Trigonometric Graphs and Identities - Spokane Public …swcontent.spokaneschools.org/.../Domain/3458/chapter14text.pdf · Chapter Test (Levels A, B, C) ... Trigonometric Graphs and

Trigonometric Graphs and IdentitiesSection 14A Section 14B

Exploring Trigonometric Graphs Trigonometric Identities

14-1 Graphs of Sine and Cosine

14-2 Graphs of Other Trigonometric Functions

14-3 Technology Lab Graph Trigonometric Identities

Connecting Algebra to Geometry Angle Relationships

14-3 Fundamental Trigonometric Identities

14-4 Sum and Difference Identities

14-5 Double-Angle and Half-Angle Identities

14-6 Solving Trigonometric Equations

Pacing Guide for 45-Minute Classes Calendar Planner®

Chapter 14DAY 1 DAY 2 DAY 3 DAY 4 DAY 5

14-1 Lesson 14-2 Lesson 14-2 Lesson Multi-Step Test PrepReady to Go On?14-3 Technology Lab

Connecting Algebra to Geometry14-3 Lesson

DAY 6 DAY 7 DAY 8 DAY 9 DAY 10

14-3 Lesson14-4 Lesson



14-6 LessonMulti-Step Test PrepReady to Go On?

Chapter 14 Test

Pacing Guide for 90-Minute Classes Calendar Planner®

Chapter 14DAY 1 DAY 2 DAY 3 DAY 4 DAY 5


14-2 LessonMulti-Step Test PrepReady to Go On?14-3 Technology Lab

Connecting Algebra to Geometry14-3 Lesson14-4 Lesson

14-4 Lesson14-5 Lesson14-6 Lesson

14-6 LessonMulti-Step Test PrepReady to Go On?Chapter 14 Test

986A Chapter 14

RESOURCE OPTIONS • RESOURCE OPTIONS • RESOURCE OPTIONS • RESOURCE O

A211NLT_c14_Interleaf.indd 986AA211NLT_c14_Interleaf.indd 986A 8/27/09 7:59:18 PM8/27/09 7:59:18 PM

Before Chapter 14 TestingDiagnose mastery of concepts in chapter.

Ready to Go On? SE pp. 1005, 1041

Multi-Step Test Prep SE pp. 1004, 1034

Section Quizzes AR

Test and Practice Generator

Prescribe intervention.

Ready to Go On? InterventionScaffolding Questions TE pp. 1004, 1034

Reteach CRB

Lesson Tutorial Videos

Before High Stakes TestingDiagnose mastery of benchmark concepts.

College Entrance Exam Practice SE p. 1041

Standardized Test Prep SE pp. 1044–1045


College Entrance Exam Practice

Before Every LessonDiagnose readiness for the lesson.

Warm Up TE


Reteach CRB

During Every LessonDiagnose understanding of lesson concepts.

Check It Out! SE

Questioning Strategies TE

Think and Discuss SE

Write About It SE

Journal TE


Reading Strategies CRB

Success for Every Learner


After Every LessonDiagnose mastery of lesson concepts.

Lesson Quiz TE

Test Prep SE



Reteach CRB

Test Prep Doctor TE

Homework Help Online

DIAGNOSE PRESCRIBE

KEY: SE = Student Edition TE = Teacher’s Edition CRB = Chapter Resource Book AR = Assessment Resources Available online Available on CD- or DVD-ROM

AssessPrior

Knowledge

FormativeAssessment

SummativeAssessment

Before Chapter 14Diagnose readiness for the chapter.

Are You Ready? SE p. 987


Are You Ready? Intervention

After Chapter 14Check mastery of chapter concepts.

Multiple-Choice Tests (Forms A, B, C)Free-Response Tests (Forms A, B, C)Performance Assessment AR

Cumulative Test AR



Reteach CRB


986B

RCE OPTIONS • RESOURCE OPTIONS • RESOURCE OPTIONS • RESOURCE OPTIONS

A211NLT_c14_Interleaf.indd 986BA211NLT_c14_Interleaf.indd 986B 8/26/09 8:16:33 PM8/26/09 8:16:33 PM

RESOURCE OPTIONS • RESOURCE OPTIONS • RESOURCE OPTIONS • RESOURCEE O

Power Presentations Teacher One Stop Premier Online EditionDynamic presentations to engage students. Complete PowerPoint® presentations for every lesson in Chapter 14.

Easy access to Chapter 14 resources and assessments. Includes lesson planning, test generation, and puzzle creation software.

Chapter 14 includes Tutorial Videos, Lesson Activities, Lesson Quizzes, Homework Help, and Chapter Project.

Lesson Resources

Technology Highlights for the Teacher

KEY: SE = Student Edition TE = Teacher’s Edition English Language Learners Spanish version available Available online Available on CD- or DVD-ROM

Before the Lesson

Prepare Teacher One Stop • Editable lesson plans• Calendar Planner • Easy access to all chapter resources

Lesson Transparencies • Teacher Tools

Teach the Lesson

Introduce Alternate Openers: Explorations Lesson Transparencies

• Warm Up • Problem of the Day

Teach Lesson Transparencies • Teaching Transparencies

Know-It Notebook™• Vocabulary• Key Concepts

Power Presentations Lesson Tutorial Videos Interactive Online Edition

• Lesson Activities• Lesson Tutorial Videos

Lab ActivitiesLab Resources Online Online Interactivities TechKeys

Practice the Lesson

Practice Chapter Resources• Practice A, B, C

Practice and Problem Solving Workbook IDEA Works!® Modifi ed Worksheets and TestsExamView Test and Practice Generator

Homework Help Online Online Interactivities Interactive Online Edition

• Homework Help

Apply Chapter Resources• Problem Solving

Practice and Problem Solving Workbook Interactive Online Edition

• Chapter Project

Project Teacher Support

After the Lesson

Reteach Chapter Resources• Reteach• Reading Strategies

Success for Every Learner

Review Interactive Answers and Solutions Solutions KeyKnow-It Notebook™

• Big Ideas• Chapter Review

Extend Chapter Resources• Challenge

986C Chapter 14

A211NLT_c14_Interleaf.indd 986CA211NLT_c14_Interleaf.indd 986C 8/26/09 8:17:25 PM8/26/09 8:17:25 PM

RCEE OPTIONS • RESOURCE OPTIONS • RESOURCE OPTIONS • RESOURCE OPTIONS

Lesson Tutorial Videos Multilingual Glossary Online InteractivitiesStarring Holt authors Ed Burger and Freddie Renfro! Live tutorials to support every lesson in Chapter 14.

Searchable glossary includes defi nitions in English, Spanish, Vietnamese, Chinese, Hmong, Korean, and 4 other languages.

Interactive tutorials provide visually engaging alternative opportunities to learn concepts and master skills.

Technology Highlights for Reaching All Learners

KEY: SE = Student Edition TE = Teacher’s Edition CRB = Chapter Resource Book Spanish version available Available online Available on CD- or DVD-ROM

Reaching All Learners

All LearnersLab ActivitiesPractice and Problem Solving Workbook Know-It Notebook

Special Needs StudentsPractice A .............................................................................CRB

Reteach.................................................................................CRB

Reading Strategies ...............................................................CRB

Are You Ready? ............................................................SE p. 987

Inclusion .......................................................................TE p. 996

IDEA Works!® Modifi ed Worksheets and TestsReady to Go On? Intervention Know-It NotebookOnline Interactivities Lesson Tutorial Videos

Developing LearnersPractice A .............................................................................CRB

Reteach.................................................................................CRB

Reading Strategies ...............................................................CRB

Are You Ready? ............................................................SE p. 987

Vocabulary Connections ..............................................SE p. 988

Questioning Strategies ...........................................................TE

Ready to Go On? Intervention Know-It NotebookHomework Help Online Online Interactivities Lesson Tutorial Videos

On-Level LearnersPractice B .............................................................................CRB

Problem Solving ..................................................................CRB



Ready to Go On? Intervention Know-It NotebookHomework Help Online Online Interactivities

Advanced LearnersPractice C .............................................................................CRB

Challenge .............................................................................CRB

Challenge Exercises ...............................................................SE

Reading and Writing Math Extend ..............................TE p. 989

Critical Thinking ........................................................TE p. 1016

Are You Ready? Enrichment

Ready To Go On? Enrichment

English Language LearnersReading Strategies ...............................................................CRB

Are You Ready? Vocabulary ........................................SE p. 987


Vocabulary Review ....................................................SE p. 1037

English Language Learners ..TE pp. 989, 993, 1018, 1046, 1047

Success for Every LearnerKnow-It NotebookSpanish Study Guide (Resumen y Repaso)Multilingual Glossary Lesson Tutorial Videos

Teaching tips to help all learners appear throughout the chapter. A few that target specific students are included in the lists below.

ENGLISH LANGUAGE LEARNERS

986D

A211NLT_c14_Interleaf.indd 986DA211NLT_c14_Interleaf.indd 986D 12/29/09 3:56:05 PM12/29/09 3:56:05 PM

RESOURCE OPTIONS • RESOURCE OPTIONS • RESOURCE OPTIONS • RESOURCEE O

Are You Ready? Ready to Go On? Test and Practice GeneratorAutomatically assess readiness and prescribe intervention for Chapter 14 prerequisite skills.

Automatically assess understanding of and prescribe intervention for Sections 14A and 14B.

Use Chapter 14 problem banks to create assessments and worksheets to print out or deliver online. Includes dynamic problems.

Ongoing Assessment

Technology Highlights for Assessment

Assessing Prior KnowledgeDetermine whether students have the prerequisite concepts and skills for success in Chapter 14.

Are You Ready? ............................SE p. 987

Warm Up .................................................................TE

Test PreparationProvide review and practice for Chapter 14 and standardized tests.

Multi-Step Test Prep ......................................SE pp. 1004, 1034

Study Guide: Review .....................................SE pp. 1036–1039

Test Tackler ....................................................SE pp. 1042–1043

Standardized Test Prep ..................................SE pp. 1044–1045

College Entrance Exam Practice ................................SE p. 1041

Countdown to Testing ........................ SE pp. C4–C27

IDEA Works!® Modifi ed Worksheets and Tests

Alternative AssessmentAssess students’ understanding of Chapter 14 concepts and combined problem-solving skills.

Chapter 14 Project .......................................................SE p. 986

Alternative Assessment ..........................................................TE

Performance Assessment ..................................................... AR

Portfolio Assessment ............................................................ AR

Lesson AssessmentProvide formative assessment for each lesson of Chapter 14.


