2 Simplifying Trig Expressions” (YouTube)
Instructions: Watch this video to get further insight into using the trig identities you have learned thus far to simplify trigonometric expressions.
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: Brightstorm’s “PrecalculusUsing Trigonometric Identities”
Link: YouTube: Brightstorm’s “PrecalculusUsing Trigonometric Identities” (YouTube)
Instructions: Watch this video to get further insight into using the trig identities you have learned thus far to simplify trigonometric expressions.
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: MrEmrah57’s “11 2 Simplifying Trig Expressions”

1.4.3 Alternate Forms of the Pythagorean Identity
Note: There are several forms of the Pythagorean Identity that use the new reciprocal trig functions. The reading listed under subunit 1.4 covers these on pages 337 and 338. Pay attention to the examples in the text and work through the “Try It Now” problems. Solutions to these problems are available at the end of the selected reading.
 Web Media: YouTube: Mindbites.com’s “Trigonometry: Simplify Fractional Trig Expression”
Link: YouTube: Mindbites.com’s “Trigonometry: Simplify Fractional Trig Expression” (YouTube)
Instructions: Watch this video and rework and problems you had with the examples on pages 337 and 338.
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: patrickJMT’s “Simplifying Trigonometric Expressions Using Identities, Example 1”
Link: YouTube: patrickJMT’s “Simplifying Trigonometric Expressions Using Identities, Example 1” (YouTube)
Instructions: Watch this video to reinforce your understanding of the use of identities to simplify trig expressions.
Watching this video should take approximately 1 minute.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.  Activity: Washington State Board for Community and Technical Colleges’ “Classwork Identities” Worksheet
Link: Washington State Board for Community and Technical Colleges’ “Classwork Identities” Worksheet (PDF)
Instructions: Complete this identity worksheet for additional practice using the identities in subunit 1.4  Assessment: Washington State Board for Community and Technical Colleges’ MyOpenMath: “Section 5.4 The Other Trig Functions Graded Homework”
Link: Washington State Board for Community and Technical Colleges’ MyOpenMath: “Section 5.4 The Other Trig Functions Graded Homework” (HTML)
Instructions: Once you have completed the readings and viewed the material for all of subunit 1.4, 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.  Assessment: Washington State Board for Community and Technical Colleges’ “Chapter 5 Quiz 3”
Link: Washington State Board for Community and Technical Colleges’ “Chapter 5 Quiz 3” (PDF)
Instructions: This quiz uses the information from subunit 1.4 as well as in the previous sections. Complete it after you have worked through the material in this subunit.
 Web Media: YouTube: Mindbites.com’s “Trigonometry: Simplify Fractional Trig Expression”

1.5 Right Triangle Trigonometry
 Reading: Lippman and Rasmussen’s Precalculus: An Investigation of Functions: “Chapter 5: Trigonometric Angles of Functions”
Lippman and Rasmussen’s Precalculus: An Investigation of Functions: “Chapter 5: Trigonometric Angles of Functions” (PDF)
Instructions: Read pages 343347 of Chapter 5 to learn about the trig functions in the context of right triangles. Note that this reading covers subunits 1.5.1 through 1.5.3.
Terms of Use: The article above is released under a Creative Commons AttributionShareAlike License 3.0 (HTML). It is attributed to Lippman & Rasmussen.
 Reading: Lippman and Rasmussen’s Precalculus: An Investigation of Functions: “Chapter 5: Trigonometric Angles of Functions”

1.5.1 Right Triangle Relationships with Sine, Cosine, and Tangent
Note: This topic is covered in the reading listed under subunit 1.5. Focus on the content of pages 343 and 344, paying particular attention to the worked examples, and completing any “Try It Now” problems in the text. The solutions to these problems are located at the end of the selected reading.
 Web Media: YouTube: Mindbites.com’s “PreCalculus: Fundamental Trigonometric Identities”
Link: YouTube: Mindbites.com’s “PreCalculus: Fundamental Trigonometric Identities” (YouTube)
Instructions: Watch this video to review right triangle trigonometry.
Watching this video should take approximately 4 minutes.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.  Web Media: YouTube: YourTeacher.com’s “Sin Cos Tan—Basic Trigonometry”
Link: YouTube: YourTeacher.com’s “Sin Cos Tan—Basic Trigonometry” (YouTube)
Instructions: Watch this video to review right triangle trigonometry. Rework any examples or problems you previously had difficulty with.
Watching this video should take approximately 4 minutes.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
 Web Media: YouTube: Mindbites.com’s “PreCalculus: Fundamental Trigonometric Identities”

1.5.2 The Cofunction Identities
Note: This topic is covered by the reading under subunit 1.5. Focus on page 345 to learn about the cofunction identities, and be sure to work through the practice problems.