Think and Discuss ...................................................................SE

Check It Out! Exercises ...........................................................SE

Write About It .........................................................................SE

Journal ....................................................................................TE

Lesson Quiz .............................................................TE

Alternative Assessment ..........................................................TE


Weekly AssessmentProvide formative assessment for each section of Chapter 14.

Multi-Step Test Prep ......................................SE pp. 1004, 1034

Ready to Go On? .............SE pp. 1005, 1035

Section Quizzes ..................................................................... AR

Test and Practice Generator ................. Teacher One Stop

Chapter AssessmentProvide summative assessment of Chapter 14 mastery.

Chapter 14 Test ..........................................................SE p. 1040

Chapter Test (Levels A, B, C) ................................................ AR• Multiple Choice • Free Response

Cumulative Test .................................................................... AR

Test and Practice Generator ................. Teacher One Stop


KEY: SE = Student Edition TE = Teacher’s Edition AR = Assessment Resources Spanish version available Available online Available on CD- or DVD-ROM

986E Chapter 14

A211NLT_c14_Interleaf.indd 986EA211NLT_c14_Interleaf.indd 986E 8/27/09 8:02:53 PM8/27/09 8:02:53 PM

RCEE OPTIONS • RESOURCE OPTIONS • RESOURCE OPTIONS • RESOURCE OPTIONS

Formal Assessment

Three levels (A, B, C) of multiple-choice and free-response chapter tests, along with a performance assessment, are available in the Assessment Resources.

A Chapter 14 Test

C Chapter 14 Test

A Chapter 14 Test

C Chapter 14 Test

B Chapter 14 Test

B Chapter 14 Test (continued)

MULTIPLE CHOICE

B Chapter 14 Test

B Chapter 14 Test (continued)

FREE RESPONSE

Chapter 14 Test

Chapter 14 Test (continued)

PERFORMANCE ASSESSMENT

986F

Name ________________________________________ Date ___________________ Class __________________

This is one of 47 sunspot drawings made by the Italian scholar Galileo Galilei in 1612.

Performance Assessment Trigonometric Graphs and Identities

Sunspots are dark areas that move across the surface of the sun. Astronomers believe sunspots represent regions that are cooler due to strong magnetic fields below the sun’s surface. Periods of high sunspot activity can create geomagnetic storms in space that disrupt communication satellites and can impair space missions.

Since 1749, astronomers have made daily records of the number of sunspots. Surprisingly, they have found that the number of sunspots over time varies in a roughly periodic cycle. So, today’s astronomers can predict periods of high and low sunspot activity.

For this activity, assume that the cycle of the number of sunspots can be modeled by a cosine function.

1. Between 1905 and 2000, the number of sunspots went through 9 cycles. What is the period of the sunspot cycle? ____________________

2. The average number of sunspots for any one year is recorded by the international sunspot number, a weighted average of many different daily observations. Between 1905 and 2000, the absolute maximum sunspot number was 190.2 and the absolute minimum was 1.4. What is the amplitude for the sunspot cycle? ____________________

3. The sunspot number was at a maximum in the year 2000. Let x represent the year number, and let y represent the sunspot number. Write a cosine function to model the sunspot cycle. (Hint: Don’t forget the phase shift and vertical shift.) ____________________

4. Use your function to approximate the number of sunspots that occurred in the year 1950. (For your information, the actual sunspot number for 1950 was 83.9.) ____________________

5. Use your function to approximate the number of sunspots that will occur in the year 2010. ____________________

6. If NASA wants to plan a space mission during the time of the lowest sunspot activity after 2010, what year should they plan for? ____________________

7. Predict the next two years after 2008 that will have 100 sunspots. ____________________

CHAPTER

14

Name ________________________________________ Date ___________________ Class __________________

Performance Assessment Teacher Support Trigonometric Graphs and Identities

PurposeThis performance task assesses the student’s ability to model real-world periodic behavior with a transformed trigonometric function.

Time30−45 minutes

GroupingIndividuals

Preparation Hints Review the characteristics of the graphs of sine and cosine functions, including frequency, period, amplitude, phase shift, and vertical shift.

Overview The student uses information about sunspot cycles to determine the period and amplitude of a cosine model. He/she puts the information together to write a cosine function, and then uses the function to solve prediction problems. Questions 4 and 5 require students to simply evaluate the function, while questions 6 and 7 require students to solve trigonometric equations either algebraically or with a graphing calculator.

Introduce the Task Hand out the assignment and allow students to read the introductory paragraph. You may want to explain that sunspot cycles are not strictly periodic because each cycle varies in length and amplitude, and there have been long periods of inactivity. As time allows before or after the performance task, have students research and graph yearly sunspot data and discuss the ways in which it is not strictly periodic. (Find actual data at the National Geophysical Data Center, www.ngdc.noaa.gov.)

Performance Indicators ______ Calculates the period of sunspot cycles given frequency. ______ Finds the amplitude of sunspot cycles given the minimum and

maximum. ______ Uses the period and amplitude to write a cosine function that

models sunspot cycles; incorporates reasonable transformations. ______ Solves trigonometric equations to find the number of sunspots in a

given year or when a given number of sunspots will occur.

Scoring Rubric Level 4: Student solves problems correctly and supports work with calculations. Level 3: Student solves problems but gives inadequate calculations. Level 2: Student solves some problems but gives no calculations to support work. Level 1: Student is not able to solve any of the problems.

CHAPTER

14

Create and customize Chapter 14 Tests. Instantly

generate multiple test versions, answer keys, and

practice versions of test items.

Test & Practice Generator

Modified chapter tests that address special learning needs are available in IDEA Works!®Modified Worksheets and Tests.

A211NLT_c14_Interleaf.indd 986FA211NLT_c14_Interleaf.indd 986F 9/11/09 5:00:47 PM9/11/09 5:00:47 PM

986 Chapter 14

Trigonometric Graphs and Identities

14A Exploring Trigonometric Graphs



14B Trigonometric Identities

Lab Graph Trigonometric Identities

14-3 Fundamental Trigonometric Identities




You can use graphs of trigonometric functions and trigonometric identi-ties to model the motion of a circle or a wheel in a variety of situations.

KEYWORD: MB7 ChProj

• Make connections among representa-tions of trigonometric functions.

• Use reasoning to solve problems involving trigonometric ratios.

A211NLS_c14opn_0986-0987.indd 986 8/10/09 10:32:35 AM

Interactivities Online ▼

Lessons 14-1 and 14-6

Lesson Tutorials Online

Lesson Tutorial Videos are available for EVERY example.

986 Chapter 14

S E C T I O N 14AExploring Trigonometric Graphs

On page 1004, students write and graph functions to model real-world

movement of tides.

Exercises designed to prepare students for success on the Multi-Step Test Prep can be found on pages 996 and 1002.

S E C T I O N 14BTrigonometric Identities

On page 1034, stu-dents apply trigono-metric identities and solve trigonometric

equations to model real-world motion of springs.

Exercises designed to prepare students for success on the Multi-Step Test Prep can be found on pages 1012, 1018, 1025, and 1032.

About the ProjectIn the Chapter Project, students write and graph functions to model the circular motion of a wheel. Students then investigate an unusual trigonometric identity.

Spinning Wheels

Project ResourcesAll project resources for teachers and students are provided online.

Materials:• ruler• graphing calculator

KEYWORD: MB7 ProjectTS

A211NLT_c14_co_986-989.indd 986A211NLT_c14_co_986-989.indd 986 8/31/09 2:46:28 PM8/31/09 2:46:28 PM

VocabularyMatch each term on the left with a definition on the right.

1. cosecant

2. cosine

3. hypotenuse

4. tangent of an angle

A. the ratio of the length of the leg adjacent the angle to the length of the opposite leg

B. the ratio of the length of the leg adjacent the angle to the length of the hypotenuse

C. the ratio of the length of the leg opposite the angle to the length of the adjacent leg

D. the ratio of the length of the hypotenuse to the length of the leg opposite the angle

E. the side opposite the right angle

Divide FractionsDivide.

5.3__5_5__2

6.3__4_1__2

7.- 3__

8_1__8

8.2__3_

- 7__4

Simplify Radical ExpressionsSimplify each expression.

9. √ � 6 · √ � 2 10. √ �� 100 - 64 11. √ � 9_

√ � 36 12. √ �� 4_

25

Multiply BinomialsMultiply.

13. (x + 11)(x + 7) 14. (y - 4)(y - 9)15. (2x - 3)(x + 5) 16. (k + 3)(3k - 3)

17. (4z - 4)(z + 1) 18. (y + 0.5)(y - 1)

Special Products of BinomialsMultiply.

19. (2x + 5) 2 20. (3y - 2) 2

21. (4x - 6)(4x + 6) 22. (2m + 1)(2m - 1)

23. (s + 7) 2 24. (-p + 4)(-p - 4)

Trigonometric Graphs and Identities 987

E

B

C

D

6_25

3_2

-3 -

8_21

2√

�

3 6

1_2 2_

5

x 2 + 18x + 77

2x2+ 7x - 15

4z2- 4

y 2 - 13y + 36

3k2+ 6k - 9

y2- 0.5y - 0.5

4x2+ 20x + 25

16x2- 36

s2+ 14s + 49

9y2- 12y + 4

4m2- 1

p 2 - 16

Are You Ready? 987

NOINTERVENE

YESENRICHDiagnose and Prescribe

ARE YOU READY? Intervention, Chapter 14

Prerequisite Skill Worksheets CD-ROM Online

Multiply and Divide Fractions Skill 47 Activity 47

Diagnose and Prescribe Online

Simplify Radical Expressions Skill 53 Activity 53

Multiply Binomials Skill 64 Activity 64

Special Products of Binomials Skill 65 Activity 65

ARE YOU READY? Enrichment, Chapter 14

WorksheetsCD-ROMOnline

OrganizerObjective: Assess students’ understanding of prerequisite skills.

Prerequisite Skills

Multiply and Divide Fractions

Simplify Radical Expressions

Multiply Binomials

Special Products of Binomials

Assessing Prior Knowledge

INTERVENTIONDiagnose and Prescribe

Use this page to determine whether intervention is necessary or whether enrichment is appropriate.

ResourcesAre You Ready? Intervention and Enrichment Worksheets

Are You Ready? CD-ROM

Are You Ready? Online

a207te_c14_co_986-989.indd 987a207te_c14_co_986-989.indd 987 12/20/05 12:07:21 PM12/20/05 12:07:21 PM

988 Chapter 14

Key Vocabulary/Vocabulario

amplitude amplitud

cycle ciclo

frequency frecuencia

period periodo

periodic function función periódica

phase shift cambio de fase

rotation matrix matriz de rotación

Vocabulary Connections

To become familiar with some of the vocabulary terms in the chapter, consider the following. You may refer to the chapter, the glossary, or a dictionary if you like.