1.5.3 Right Triangle Trig Applications
Note: This topic is covered by the reading under subunit 1.5. Focus on pages 345347. Work through the examples and “Try It Now” problems to get a working knowledge of the applications of right triangle trigonometry. The solutions to these are available at the end of the selected reading.
 Web Media: YouTube: Mindbites.com’s “Trig: Word Problems—Sine, Cosine & Tangent”
Link: YouTube: Mindbites.com’s “Trig: Word Problems—Sine, Cosine & Tangent” (YouTube)
Instructions: Watch the video, trying any problems on your own and checking the video for accuracy.
Watching this video should take approximately 4 minutes.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.  Web Media: YouTube: patrickJMT’s “Trigonometry Word Problem, Finding the Height of a Building, Example 1”
Link: YouTube: patrickJMT’s “Trigonometry Word Problem, Finding the Height of a Building, Example 1” (YouTube)
Instructions: Watch the video, trying any problems on your own and checking the video for accuracy.
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: Brightstorm’s “Angle of Elevation and Depression”
Link: YouTube: Brightstorm’s “Angle of Elevation and Depression” (YouTube)
Instructions: Watch the video, trying any problems on your own and checking the video for accuracy.
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: sgordo3398’s “Section 5.5 #7”
Link: YouTube: sgordo3398’s “Section 5.5 #7” (YouTube)
Instructions: This optional video will be useful to solve the problems in the assignment at the end of the subunit.
Watching this video should take approximately 2 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 5.5 Graded Homework”
Link: Washington State Board for Community and Technical Colleges’ MyOpenMath: “Section 5.5 Graded Homework” (HTML)
Instructions: Once you have completed the readings and viewed the material for all of subunit 1.4, 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.  Assessment: Washington State Board of Community and Technical Colleges’ “Chapter 5 Quiz 4”
Link: Washington State Board of Community and Technical Colleges’ “Chapter 5 Quiz 4” (PDF)
Instructions: This quiz uses the information from Subunit 1.5 as well as in the previous sections. Complete it after you have worked through the material in this subunit.
 Web Media: YouTube: Mindbites.com’s “Trig: Word Problems—Sine, Cosine & Tangent”

Unit 1 Review
 Assessment: Washington State Board for Community and Technical Colleges’ MyOpenMath: “Chapter 5 Review”
Link: Washington State Board for Community and Technical Colleges’ MyOpenMath: “Chapter 5 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 will provide a review for your unit assessment.
 Assessment: Washington State Board for Community and Technical Colleges’ MyOpenMath: “Chapter 5 Review”

Unit 1 Assessment
 Assessment: Washington State Board for Community and Technical Colleges’ “Chapter 5 Exam 1a”
Link: Washington State Board for Community and Technical Colleges’ “Chapter 5 Exam 1a” (PDF)
Instructions: Complete this assessment when you have completed all of the material in Unit 1. You should complete the assessment without the use of a calculator.
 Assessment: Washington State Board for Community and Technical Colleges’ “Chapter 5 Exam 1a”

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.
Unit 2 Time Advisory show close
This unit continues the study of the trigonometric functions with an indepth 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 Learning Outcomes show close

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 353365 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.
Terms of Use: The article above is released under a Creative Commons AttributionShareAlike License 3.0 (HTML). It is attributed to Lippman & Rasmussen.
 Reading: Lippman and Rasmussen’s Precalculus: An Investigation of Functions: “Chapter 6: Periodic Functions”

2.1.1 Periodic Functions
Note: Periodic Functions are covered in the reading listed under subunit 2.1; pay attention in particular to pages 353355, 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.
 Web Media: YouTube: Khan Academy’s “Graphing Trig Functions”
Link: YouTube: Khan Academy’s “Graphing Trig Functions” (YouTube)
Instructions: Watch this video to get a basic understanding of what the graphs of the trig functions look like.
Watching this video should take approximately 10 minutes.
Terms of Use: This resource is licensed under a Creative Commons AttributionNonCommercialNoDerivs 3.0 Unported License. It is attributed to the Khan Academy.
 Web Media: YouTube: Khan Academy’s “Graphing Trig Functions”

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.
 Web Media: YouTube: patrickJMT’s “Graphing Trigonometric Functions  Graph Transformations P1”

2.1.5.1 Midline
Notes: Read pages 358360 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 rework 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.
 Web Media: YouTube: MrGibsonrhs’s “Writing an Equation of a Sin/Cos Function When Given the Graph”

2.1.5.2 Transformations of Sine and Cosine
Note: This topic is covered by the reading beneath subunit 2.1. In particular, pages 361363 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 AttributionNonCommercialNoDerivs 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.
 Web Media: YouTube: Brightstorm’s “Precalculus: Transforming the Graphs of Sine and Cosine”