1. What does the word amplify mean? What might the amplitude of a pendulum swing refer to?

2. What does a cycle refer to in everyday language? Give examples of cyclical phenomena.

3. Give an example of something that occurs frequently. To describe how often something occurs, like brushing our teeth, we can say “we brush twice a day.” Describe the frequency of your example.

4. What does period mean in everyday language? What might a periodic function refer to?

5. What result might you expect from using a rotation matrix ?

You will study • problems involving

trigonometric functions.

• factoring to solve trigonometric equations.

• trigonometric function models of real-world problems.

• solving trigonometric equations by using algebra and graphs.

You can use the skills in this chapter • in your future math classes,

particularly Calculus.

• in other classes, such as Physics, Biology, and Economics.

• outside of school to observe cyclical patterns and make conjectures.

In previous chapters, you • solved problems involving

triangles and trigonometric ratios.

• factored to solve quadratic equations.

• applied function models to solve real-world problems.

• solved equations by using algebra and graphs.

988 Chapter 14

Study Guide: Preview

OrganizerObjective: Help students organize the new concepts they will learn in Chapter 14.

Online EditionMultilingual Glossary

Resources

PuzzleView

Multilingual Glossary Online

KEYWORD: MB7 Glossary

Answers toVocabulary ConnectionsPossible answers:

1. to increase in amount or impor-tance; a measure of the size or amount of its swing

2. something that repeats itself or returns to the same point; The yearly cycle refers to the move-ment of Earth around the Sun or the sequence of the seasons. A washing machine cycle refers to a sequence of operations—filling, agitation, rinsing, and spinning.

3. blinking; 15 blinks per minute

4. a length of time; a function that repeats at regular intervals

5. a rotation of whatever is oper-ated on by the rotation matrix

C H A P T E R

14

A211NLT_c14_co_986-989.indd 988A211NLT_c14_co_986-989.indd 988 8/31/09 2:47:17 PM8/31/09 2:47:17 PM

Trigonometric Graphs and Identities 989

Study Strategy: Prepare for Your Final ExamMath is a cumulative subject, so your final exam will probably cover all of the material that you have learned from the beginning of the course. Preparation is essential for you to be successful on your final exam. It may help you to make a study timeline like the one below.

Try This

1. Create a timeline that you will use to study for your final exam.

1 week before the f ina l :1 week before the f ina l :• Take the practice exam and check it. • Take the practice exam and check it.

For each problem I miss, find two or For each problem I miss, find two or three similar ones and work those.three similar ones and work those.

• Work with a friend in the class to quiz • Work with a friend in the class to quiz each other on formulas, postulates, each other on formulas, postulates, and theorems from my list.and theorems from my list.

2 weeks before the f ina l :2 weeks before the f ina l :• Look at previous exams and homework to • Look at previous exams and homework to

determine areas I need to focus on; rework determine areas I need to focus on; rework problems that were incorrect or incomplete.problems that were incorrect or incomplete.

• Make a list of all formulas and theorems that • Make a list of all formulas and theorems that I need to know for the final.I need to know for the final.

• Create a practice exam using problems from • Create a practice exam using problems from the book that are similar to problems from the book that are similar to problems from each exam.each exam.

1 day before the f ina l :1 day before the f ina l :• Make sure I have pencils and a • Make sure I have pencils and a

calculator (check batteries!)calculator (check batteries!).

Reading and Writing Math 989

OrganizerObjective: Help students apply strategies to understand and retain key concepts.

Online Edition

Study Strategy:


Prepare for Your Final ExamDiscuss Students benefit from adequate preparation for a final exam. Planning a study strategy as early as possible can help reduce the anxiety of taking a final exam as it draws nearer and allow students to concentrate on course content.

Extend As students work through Chapter 14, have them create a timeline that they will use to study for the Chapter 14 Test. Have them consider how that timeline could be generalized into one that could be used for their final exam. In particular, have students think about how the schedule of the timeline would have to be altered for them to prepare for a test that covers all of the material in the course.

Answers to Try This 1. Check students’ work.

C H A P T E R

14

Reading Connection

Trig or Treatby Y.E.O. AdrianThis is a comprehensive and extremely useful guide to proving trigonometric identities. Identities are presented in the form of games for readers to play, with helpful suggestions on how to relate the identity to what has come before, how to analyze it, and, if needed, how to complete the proof. Activity The use of mnemonic devices in trigonometry usually begins and ends with SOHCAHTOA. Challenge students to come up with similar methods for remembering more complex formulations–the Law of Sines, for example, or the double-angle and half-angle identities.

A211NLT_c14_co_986-989.indd 989A211NLT_c14_co_986-989.indd 989 9/2/09 4:45:46 AM9/2/09 4:45:46 AM

Exploring Trigonometric Graphs

One-Minute Section PlannerLesson Lab Resources Materials

Lesson 14-1 Graphs of Sine and Cosine• Recognize and graph periodic and trigonometric functions.

□ SAT-10 □ NAEP □✔ ACT □ SAT □ SAT Subject Tests

Algebra Lab Activities14-1 Algebra Lab

Requiredgraphing calculator

Lesson 14-2 Graphs of Other Trigonometric Functions• Recognize and graph trigonometric functions.


Algebra Lab Activities14-2 Algebra Lab


MK = Manipulatives Kit

SECTION

14A

990A Chapter 14

a207te_c14_s&o_990A-990B.indd 990Aa207te_c14_s&o_990A-990B.indd 990A 12/20/05 12:05:11 PM12/20/05 12:05:11 PM

990B

PERIODIC FUNCTIONSLesson 14-1A periodic function is a function that repeats its output values in regular intervals. The regular intervals are called cycles and the length of one cycle is the periodof the function. Using function notation, a function f (x) has period P if f (x) = f (x + P) for all values of x. Consequently, for all integers n, f (x) = f (x + nP).

The graph shows a function f (x) with period P. For any particular value x 0 , the graph demonstrates how f ( x 0) = f ( x 0 + P) = f ( x 0 + 2P) = f ( x 0 + 3P), and so on.

This means that the graph of f (x) coincides with itself after a horizontal translation of P units.

THE GRAPHS OF THE SINE AND COSINE FUNCTIONSLesson 14-1The sine and cosine functions are periodic functions with period 2π. As students work with the graphs of these functions, they should begin to recognize several important characteristics of the graphs.

• The sine and cosine functions both have graphs that are smooth, nonlinear curves.

• The sine and cosine functions both have amplitude 1.• The graph of y = sin x passes through the origin.• The graphs of y = sin x and y = cos x have identical

shapes. The graph of y = cos x is a translation π__2 units

to the left of the graph of y = sin x.

This last point is especially important. In algebraic terms, it states that cos x = sin (x + π__

2) . This relation-ship can be verified by using the identity sin (A + B) =sin A cos B + cos A sin B, which students will learn in Lesson 14-4.

2-1

1

-2 -

y = sin x

y = cos x = sin (x + )2

Students should be aware that knowing the basic shapes of the graphs of the sine and cosine functions can help them remember the values of these func-tions. For example, students sometimes have difficulty remembering whether sin π__

2 = 0 or cos π__2 = 0. A quick

sketch of either graph should help them see that the latter equation is correct.

THE GRAPHS OF THE TANGENT AND COTANGENT FUNCTIONSLesson 14-2The tangent function has period π. This period is less than that of the sine or cosine function. Thus, the graph of the tangent function repeats more frequently on a given interval than the sine or cosine function. The function y = tan x is undefined at x = π__

2 + nπ, where n is an integer. Therefore, the graph of y = tan x has vertical asymptotes at these values of x.

The graph of y = tan x demonstrates an interesting feature of the function. As x approaches π__

2 from the left (i.e., x < π__

2 ), tan x approaches infinity; as x approaches π__2 from the right (i.e., x > π__

2 ), tan x approaches negative infinity. The same is true at each vertical asymptote.

Once students are familiar with the graph of the tan-gent function, they can sketch the graph of the cotan-gent function by recalling the reciprocal relationship of the functions. In particular, where the graph of y = tan x has an x-intercept, the graph of y = cot x has an asymptote. Where the graph of y = tan x has an asymptote, the graph of y = cot x has an x-intercept.

Math Background

A211NLT_c14_s&o_990A-990B.indd 990BA211NLT_c14_s&o_990A-990B.indd 990B 8/31/09 2:49:10 PM8/31/09 2:49:10 PM

990 Chapter 14 Trigonometric Graphs and Identities


Why learn this?Periodic phenomena such as sound waves can be modeled with trigonometric functions. (See Example 3.)

ObjectiveRecognize and graph periodic and trigonometric functions.

Vocabularyperiodic functioncycleperiodamplitudefrequencyphase shift

Periodic functions are functions that repeat exactly in regular intervals called cycles .The length of the cycle is called its period .Examine the graphs of the periodic function and nonperiodic function below. Notice that a cycle may begin at any point on the graph of a function.

Periodic Not Periodic

1E X A M P L E Identifying Periodic Functions

Identify whether each function is periodic. If the function is periodic, give the period.

A

The pattern repeats exactly, so the function is periodic. Identify the period by using the start and finish of one cycle.This function is periodic with period 2.


1a. 1b.

B

Although there is some symmetry, the pattern doesnot repeat exactly.This function is not periodic.

periodic; 3not periodic

Introduce1

Explorations and answers are provided in Alternate Openers: Explorations Transparencies.

KEYWORD: MB7 Resources

MotivateStudents have modeled many types of behavior that exhibit growth or decay. However, a great many behaviors are periodic, that is, they repeat over a specific period of time. Have students begin to consider periodic functions by brain-storming examples of periodic behavior. Possible answers: temperatures and weather, tides and waves, and orbits of planets

990 Chapter 14

14-1 OrganizerPacing: Traditional 1 day

Block 1 __ 2 day

Objectives: Recognize and graph periodic and trigonometric functions.

Algebra LabIn Algebra Lab Activities

Online EditionGraphing Calculator, TutorialVideos, Interactivity

Warm UpEvaluate.

1. sin π _ 6 0.5 2. cos π

_ 2 0

3. cos π _ 3 0.5 4. sin π

_ 4

√ �

2 _

2

Find the measure of the reference angle for each given angle.

5. 14 5 ◦ 3 5 ◦ 6. 31 7 ◦ 4 3 ◦

Also available on transparency

Student: I think I’ve done enough problems about frequency.

Teacher: How do you know?

Student: My brain hertz.