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 363365 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.
 Web Media: YouTube: Brightstorm’s “Precalculus: More Transformations of Sine and Cosine”

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 Commons AttributionShareAlike License 3.0 (HTML). It is attributed to Lippman & Rasmussen.
 Reading: Lippman and Rasmussen’s Precalculus: An Investigation of Functions: “Chapter 6: Periodic Functions”

2.2.1 The Graph of the Tangent Function
Note: To understand the graph of the tangent function, focus on pages 369372 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.
 Assessment: Washington State Board for Community and Technical Colleges’ MyOpenMath: “Section 6.2 Graded Homework”

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 Commons AttributionShareAlike License 3.0 (HTML). It is attributed to Lippman & Rasmussen.
 Reading: Lippman and Rasmussen’s Precalculus: An Investigation of Functions: “Chapter 6: Periodic Functions”

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 AttributionNonCommercialNoDerivs 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.
 Web Media: YouTube: Khan Academy’s “Inverse Trig Functions: Arcsin” and “Inverse Trig Functions: Arctan”

2.3.2 Evaluating Inverses with the Unit Circle or a Calculator
Note: Work through Example 1 through Example 6 on pages 380383 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.
 Web Media: YouTube: bwolky’s “Section 6.3 #3” and “Section 6.3 #4”

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.
 Assessment: Washington State Board for Community and Technical Colleges’ MyOpenMath: “Section 6.3 Graded Homework”

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 387394 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 Commons AttributionShareAlike License 3.0 (HTML). It is attributed to Lippman & Rasmussen.
 Reading: Lippman and Rasmussen’s Precalculus: An Investigation of Functions: “Chapter 6: Periodic Functions”
 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 387389 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 rework 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 rework 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.
 Web Media: YouTube: MathTV’s “Trig: Solving Equations 1”

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.
 Web Media: YouTube: bwolky’s “Section 6.3 #7” and “Section 6.3 #8”

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
 Web Media: YouTube: bwolky’s “Section 6.3 #9”

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 Commons AttributionShareAlike License 3.0 (HTML). It is attributed to Lippman & Rasmussen.
 Reading: Lippman and Rasmussen’s Precalculus: An Investigation of Functions: “Chapter 6: Periodic Functions”

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 397398 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.
 Web Media: YouTube: patrickJMT’s “Right Triangles and Trigonometric Functions”

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 AttributionNonCommercialNoDerivs 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.
 Web Media: YouTube: Khan Academy’s “Ferris Wheel Trig Problem”

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.
 Assessment: Washington State Board for Community and Technical Colleges’ MyOpenMath: “Chapter 6 Review”

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.
 Assessment: Washington State Board for Community and Technical Colleges’ “Chapter 6 Exam”

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.
Unit 3 Time Advisory show close
This unit builds on the equationsolving 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 Learning Outcomes show close

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 AttributionShareAlike License 3.0 (HTML). It is attributed to Lippman & Rasmussen.
 Reading: Lippman and Rasmussen’s Precalculus: An Investigation of Functions: “Chapter 7: Trigonometric Equations and Identities”

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.
 Web Media: YouTube: MathTV’s “Trig: Solving Equations 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 417430. 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 AttributionShareAlike License 3.0 (HTML). It is attributed to Lippman & Rasmussen.
 Reading: Lippman and Rasmussen’s Precalculus: An Investigation of Functions: “Chapter 7: Trigonometric Equations and Identities”

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.
 Web Media: YouTube: patrickJMT’s “Using the Sum and Difference Identities for Sine, Cosine, and Tangent, Ex 1”

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 redo 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.
 Web Media: YouTube: bullcleo’s “Sum to Product and Product to Sum Identities”

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 AttributionShareAlike License 3.0 (HTML). It is attributed to Lippman & Rasmussen.
 Reading: Lippman and Rasmussen’s Precalculus: An Investigation of Functions: “Chapter 7: Trigonometric Equations and Identities”

3.3.1 Using the Double Angle Identities
Note: This topic is covered by the reading assigned below subunit 3.3. Focus on pages 431436 and work through all problems in the text to begin to grasp the double angle identities.
 Web Media: YouTube: davidmpeabody’s “RHS Pre CalculusDouble 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.
 Web Media: YouTube: davidmpeabody’s “RHS Pre CalculusDouble Angle Identities”

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 436441 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.
 Web Media: YouTube: patrickJMT’s “Power Reducing Formulas for Sine and Cosine, Example 1”

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 AttributionShareAlike License 3.0 (HTML). It is attributed to Lippman & Rasmussen.
 Reading: Lippman and Rasmussen’s Precalculus: An Investigation of Functions: “Chapter 7: Trigonometric Equations and Identities”