A211NLT_c14_l1_990-997.indd 990A211NLT_c14_l1_990-997.indd 990 8/31/09 2:51:25 PM8/31/09 2:51:25 PM

14-1 Graphs of Sine and Cosine 991

The trigonometric functions that you studied in Chapter 13 are periodic. You can graph the function f (x) = sin x on the coordinate plane by using y-values from points on the unit circle where the independent variable x represents the angle θ in standard position.

x(= θ) y

π

_3

√

�

3_2

5π

_6

1_2

4π

_3

-

√

�

3_2

11π

_6

-

1_2

Similarly, the function f (x) = cos x can be graphed on the coordinate plane by using x-values from points on the unit circle.

The amplitude of sine and cosine functions is half of the difference between the maximum and minimum values of the function. The amplitude is always positive.

FUNCTION y = sin x y = cos x

GRAPH

DOMAIN ⎧ ⎨

⎩ x⎥ x � �

⎫ ⎬

⎭

⎧ ⎨

⎩ x⎥ x � �

⎫

⎬

⎭

RANGE ⎧ ⎨

⎩ y⎥ -1 ≤ y ≤ 1

⎫

⎬

⎭

⎧

⎨

⎩ y⎥ -1 ≤ y ≤ 1

⎫

⎬

⎭

PERIOD 2π 2π

AMPLITUDE 1 1

Characteristics of the Graphs of Sine and Cosine

You can use the parent functions to graph transformations y = a sin bx and y = a cos bx. Recall that a indicates a vertical stretch (⎪a⎥ > 1) or compression (0 < ⎪a⎥ < 1) , which changes the amplitude. If a is less than 0, the graph is reflected across the x-axis. The value of b indicates a horizontal stretch or compression, which changes the period.

For the graphs of y = a sin bx or y = a cos bx where a ≠ 0 and x is in radians,

• the amplitude is ⎪a⎥.

• the period is 2π

_⎪b⎥

.

Transformations of Sine and Cosine Graphs

The graph of the sine function passes through the origin.The graph of the cosine function has y-intercept 1.

Lesson 14-1 991

Example 1


A.

yes; π

B.

no

Additional Examples

INTERVENTION Questioning Strategies

EXAMPLE 1

• How much of the curve do you need to see before you can con-clude that it is periodic?

• How can you determine the period?

Kinesthetic Have students trace the circumference of the unit circle with a finger

and note where the y-coordinates are positive, negative, increasing, and decreasing and then trace along the graph of sine to make the connec-tion with the y-coordinate. Repeat for cosine.

Guided InstructionIntroduce the graph of sine and cosine by first drawing the unit circle and choosing values with which to plot points. Be sure to include the quadrantal angle values, as these correspond to maxima, minima, and x-intercepts.

Discuss how the types of transformations that students learned in previous chapters are applied to graphs of sine and cosine.

Teach2

Through Multiple Representations

Have students work in groups to create large drawings of the unit circle to post in the classroom. Ask students to discuss the connection between the sine and the cosine functions and coordinates of points on the unit circle.

A211NLT_c14_l1_990-997.indd 991A211NLT_c14_l1_990-997.indd 991 9/1/09 1:35:06 AM9/1/09 1:35:06 AM


2E X A M P L E Stretching or Compressing Sine and Cosine Functions

Using f (x) = sin x as a guide, graph the function g (x) = 3 sin 2x. Identify the amplitude and period.

Step 1 Identify the amplitude and period.

Because a = 3, the amplitude is ⎪a⎥ = ⎪3⎥ = 3.

Because b = 2, the period is 2π

_⎪b⎥

= 2π

_⎪2⎥

= π.

Step 2 Graph.

The curve is vertically stretched by a factor of 3 and horizontally compressed by a factor of 1__

2 .

The parent function f has x-intercepts at multiples of π and g has x-intercepts at multiples of π

__2 .

The maximum value of g is 3, and the minimum value is -3.

2. Using f (x) = cos x as a guide, graph the function h(x) = 1__

3 cos 2x. Identify the amplitude and period.

Sine and cosine functions can be used to model real-world phenomena, such as sound waves. Different sounds create different waves. One way to distinguish sounds is to measure frequency.Frequency is the number of cycles in a given

unit of time, so it is the reciprocal of the period of a function.

Hertz (Hz) is the standard measure of frequency and represents one cycle per second. For example, the sound wave made by a tuning fork for middle A has a frequency of 440 Hz. This means that the wave repeats 440 times in 1 second.

3E X A M P L E Sound Application

Use a sine function to graph a sound wave with a period of 0.005 second and an amplitude of 4 cm. Find the frequency in hertz for this sound wave.

Use a horizontal scale where one unit represents 0.001 second. The period tells you that it takes0.005 seconds to complete one full cycle. The maximum and minimum values are given by the amplitude.

frequency = 1_period

= 1_0.005

= 200 Hz

The frequency of the sound wave is 200 Hz.

3. Use a sine function to graph a sound wave with a period of 0.004 second and an amplitude of 3 cm. Find the frequency in hertz for this sound wave.

2.

amplitude: 1_3 ; period: π

3.

frequency: 250 Hz

992 Chapter 14

Example 2

Using f (x) = sin x as a guide, graph the function

g (x) = 1 _ 2 sin ( 1 _

2 x) . Identify

the amplitude and period.

amplitude: 1 _ 2 ; period: 4π

Example 3

Use a sine function to graph a sound wave with a period of 0.002 s and an amplitude of 3 cm. Find the frequency in hertz for this sound wave.

frequency: 500 Hz

Additional Examples


EXAMPLE 2

• How do horizontal or vertical com-pressions of sine and cosine graphs compare to these transformations of other functions?

EXAMPLE 3

• What are the frequencies of the parent curves f (x) = sin x and g (x) = cos x?

Science Link You may wish to have students reinforce their understanding of amplitude and

frequency by investigating the meanings of these terms in broadcast media. Radio signals are broadcast on AM (amplitude modulation) and FM (frequency modula-tion) bands.



Sine and cosine can also be translated as y = sin (x - h) + k and y = cos (x - h ) + k. Recall that a vertical translation by k units moves the graph up (k > 0) or down (k < 0) .

A phase shift is a horizontal translation of a periodic function. A phase shift of h units moves the graph left (h < 0) or right (h > 0) .

4E X A M P L E Identifying Phase Shifts for Sine and Cosine Functions

Using f (x) = sin x as a guide, graph g (x) = sin (

x + π __2 )

. Identify the x-intercepts and phase shift.

Step 1 Identify the amplitude and period.

Amplitude is ⎪a⎥ = ⎪1⎥ = 1.

The period is 2π

_⎪b⎥

= 2π

_⎪1⎥

= 2π.

Step 2 Identify the phase shift.

x + π _2

= x - (

-

π

_2 )

Identify h.

Because h = -

π

_2

, the phase shift is π _2

radians to the left.

All x-intercepts, maxima, and minima of f (x) are shifted π _2

units to the left.

Step 3 Identify the x-intercepts.

The first x-intercept occurs at -

π

__2

. Because sin x has two x-intercepts in each period of 2π, the x-intercepts occur at - π

__2 + nπ, where n is

an integer.

Step 4 Identify the maximum and minimum values.

The maximum and minimum values occur between the x-intercepts. The maxima occur at 2πn and have a value of 1. The minima occur at π + 2πn and have a value of -1.

Step 5 Graph using all of the information about the function.

4. Using f (x) = cos x as a guide, graph g (x) = cos (x - π) . Identify the x-intercepts and phase shift.

You can combine the transformations of trigonometric functions. Use the values of a, b, h, and k to identify the important features of a sine or cosine function.

Amplitude

Period Vertical shift

Phase shift

The repeating pattern is maximum, intercept, minimum, intercept,…. So intercepts occur twice as often as maximum or minimum values.

4.

x-intercepts: π _2 + πn;

phase shift: π right

Lesson 14-1 993

It is worth noting that an expression such as sin π_

15(t - 7.5) is equivalent

to sin ⎧ ⎨

⎩ π _ 15

(t - 7.5) ⎫ ⎬

⎭ and not to

(sin π _ 15

) · (t - 7.5) . The use of the

extra brackets may be cumbersome, but it serves to eliminate confusion.

Example 4

Using f (x) = sin x as a guide,

graph g (x) = sin (x – π _ 4 ) .

Identify the x-intercepts and phase shift.

x-intercepts: π _ 4 + nπ; phase

shift: π _ 4 units to the right

Additional Examples


EXAMPLE 4

• How could sine be shifted so that its graph looks like that of cosine?

Reading Math Point out to students the word sinusoidal. Discuss with

them the meaning “of, relating to, or varying according to a sine curve.” Also note that a cosine function, because it is shaped similar to a sine function, is sinusoidal.




5E X A M P L E Entertainment Application

The Ferris wheel at the landmark Navy Pier in Chicago takes 7 minutes to make one full rotation. The height H in feet above the ground of one of the six-person gondolas can be modeled by H (t) = 70 sin 2π

___

7 (t - 1.75)

+ 80, where t is time in minutes.

a. Graph the height of a cabin for two complete periods.

H(t) = 70 sin 2π

_7

(t - 1.75) + 80 a = 70, b = 2π

___7

, h = 1.75, k = 80

Step 1 Identify the important features of the graph.

Amplitude: 70

Period: 2π

_⎪b⎥

= 2π

_

⎪ 2π

___7 ⎥

= 7

The period is equal to the time required for one full rotation.

Phase shift: 1.75 minutes right

Vertical shift: 80

There are no x-intercepts.

Maxima: 80 + 70 = 150 at 3.5 and 10.5

Minima: 80 - 70 = 10 at 0, 7, and 14


b. What is the maximum height of a cabin?

The maximum height is 80 + 70 = 150 feet above the ground.

5. What if...? Suppose that the height H of a Ferris wheel can be modeled by H(t) = -16 cos π

__45t + 24, where t is the time

in seconds.

a. Graph the height of a cabin for two complete periods.

b. What is the maximum height of a cabin?

THINK AND DISCUSS 1. DESCRIBE how the frequency and period of a periodic function are

related. How does this apply to the graph of f (x) = cos x?

2. EXPLAIN how the maxima and minima are related to the amplitude and period of sine and cosine functions.

3. GET ORGANIZED Copy and complete the graphic organizer. For each type of transformation, give an example and state the period.

40 ft

5a.

994 Chapter 14

Assess After the Lesson14-1 Lesson Quiz, TE p. 997

Alternative Assessment, TE p. 997

Monitor During the LessonCheck It Out! Exercises, SE pp. 990–994Questioning Strategies, TE pp. 991–994

Diagnose Before the Lesson14-1 Warm Up, TE p. 990

and INTERVENTIONSummarizeThe graphs of f (x) = sin x and g (x) = cos x are periodic, which is not sur-prising because they represent the move-ment of a point around a circle. They may be transformed in the same way as other previously studied graphs. In the function f (x) = a sin b (x - h) + k, a and b are used to identify the amplitude and the period. A horizontal translation of a sine or cosine function is called a phase shift.

Close3 Answers to Think and Discuss 1. The frequency of a periodic function is

the reciprocal of the period. The period of f (x) = cos x is 2π, and the frequency

is 1 _ 2π

.

2. The period of sine and cosine func-tions tells how often the maxima and minima occur. The amplitude affects the value of the maxima and minima.

3. See p. A14.

Example 5

The number of people, in thousands, employed in a resort town can be modeled by

g (x) = 1.5 sin π _ 6 (x + 2) + 5.2,

where x is the month of the year.

A. Graph the number of people employed in the town for one complete period.

B. What is the maximum number of people employed? 6700

Additional Examples


EXAMPLE 5

• How can you find the radius of the Ferris wheel from the function that describes the wheel’s height?



ExercisesExercises14-1KEYWORD: MB7 14-1

KEYWORD: MB7 Parent

GUIDED PRACTICE 1. Vocabulary Periodic functions repeat in regular intervals called −−− ? .

(cycles or periods)

SEE EXAMPLE 1 p. 990


2. 3.


Using f (x) = sin x or f (x) = cos x as a guide, graph each function. Identify the amplitude and the period.

4. f (x) = 2 sin 1_2

x 5. h(x) = 1_4

cos x 6. k(x) = sin πx


7. Sound Use a sine function to graph a sound wave with a period of 0.01 second and an amplitude of 6 in. Find the frequency in hertz for this sound wave.


Using f (x) = sin x or f (x) = cos x as a guide, graph each function. Identify the x-intercepts and the phase shift.

8. f (x) = sin (x + 3π

_2 ) 9. g(x) = cos (x - π

_2 ) 10. h(x) = sin (x - π

_4 )


11. Recreation The height H in feet above the ground of the seat of a playground swing can be modeled by H(θ) = -4 cos θ + 6, where θ is the angle that the swing makes with a vertical extended to the ground. Graph the height of a swing’s seat for 0° ≤ θ ≤ 90°. How high is the swing when θ = 60°?

PRACTICE AND PROBLEM SOLVING

For See Exercises Example

12–13 1 14–17 2 18 3 19–22 4 23 5

Independent Practice Identify whether each function is periodic. If the function is periodic, give the period.

12. 13.

Using f (x) = sin x or f (x) = cos x as a guide, graph each function. Identify the amplitude and period.

14. f (x) = 4 cos x 15. g(x) = 3_2

sin x 16. g(x) = -cos 4x 17. j(x) = 6 sin 1_3

x

18. Sound Use a sine function to graph a sound wave with a period of 0.025 seconds and an amplitude of 5 in. Find the frequency in hertz for this sound wave.

Skills Practice p. S30Application Practice p. S45

Extra Practice

cycles

periodic; 5 not periodic

periodic; 2πnot periodic

4 ft

100 Hz

40 Hz

7.

8.

x-intercepts: nπ;

phase shift: 3π

_ 2 left

9.

x-intercepts: nπ;

phase shift: π _ 2 right

10.

x-intercepts: π _ 4 + πn;

phase shift: π _ 4 right

11, 14–18. For graphs, see p. A50.

14. amplitude: 4; period: 2π

15. amplitude: 3 _ 2 ; period: 2π

16. amplitude: 1; period: π _ 2


Lesson 14-1 995

ExercisesExercises


14–1

Assignment Guide

Assign Guided Practice exercises as necessary.

If you finished Examples 1–2 Basic 12–17, 29–32 Average 12–17, 29–32 Advanced 12–17, 29–32, 44

If you finished Examples 1–5 Basic 12–37, 39–43, 48–53 Average 12–44, 48–53 Advanced 12–53

Homework Quick CheckQuickly check key concepts.Exercises: 12, 14, 20, 28

Answers 4.

amplitude: 2; period: 4π

5.

amplitude: 1 _ 4 ; period: 2π

6.

amplitude: 1; period: 2

a207te_c14_l1_990-997.indd 995a207te_c14_l1_990-997.indd 995 6/7/06 10:38:27 AM6/7/06 10:38:27 AM


Using f (x) = sin x or f (x) = cos x as a guide, graph each function. Identify the x-intercepts and phase shift.

19. f (x) = sin (x + π) 20. h(x) = cos (x - 3π)

21. g(x) = sin (x + 3π_4 ) 22. j(x) = cos (x + π_

4 ) 23. Oceanography The depth d in feet of the water in a bay at any time is given by

d(t) = 3__2 sin (5π

___31

t) + 23, where t is the time in hours. Graph the depth of the water.

What are the maximum and minimum depths of the water?

24. Medicine The figure shows a normal adult electrocardiogram, known as an EKG. Each cycle in the EKG represents one heartbeat.

a. What is the period of one heartbeat?

b. The pulse rate is the number of beats in one minute. What is the pulse rate indicated by the EKG?

c. What is the frequency of the EKG?

d. How does the pulse rate relate to the frequency in hertz?

Determine the amplitude and period for each function. Then describe the transformation from its parent function.

25. f (x) = sin (x + π_4 ) - 1 26. h(x) = 3_

4 cos π_

4x

27. h(x) = cos (2πx) - 2 28. j(x) = -3 sin 3x

Estimation Use a graph of sine or cosine to estimate each value.

29. sin 160° 30. cos 50° 31. sin 15° 32. cos 95°

Write both a sine and a cosine function for each set of conditions.

33. amplitude of 6, period of π 34. amplitude of 1_4

, phase shift of 2_3

π left

Write both a sine and a cosine function that could be used to represent each graph.

35. 36.

37. This problem will prepare you for the Multi-Step Test Prep on page 1004.

The tide in a bay has a maximum height of 3 m and a minimum height of 0 m. It takes 6.1 hours for the tide to go out and another 6.1 hours for it to come back in. The height of the tide h is modeled as a function of time t.

a. What are the period and amplitude of h? What are the maximum and minimum values?

b. Assume that high tide occurs at t = 0. What are h(0) and h(6.1) ?

c. Write h in the form h(t) = a cos bt + k.

An EKG measures the electrical signals that control the rhythm of a beating heart. EKGs are used to diagnose and monitor heart disease.

Medicine

≈ 0.8 s

75 beats/min1.25 Hz

f (x) = 6 sin 2x ; f (x) = 6 cos 2x

period: 12.2; amplitude: 1.5; max.: 3; min.: 0

b. h(0)= 3; h(6.1)

= 0h(t) = 1.5 cos 2π

_12.2

t + 1.5

≈ 0.3 ≈ 0.6 ≈ 0.25 ≈ -0.1

996 Chapter 14

Inclusion In Exercise 23, to find the minimum and maximum values of the

sine function, remind students that the range of sin (x) is from -1 to 1. Then they simply note that the extreme values will be 3 _ 2 (-1) + 23 and 3 _

2 (1) + 23.

Exercise 37 involves graphing and inter-preting sine func-

tions. This exercise prepares students for the Multi-Step Test Prep on page 1004.

Answers19–22. See p. A50.

23.

max.: 24.5 ft; min.: 21.5 ft

24d. The pulse rate is measured in beats per minute and the fre-quency is measured in cycles, or beats, per second. They both measure the same quantity.

25–28. See p. A50.

34. f (x) = 1 _ 4 sin

(

x + 2π

_ 3

)

35. f (x) = -4 sin 2x ;

g (x) = 4 cos 2 (

x + π _ 4 )

36. f (x) = -

1 _ 4 sin 1 _

2 x + 1;

g (x) = 1 _ 4 cos 1 _

2 (x - π) + 1

14-1 RETEACH14-1 READING STRATEGIES

14-1 PRACTICE C

14-1 PRACTICE B

14-1 PRACTICE A

a207te_c14_l1_990-997.indd 996a207te_c14_l1_990-997.indd 996 12/27/05 5:21:46 PM12/27/05 5:21:46 PM


38. Critical Thinking Given the amplitude and period of a sine function, can you find its maximum and minimum values and their corresponding x-values? If not, what information do you need and how would you use it?

39. Write About It What happens to the period of f (x) = sin bθ when b > 1? b < 1? Explain.

40. Which trigonometric function best matches the graph?

y = 1_2

sin x y = 1_2

sin 2x

y = 2 sin x y = 2 sin 1_2

x

41. What is the amplitude for y = -4 cos 3πx?

-4 4

3 3π

42. Based on the graphs, what is the relationship between f and g?

f has twice the amplitude of g.

f has twice the period of g.

f has twice the frequency of g.

f has twice the cycle of g.

43. Short Response Using y = sin x as a guide, graph y = -4 sin 2 (x - π) on the interval [0, 2π] and describe the transformations.

CHALLENGE AND EXTEND 44. Graph f (x) = Si n -1 x and g (x) = Co s -1 x. (Hint: Use what you learned about graphs of

inverse functions in Lesson 9-5 and inverse trigonometric functions in Lesson 13-4.)

Consider the functions f (θ) = 1__2 sin θ and g (θ) = 2 cos θ for 0° ≤ θ ≤ 360°.

45. On the same set of coordinate axes, graph f (θ) and g (θ) .

46. What are the approximate coordinates of the points of intersection of f (θ) and g (θ) ?

47. When is f (θ) > g (θ) ?

SPIRAL REVIEWUse interval notation to represent each set of numbers. (Lesson 1-1)

48. -7 < x ≤ 5 49. x ≤ -2 or 1 ≤ x < 13 50. 0 ≤ x ≤ 9

51. Flowers Adam has $100 to purchase a combination of roses, lilies, and carnations. Roses cost $6 each, lilies cost $2 each, and carnations cost $4 each. (Lesson 3-5)

a. Write a linear equation in three variables to represent this situation.

b. Complete the table.

Use the given measurements to solve �ABC. Round to the nearest tenth. (Lesson 13-6)

52. b = 20, c = 11, m∠A = 165° 53. a = 11.9, b = 14.7, c = 26.1

Roses 6 3 7

Lilies 8 5

Carnations 11 15 13

76° < θ < 256°

(-7, 5]

[0, 9]49. (-∞, -2] or[1, 13)

6r + 2� + 4c = 100

4

10 3

18

a = 30.8; m∠B = 9.7°; m∠C = 5.3°

m∠A = 10°; m∠B = 12.4°; m∠C = 157.6

46. (256°, -0.485)

and (76°, 0.485)

43.

phase shift π right, horizontal compression, vertical stretch, and reflection across the x-axis; amplitude: 4; period: π;x-intercepts: 0, π

_2 ,

π, 3π

_2 , and 2π;

max.: 4; min.: -4

a207se_c14l01_0990_0997.indd 997 2/13/07 4:10:56 PM

Lesson 14-1 997

If students have difficulty with Exercise 40, they can

test a key point on the curve, such as (π, 2) , which fits only choice D.