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.
 Assessment: Washington State Board for Community and Technical Colleges’ MyOpenMath: “Section 7.4 Graded Homework”

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.
 Assessment: Washington State Board for Community and Technical Colleges’ MyOpenMath: “Chapter 7 Review”

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.
 Assessment: Washington State Board for Community and Technical Colleges’ “Chapter 7 Exam B”

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 nonright triangles, which is what you will need to solve the problem above.
Unit 4 Time Advisory show close
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 Learning Outcomes show close

4.1 Nonright 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 AttributionShareAlike License 3.0 (HTML). It is attributed to Lippman & Rasmussen.
 Reading: Lippman and Rasmussen’s Precalculus: An Investigation of Functions: “Chapter 8: Further Applications of Trigonometry”

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 “PreCalculus: The Law of Sines”
Link: YouTube: Mindbites.com’s “PreCalculus: 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.
 Web Media: YouTube: Mindbites.com’s “PreCalculus: The Law of Sines”

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 NonRight Triangles: Law of Sines and Cosines Graded Homework”
Link: Washington State Board for Community and Technical Colleges’ MyOpenMath: “Section 8.1 NonRight 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 NonRight Triangles: Law of Sines and Cosines Graded Homework”
Link: Washington State Board for Community and Technical Colleges’ MyOpenMath: “Section 8.1 Applications NonRight 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.
 Web Media: YouTube: AlRichards314’s “The Cosine Law”

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 AttributionShareAlike License 3.0 (HTML). It is attributed to Lippman & Rasmussen.
 Reading: Lippman and Rasmussen’s Precalculus: An Investigation of Functions: “Chapter 8: Further Applications of Trigonometry”

4.2.1 Defining Polar Coordinates
Note: This topic is covered by the reading assigned below subunit 4.2. Focus on pages 467468 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.
 Web Media: patrickJMT’s “Polar Coordinates: Basic Graphing”

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.
 Web Media: YouTube: patrickJMT’s “Converting between Polar and Rectangular Cartesian) Coordinates, Ex 3”

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 470475, 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 AttributionNonCommercialNoDerivs 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.
 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”

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 AttributionShareAlike License 3.0 (HTML). It is attributed to Lippman & Rasmussen.
 Reading: Lippman and Rasmussen’s Precalculus: An Investigation of Functions: “Chapter 8: Further Applications of Trigonometry”

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 3The 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.
 Web Media: YouTube: derekownes’ “Complex Numbers, Part 3 The Complex Plane”

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.
 Web Media: YouTube: patrickJMT’s “Adding and Subtracting Complex (Imaginary) Numbers”

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.
 Web Media: YouTube: waszel’s “The Polar Form of Complex Numbers”

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.
 Web Media: YouTube: MathDoctorBob’s “Example of De Moivre's Theorem”

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 491502 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 AttributionShareAlike License 3.0 (HTML). It is attributed to Lippman & Rasmussen.
 Reading: Lippman and Rasmussen’s Precalculus: An Investigation of Functions: “Chapter 8: Further Applications of Trigonometry”

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 “PreCalculus: Finding Vector Magnitude & Direction”
Link: YouTube: Mindbites.com’s “PreCalculus: 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.
 Web Media: YouTube: Mindbites.com’s “PreCalculus: Finding Vector Magnitude & Direction”

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
 Web Media: YouTube: patrickJMT’s “Finding the Components of a Vector, Ex 1”

4.4.2.3 Alternate Notation for Component Form
Note: Read pages 494496 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 redo 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 redo 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.
 Web Media: YouTube: patrickJMT’s “Vector Basics: Drawing Vectors/Vector Addition”

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 AttributionShareAlike License 3.0 (HTML). It is attributed to Lippman & Rasmussen.
 Reading: Lippman and Rasmussen’s Precalculus: An Investigation of Functions: “Chapter 8: Further Applications of Trigonometry”

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.
 Web Media: YouTube: sgordo3398’s “DSCF2359”

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 rework 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 rework 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.
 Web Media: YouTube: math2bprof’s “Parametric Curves Part 1”

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.
 Assessment: Washington State Board for Community and Technical Colleges’ MyOpenMath: “Section 8.5 Parametric Equations Graded Homework”

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.
 Assessment: Washington State Board for Community and Technical Colleges’ MyOpenMath: “Chapter Review Worksheet”

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.
 Assessment: Washington State Board for Community and Technical Colleges’ “Chapter 4 Exam”

Final Exam
 Final Exam: The Saylor Foundation’s “MA003 Final Exam”
Link: The Saylor Foundation’s “MA003 Final Exam”
Instructions: You must be logged into your Saylor Foundation School account in order to access this exam. If you do not yet have an account, you will be able to create one, free of charge, after clicking the link.
 Final Exam: The Saylor Foundation’s “MA003 Final Exam”