Answers 38. No; possible answer: you would

also need to know whether there is a phase shift or vertical shift. The phase shift and period give the location of the maxima and minima. The amplitude and verti-cal shift give the values of the maxima and minima.

39. The period decreases for b > 1 and increases for b < 1 because

the period is given by 2π

_ b .

44, 45. See p. A50.

JournalHave students describe how to iden-tify the period and the phase shift of a sine or cosine function.

Have students explain and give examples of the roles of a, b, h, and k for f (x) = a cos b (x - h) + k.

14-1

1. Using f (x) = cos x as a guide, graph g (x) = 1.5cos 2x.

Suppose that the height, in feet, above ground of one of the cabins of a Ferris wheel at t minutes is modeled by

H (t) = 30 sin ( 2π

_ 3 (t - 3 _

4 ) ) + 36.

2. Graph the height of the cabin for two complete revolutions.

3. What is the radius of this Ferris wheel? 30 ft


14-1 CHALLENGE14-1 PROBLEM SOLVING



The tangent and cotangent functions can be graphed on the coordinate plane. The tangent function is undefined when θ = π

__2 + πn, where n is an integer. The

cotangent function is undefined when θ = πn. These values are excluded from the domain and are represented by vertical asymptotes on the graph. Because tangent and cotangent have no maximum or minimum values, amplitude is undefined.

To graph tangent and cotangent, let the variable x represent the angle θ in standard position.

FUNCTION y = tan x y = cot x

GRAPH

DOMAIN

⎧

⎨

⎩ x⎥ x ≠ π

_2

+ πn,

where n is an integer ⎫ ⎬

⎭

⎧

⎨

⎩ x⎥ x ≠ πn,


⎭

RANGE ⎧ ⎨

⎩ y⎥ -∞ < y < ∞

⎫

⎬

⎭

⎧ ⎨

⎩ y⎥ -∞ < y < ∞

⎫

⎬

⎭

PERIOD π π

AMPLITUDE undefined undefined

Characteristics of the Graphs of Tangent and Cotangent

Like sine and cosine, you can transform the tangent function.

For the graph of y = a tan bx, where a ≠ 0 and x is in radians,

• the period is π

_⎪b⎥

. • the asymptotes are located at x = π

_2⎪b⎥

+ πn_⎪b⎥

,where n is an integer.

Transformations of Tangent Graphs


ObjectiveRecognize and graph trigonometric functions.

Why learn this?You can use the graphs of reciprocal trigonometric functions to model rotating objects such as lights. (See Exercise 25.)

A211NLS_c14l02_0998_1003.indd 999 8/14/09 9:50:49 AM

998 Chapter 14

Introduce1



MotivateIn the previous lesson, students investigated the sine and cosine functions, which can be used to model certain types of periodic behavior. The other four trigonometric functions also have graphs that are periodic and based on the unit circle values. The graphs of sine and cosine can be used to sketch graphs of tangent, cotangent, secant, and cosecant.

14-2 OrganizerPacing: Traditional 2 days

Block 1 day

Objectives: Recognize and graph trigonometric functions.

Algebra LabIn Algebra Lab Activities

Online EditionGraphing Calculator, Tutorial Videos

Warm UpIf sin A = 3 _

5 , evaluate:

1. cos A 4 _ 5 2. tan A 3 _

4

3. cot A 4 _ 3 4. sec A 5 _

4

5. csc A 5 _ 3


Q: How do you know when trig class is almost over?

A: When time starts getting asymptotically close to the end of the period.


14-2 Graphs of Other Trigonometric Functions 999

1E X A M P L E Transforming Tangent Functions

Using f (x) = tan x as a guide, graph g (x) = tan 2x. Identify the period, x-intercepts, and asymptotes.

Step 1 Identify the period.

Because b = 2, the period is π

_⎪b⎥

= π

_⎪2⎥

= π _2

.


An x-intercept occurs at x = 0. Because the period is π _2

, the x-intercepts occur at π

_2

n, where n is an integer.

Step 3 Identify the asymptotes.

Because b = 2, the asymptotes occur at

x = π

_2⎪2⎥

+ πn_⎪2⎥

, or x = π _4

+ πn_2

.


1. Using f (x) = tan x as a guide, graph g(x) = 3 tan 1_2

x. Identify the period, x-intercepts, and asymptotes.

For the graph of y = a cot bx, where a ≠ 0 and x is in radians,

• the period is π

_⎪b⎥

. • the asymptotes are located at x = πn_⎪b⎥

,where n is an integer.

Transformations of Cotangent Graphs

2E X A M P L E Graphing the Cotangent Function

Using f (x) = cot x as a guide, graph g(x) = cot 0.5x. Identify the period,x-intercepts, and asymptotes.


Because b = 0.5, the period is π

_⎪b⎥

= π

_⎪0.5⎥

= 2π.


An x-intercept occurs at x = π. Because the period is 2π, the x-intercepts occur at x = π + 2πn, where n is an integer.


Because b = 0.5, the asymptotes occur at

x = πn

_⎪0.5⎥

= 2πn.


g(x) = tan2x

1.

period: 2π; x-intercepts:2πn; asymptotes: π + 2πn

A211NLS_c14l02_0998_1003.indd 999 8/14/09 9:50:49 AM

Lesson 14-2 999

Students may forget to include nega-tive values of n when determining the domain of tangent and cotan-gent functions. Remind students that n is an integer, not a natural number.

Example 1

Using f (x) = tan x as a guide,

graph g (x) = 1 _ 2 tan

(

1 _ 3 x

)

.

Identify the period, x-intercepts, and asymptotes.

period: 3π; x-intercepts: 3πn;

asymptotes: x = 3π

_ 2 + 3πn

Additional Examples


EXAMPLE 1

• What determines the location of the asymptotes on the graph of a tangent function?

Guided InstructionIntroduce the graphs of tangent and cotan-gent by first drawing the unit circle and choosing values with which to plot points. Be sure to include the angle values that correspond to x-intercepts and asymptotes.

Teach2

Through Modeling

Have students plot the sine curve, then have them carefully construct the cosecant curve by taking a series of points from the sine curve and plotting the reciprocal of each point.



2. Using f (x) = cot x as a guide, graph g (x) = -cot 2x. Identify the period, x-intercepts, and asymptotes.

Recall that sec θ = 1____cos θ

. So, secant is undefined where cosine equals zero and the graph will have vertical asymptotes at those locations. Secant will also have the same period as cosine. Sine and cosecant have a similar relationship. Because secant and cosecant have no absolute maxima or minima, amplitude is undefined.

FUNCTION y = sec x y = csc x

GRAPH

DOMAIN ⎧ ⎨

⎩ x⎥ x ≠ π

_2

+ πn,


⎭

⎧ ⎨

⎩ x⎥ x ≠ πn,


⎭

RANGE ⎧ ⎨

⎩ y⎥ y ≤ -1, or y ≥ 1

⎫ ⎬

⎭

⎧ ⎨

⎩ y⎥ y ≤ -1, or y ≥ 1

⎫

⎬

⎭

PERIOD 2π 2π

AMPLITUDE undefined undefined

Characteristics of the Graphs of Secant and Cosecant

You can graph transformations of secant and cosecant by using what you learned in Lesson 14-1 about transformations of graphs of cosine and sine.

3E X A M P L E Graphing Secant and Cosecant Functions

Using f (x) = cos x as a guide, graph g (x) = sec 2x. Identify the period and asymptotes.


Because sec 2x is the reciprocal of cos 2x, the graphs will have the same period.

Because b = 2 for cos 2x, the period is 2π

_⎪b⎥

= 2π

_⎪2⎥

= π.


Because the period is π, the asymptotes occur at x = π

_2⎪2⎥

+ π_⎪2⎥

n = π _4

+ π_2

n,

where n is an integer.


3. Using f (x) = sin x as a guide, graph g (x) = 2 csc x. Identify the period and asymptotes.

2.

period: π _2 ;

x-intercepts: π _4 + π

_2

n;

asymptotes: π _2

n

3.

period: 2π;asymptotes: πn

1000 Chapter 14

Example 2

Using f (x) = cot x as a guide,

graph g (x) = 1 _ 2 cot 3x. Identify

the period, x-intercepts, and asymptotes.

period: π _ 3 ; x-intercepts: π

_ 6 + π

_ 3 n;

asymptotes: x = π _ 3 n

Example 3

Using f (x) = cos x as a guide,

graph g (x) = 1 _ 2 sec ( 1 _

2 x) .

Identify the period and asymptotes.

period: 4π; asymptotes: x = π + 2πn

Additional Examples


EXAMPLE 2

• Compare the graph of cotangent to the graph of tangent.

EXAMPLE 3

• What determines the location of the asymptotes on the graph of a secant or cosecant function?





and INTERVENTION SummarizeRemind students that the cotangent, secant, and cosecant functions are, respectively, reciprocals of the tangent, cosine, and sine functions. They are subject to all of the same transformations as any other curve.

Close3



14-2KEYWORD: MB7 14-2

GUIDED PRACTICESEE EXAMPLE 1 p. 999

Using f (x) = tan x as a guide, graph each function. Identify the period, x-intercepts, and asymptotes.

1. k(x) = 2 tan (3x) 2. g(x) = tan 1_4

x 3. h(x) = tan 2πx


Using f (x) = cot x as a guide, graph each function. Identify the period, x-intercepts, and asymptotes.

4. j(x) = 0.25 cot x 5. p(x) = cot 2x 6. g (x) = 3_2

cot x


Using f (x) = cos x or f (x) = sin x as a guide, graph each function. Identify the period and asymptotes.

7. g(x) = 1_2

sec x 8. q(x) = sec 4x 9. h(x) = 3 csc x



10–13 1 14–16 2 17–19 3

Independent Practice Using f (x) = tan x as a guide, graph each function. Identify the period, x-intercepts, and asymptotes.

10. p(x) = tan 3_2

x 11. g(x) = tan (x + π _4 )

12. h(x) = 1_2

tan 4x 13. j(x) = -2 tan π _2

x


14. h(x) = 4 cot x 15. g(x) = cot 1_4

x 16. j(x) = 0.1 cot x


17. g(x) = -sec x 18. k(x) = 1_2

csc x 19. h(x) = csc (-x)

KEYWORD: MB7 Parent

THINK AND DISCUSS 1. EXPLAIN why f (x) = sin x can be used to graph g (x) = csc x.

2. EXPLAIN how the zeros of the cosine function relate to the vertical asymptotes of the graph of the tangent function.

3. GET ORGANIZEDCopy and complete the graphic organizer.

ExercisesExercises


Extra Practice

Lesson 14-2 1001


Answers to Think and DiscussPossible answers:

1. Cosecant is the reciprocal of the sine function, so the graphs are related.

2. By using the unit circle, you can see that tangent is undefined at the same values where cosine is equal to 0. An undefined value corresponds to an asymptote, so the zeros of cosine correspond to the asymptotes of tangent.

3. See p. A14.

Assignment Guide


Basic 10-33, 35-40, 51-59 Average 10-43, 51-59 Advanced 10-59


Answers

1–19. See p. A50.

14-2 ExercisesExercises

I have my advanced students read a chapter or excerpts about the development of the six trigonometric functions over time from the book Trigonometric Delights by Eli Maor. Students are usually quite surprised to find out that the trigonometric functions were not all discovered at one time.

Mary Lane BlomquistKewaskum, WI

Teacher to Teacher



The Greek gnomon was a tall staff, but gnomon is also the part of a sundial that casts a shadow. Based on the variation of shadows at high noon, a gnomon can be used to determine the day of the year, in addition to the time of day.

Math History

Find four values for which each function is undefined.

21. f (θ) = tan θ 22. g(θ) = cot θ 23. h(θ) = sec θ 24. j(θ) = csc θ

25. Law Enforcement A police car is parked on the side of the road next to a building. The flashing light on the car is 6 feet from the wall and completes one full rotation every 3 seconds. As the light rotates, it shines on the wall. The equation representing the distance a

in feet is a(t) = 6 sec (2__3

πt).

a. What is the period of a(t)?

b. Graph the function for 0 ≤ t ≤ 3.

c. Critical Thinking Identify the location of any asymptotes. What do the asymptotes represent?

26. Math History The ancient Greeks used a gnomon, a type of tall staff, to tell the time of day based on the lengths of shadows and the altitude θ of the sun above the horizon.

a. Use the figure to write a cotangent function that can be used to find the length of the shadow s in terms of the height of the gnomon h and the angle θ.

b. Graph your answer to part a for a gnomon of height 6 ft.

Complete the table by labeling each function as increasing or decreasing.

0 < x <

π

_2

π

_2

< x < π π < x <

3π

_2

3π

_2

< x < 2π

27. sinx

28. cscx

29. cosx

30. secx

31. tanx

32. cotx

33. Critical Thinking Based on the table above, what do you observe about the increasing/decreasing relationship between reciprocal pairs of trigonometric functions?


Between 1:00 P.M. (t = 1) and 6:00 P.M. (t = 6) , the height (in meters) of the tide in a bay is modeled by h(t) = 0.4 csc 5π

___31

t.

a. Graph the function for the range 1 ≤ t ≤ 6.

b. At what time does low tide occur?

c. What is the height of the tide at low tide?

d. What is the maximum height of the tide during this time span? When does this occur?

21. π

_2 ; 3π

_

2 ; -

π

_2 ; 5π

_

2

22. -π; 0; π; 2π

23. π

_2 ; 3π

_

2 ; -

π

_2

; 5π

_2

24. -π; 0; π; 2π

0.4 m

about 3.95 m; 6:00 P.M.

about 3:06 P.M. (t = 3.1)

Possible answer: The asymptotes represent the time when the light shines parallel to the wall.

3 s

s = h cot θ

t = 3_4

and t = 9_4

;

33.Possible answer: For reciprocal pairs of trigonometric functions, when one increases, the other decreases and vice versa.

inc. dec. dec. inc.

dec. inc. inc. dec.

dec. dec. inc. inc.

inc. inc. dec. dec.

inc. inc. inc. inc.

dec. dec. dec. dec.

1002 Chapter 14

Exercise 20 involves graphing and inter-preting cosecant

functions. This exercise prepares students for the Multi-Step Test Prep on page 1004.

Answers 20a.

25b.

26b.


14-2 PRACTICE C

14-2 PRACTICE B

14-2 PRACTICE A



34. Critical Thinking How do the signs (whether a function is positive or negative) of reciprocal pairs of trigonometric functions relate?

35. Write About It Describe how to graph f (x) = 3 sec 4x by using the graph of g(x) = 3 cos 4x.

36. Which is NOT in the domain of y = cot x?

- π _2

0 π

_2

3π

_2

37. What is the range of f(x) = 3 csc 2θ?⎧ ⎨

⎩ y⎥ y ≤ -1 or y ≥ 1

⎫

⎬

⎭

⎧ ⎨

⎩ y⎥ y ≤ -2 or y ≥ 2

⎫

⎬

⎭

⎧ ⎨

⎩ y⎥ y ≤ -3 or y ≥ 3

⎫

⎬

⎭

⎧ ⎨

⎩ y⎥ y ≤ - 1_

2 or y ≥ 1_

2

⎫

⎬

⎭

38. Which could be the equation of the graph?

y = tan 2x y = 2 tan x

y = cot 2x y = 2 cot x

39. What is the period of y = tan 1_2

x?π

_2

2π

π 4π

40. The graph of which function has a period of 2π ___

3 and an asymptote at x = π

__2?

y = sec 3_2

x y = csc 3_2

x

y = sec 3x y = csc 3x

CHALLENGE AND EXTENDDescribe the period, local maximum and minimum values, and phase shift.

41. f (x) = 4 - 3 csc π(x -1) 42. g(x) = 4 cot 1_2

(x - π _2 ) 43. h(x) = 0.5 sec 2 (x + π

_4 )

44. f (x) = 9 + 2 tan 3 (x + π) 45. g(x) = 0.62 + 0.76 sec x 46. h(x) = csc π _2

(x + 5_7)

Graph each trigonometric function and its inverse. Identify the domain and range of the corresponding inverse function.

47. f (x) = Sec x for 0 ≤ x ≤ π and x ≠ π _2

48. f (x) = Tan x for - π _2

< x < π _2

49. g (x) = Csc x for - π _2

≤ x ≤ π _2

and x ≠ 0 50. g (x) = Cot x for 0 < x < π

SPIRAL REVIEWFind the additive and multiplicative inverse for each number. (Lesson 1-2)

51. - 1_10

52. 0.2 53. -3 √ 5 54. 4_9

55. Technology Marjorie’s printer prints 30 pages per minute. How many pages does Marjorie’s printer print in 22 seconds? (Lesson 2-2)

Convert each measure from degrees to radians or from radians to degrees. (Lesson 13-3)

56. 45° 57. 3π

_4

radians 58. 225° 59. - π _3

radiansπ

_4 radians 135°

5π

_4 radians -60°

1_10

; -10 -0.2; 5

53. 3 √

�

5 ; -

√

�

5_15

-

4_9 ; 9_

4

11 pages

Lesson 14-2 1003

Identifying stretches and compressions of tangent and

cotangent graphs can be difficult. If students have difficulty with Exercise 38, they should focus on the shape and the period. The shape and period indicate that the correct answer is choice B.

Answers34, 35, 41–50. See p. A51.

JournalHave students explain how the domain and range of each of the six trigonometric graphs relates to right triangles.

Give students the graphs of the

functions f (x) = 2 csc ( 3 _ 4 x) ,

g (x) = cot (x + π _ 4 ) , and

h (x) = 3 sec ( π _ 2 x) , and ask them to

write the function rule.

14-2

1. Using f (x) = tan x as a guide,

graph g (x) = -tan ( 1 _ 2 x) .

Identify the period, x-intercepts, and asymptotes.

period: 2π; x-intercepts: 2πn; asymptotes: x = π + 2πn

2. Using f (x) = sin (x) as a guide,

graph g (x) = 1 _ 3 csc ( 1 _

3 x) .

Identify the period and asymptotes.

period: 6π; asymptotes: x = 3πn


14-2 CHALLENGE14-2 PROBLEM SOLVING


Trigonometric GraphsThe Tide Is Turning Tides are caused by several factors, but the main factor is the gravitational pull of the Moon. As the Moon revolves around Earth, the Moon causes large bodies of water to swell toward it resulting in rising and falling tides. You can use trigonometric functions to develop a model of a simplified tide.

1. The highest tides in the world have been measured at the Bay of Fundy, in Nova Scotia, Canada. As shown in the table, high tides in the bay can reach heights of 16.3 m. Assume that it takes 6.25 hours for the tide to completely retreat and then another 6.25 hours for the tide to come back in. Write a periodic function based on the cosine function that models the height of the tide over time.

2. What are the amplitude, period, maximum and minimum values, and phase shift of the function?

3. Graph the function.

4. At time t = 0, the tide is at 16.3 m. What is the tide’s height after 3 hours? after 9 hours?

5. Will a high tide occur at the same time each day at the Bay of Fundy? Why or why not?

6. It is possible to write a function that models the height of the tide based on the sine function. What is the function? What is the phase shift?

Tides at the Bay of Fundy

Time (h) Height (m)

High Tide t = 0 16.3

Low Tide t = 6.25 0


SECTION 14A

amplitude: 8.15; period: 12.5; maximum: 16.3; minimum: 0; phase shift: none

about 8.66 m; about 6.62 m

yes; h(t) = 8.15 sin(

4π

_25

t + π _2 )

+ 8.15; π _2

left

h(t) = 8.15 cos 4π

_25

t + 8.15

5. No; the period of the function is 12.5 h, which is not a factor of 24 h.

OrganizerObjective: Assess students’ ability to apply concepts and skills in Lessons 14-1 and 14-2 in a real-world format.

Online Edition

ResourcesAlgebra II Assessments

www.mathtekstoolkit.org

Problem Text Reference

1–6 Lesson 14-1

Answers 3.

INTERVENTION

Scaffolding Questions 1. What information gives the period of the

tide function? the time between high and low tide What information gives the amplitude of the tide function? the heights of high and low tide

2. How do you know there is no phase shift? High tide occurs exactly at t = 0.

3. What are ways that you can check that your graph is correct? Possible answer: Check for a maximum at t = 0; minimum value should be 0, so the graph should touch, but not cross, the x-axis.

4. How can you make a rough estimate of the height of the tide at t = 3? It is half-way between16.3 m and 0 m, about 8 m.

5. If the high tide occurred at the same time each day, what would be true of the period? It would be a divisor of 24.

6. What phase shift makes a sine graph

equivalent to a cosine graph? π _ 2 left

ExtensionAt what time is the height of the tide exactly 8.15 m? 3.125 hr, or 3 hr 7.5 min


S E C T I O N

14A

1004 Chapter 14

a207te_c14_m&r_1004-1005.indd 1004a207te_c14_m&r_1004-1005.indd 1004 12/20/05 12:08:40 PM12/20/05 12:08:40 PM

Ready to Go On? 1005

Quiz for Lessons 14-1 Through 14-2

14-1 Graphs of Sine and CosineIdentify whether each function is periodic. If the function is periodic, give the period.

1. 2.

3. 4.


5. f (x) = sin 4x 6. g(x) = -3 sin x 7. h(x) = 0.25 cos πx


8. f (x) = cos (x - 3π

_

2 ) 9. g(x) = sin(x - 3π

_

4 ) 10. h(x) = cos (x + 5π

_

4 ) 11. The torque τ applied to a bolt is given by τ (x) = Fr sin x, where r is the length

of the wrench in meters, F is the applied force in newtons, and x is the angle between F and r in radians. Graph the torque for a 0.5 meter wrench and a force of 500 newtons for 0 ≤ x ≤ π

__2 . What is the torque for an angle of π

__3 ?

14-2 Graphs of Other Trigonometric FunctionsUsing f (x) = tan x as a guide, graph each function. Identify the period, x-intercepts, and asymptotes.

12. f (x) = 1_2

tan 4x 13. g(x) = -2 tan 1_2

x 14. h(x) = tan 1_2

πx


15. g(x) = -2 cot x 16. h(x) = cot 0.5x 17. j(x) = cot 4x


18. f (x) = -2 sec x 19. g(x) = 1_4

csc x 20. h(x) = sec πx

SECTION 14A

not periodic periodic; 2π

periodic; 4 not periodic


OrganizerObjective: Assess students’ mastery of concepts and skills in Lessons 14-1 and 14-2.

Online Edition

ResourcesAssessment Resources

Section 14A Quiz


INTERVENTION

ResourcesReady to Go On? Intervention and Enrichment Worksheets

Ready to Go On? CD-ROM

Ready to Go On? Online


NOINTERVENE

YESENRICH

READY TO GO ON? Intervention, Section 14AReady to Go On?

Intervention Worksheets CD-ROM Online

Lesson 14-1 14-1 Intervention Activity 14-1 Diagnose and Prescribe OnlineLesson 14-2 14-2 Intervention Activity 14-2

Diagnose and Prescribe

READY TO GO ON? Enrichment, Section 14A


S E C T I O N

14A

A211NLT_c14_m&r_1004-1005.indd 1005A211NLT_c14_m&r_1004-1005.indd 1005 9/1/09 1:42:01 AM9/1/09 1:42:01 AM

Trigonometric Identities

One-Minute Section PlannerLesson Lab Resources Materials

14-3 Technology Lab Graph Trigonometric Identities• Use a graphing calculator to compare graphs and make conjectures

about trigonometric identities.


Technology Lab Activities14-3 Lab Recording Sheet


Lesson 14-3 Fundamental Trigonometric Identities• Use fundamental trigonometric identities to simplify and rewrite

expressions and to verify other identities.

□ SAT-10 □ NAEP □✔ ACT □ SAT □✔ SAT Subject Tests

Technology Lab Activities14-3 Technology Lab

Optionalgraphing calculator

Lesson 14-4 Sum and Difference Identities• Evaluate trigonometric expressions by using sum and difference

identities.

• Use matrix multiplication with sum and difference identities to perform rotations.



Lesson 14-5 Double-Angle and Half-Angle Identities• Evaluate and simplify expressions by using double-angle and half-

angle identities.

□ SAT-10 □ NAEP □ ACT □ SAT □✔ SAT Subject Tests


Lesson 14-6 Solving Trigonometric Equations• Solve equations involving trigonometric functions.



MK = Manipulatives Kit

SECTION

14B

1006A Chapter 14

a207te_c14_s&o_1006A-1006B.indd 1006Aa207te_c14_s&o_1006A-1006B.indd 1006A 12/20/05 12:07:29 PM12/20/05 12:07:29 PM

1006B

IDENTITIESLesson 14-3An identity is an equation that is true for all values of the variable(s). Identities may be indicated by ≡, such as 5 (x – 3) ≡ 5x + 15. Students have seen identities, although the relationships may not have been called identities. For example, the rule for factoring a differ-ence of two squares, a 2 – b 2 = (a + b) (a – b) , is an identity since the equation is true for all a and b.

Students should understand that to prove an equation is an identity, they must begin with the expression on one side of the equation and perform a series of alge-braic manipulations to arrive at the expression on the other side. Each step must be justified by a property, a definition, or a previously proven identity.

TRIGONOMETRIC IDENTITIESLesson 14-3The identities sin (–θ) = –sin θ and cos (–θ) = cos θ describe fundamental characteristics of the sine and cosine functions.

• When f (x) = –f (x) , f is an odd function, and its graph has 180° rotational symmetry about the origin. The sine function is odd.

• When f (–x) = f (x) , f is an even function, and its graph is symmetric about the y-axis. The cosine function is even.

Students may try to memorize all of the trigonometric identities. Point out that if they remember the following information, they can derive most identities by using algebra.• Definitions of the trigonometric functions• y = sin x is odd; y = cos x is even.• sin 2 θ + cos 2 θ = 1• sin (A + B) = sin A cos B + cos A sin B• cos (A + B) = cos A cos B – sin A sin B• The values of sin θ and cos θ when θ is a multiple

of π __ 2

For example, to find sin (A – B) or cos (A – B) , use the sum identities above and replace B with –B; or find double angle identities by replacing B with A.

SUM AND DIFFERENCE IDENTITIESLesson 14-4The following proof of the sum identities uses complex numbers and vector operations. The complex number z = cos A + isin A is represented by the point Z (cos A, sin A) and by the vector <cos A, sin A>. Note that Z is on the unit circle. The magnitude of the vector is 1, and its angle with the x-axis is A.

The complex number w = cos B + isin B is repre-sented by W(cos B, sin B) and <cos B, sin B>. W is on the unit circle and the magnitude of the vector is 1; its angle with the x-axis is B.

Z (cos A, sin A)

W (cos B, sin B)

AB

As discussed in the Math Background for Section 5B, when two vectors are multiplied, the magnitude of the resulting vector is the product of the magnitudes of the original vectors. Also, the angle of the resulting vector is the sum of the angles of the original vectors.

So the vector <cos A, sin A> <cos B, sin B> has mag-nitude 1, and the corresponding point is on the unit circle. Also, the angle formed by this vector and the x-axis is A + B, so the coordinates of that point are (cos (A + B) , sin (A + B) ) . In other words, zw = cos (A + B) + isin (A + B) .

But zw is also equal to (cos A + isin A) (cos B + isin B) . Expanding this product gives

(cos A cos B – sin A sin B) + i (cos A sin B + sin A cos B) .

Equating both expressions for zw gives

cos (A + B) + isin (A + B) = (cos A cos B – sin A sin B) + i (cos A sin B + sin A cos B)

which proves the sum identities for sine and cosine.

Math Background

A211NLT_c14_s&o_1006A-1006B.indd1006B 1006BA211NLT_c14_s&o_1006A-1006B.indd1006B 1006B 8/31/09 3:07:27 PM8/31/09 3:07:27 PM


Graph Trigonometric IdentitiesYou can use a graphing calculator to compare graphs and make conjectures about trigonometric identities.

Activity

Determine whether si n 2 x_______1 - cos x = 1 + cos x is a possible identity.

If the equation is an identity, there should be no visible difference in the graphs of the left- and right-hand sides of the equation.

1 Enter si n 2 x_______1 - cos x

as Y1 and 1 + cos x as Y2. For Y2, select the mode represented by the 0 with a line through it. This will help you see the path of the graph.

2 Set the graphing window by using and 7:ZTrig.

3 Watch the calculator as the graphs are generated. As Y2 is being graphed, a circle will move along the path of the graph.

4 The path of the circle, Y2, traced the graph of Y1. The graphs appear to be the same.

Because the graphs appear to be identical, si n 2 x_______1 - cos x

= 1 + cos x is most likely an identity. Use algebra to confirm.

Try This

1. Make a Conjecture Determine whether sec x - tan x sin x = cos x is a possible identity.

2. Prove or disprove your answer to Problem 1 by using algebra.

3. Make a Conjecture Determine whether 1 + tan x_______1 + cot x

= tan x is a possible identity.

4. Prove or disprove your answer to Problem 3 by using algebra.

14-3

Use with Lesson 14-3

KEYWORD: MB7 Lab14

It is a possible identity.

It is a possible identity.

Organizer

Pacing: Traditional 1 __ 2 dayBlock 1 __ 4 day

Objective: Use a graphing calculator to compare graphs and make conjectures about trigonometric identities.

Materials: graphing calculator

Online EditionGraphing Calculator, TechKeys

ResourcesTechnology Lab Activities

14-3 Lab Recording Sheet

TeachDiscussDiscuss the limitations of using a graph to determine whether an equation is an identity. Include topics such as domain and viewing windows.

CloseKey ConceptWhen you graph both sides of a trigonometric equation together and their graphs coincide, then the equation is most likely an identity.

AssessmentJournal Have students explain why graphing shows that an equation is only most likely an identity.

Use with Lesson 14-3

Answers to Try This 2. sec x - tan x sin x

= 1 _ cos x - si n 2 x _ cos x

= 1 - si n 2 x _ cos x

= co s 2 x _ cos x = cos x

4. 1 + tan x

_ 1 + cot x

=

1 + sin x ____ cos x _ 1 + cos x ____

sin x

= cos x + sin x _________ cos x

_ sin x + cos x _________

sin x

= cos x + sin x

__ cos x · sin x __ sin x + cos x

= sin x _ cos x = tan x

1006 Chapter 14


a207te_c14_lab_1006.indd 1006a207te_c14_lab_1006.indd 1006 6/7/06 10:42:23 AM6/7/06 10:42:23 AM

Angle Relationships

Angle relationships in circles and polygons can be used to solve problems.

Try This

Solve each problem. Round each answer to the nearest hundredth.

1. A circle is inscribed in an equilateral triangle with 8 in. sides. What is the diameter of the circle? What is the altitude of the triangle?

2. An isosceles right triangle is inscribed in a semicircle with a radius of 20 cm. What are the lengths of the three sides of the triangle?

3. The interior angles of a regular polygon each measure 150°. If this polygon is inscribed in a circle with a 10 in. diameter, how long is each side of the polygon?

4. Use the figure to find the side lengths of all three shaded triangles if the diameter of the circle is 10 cm. Round to the nearest hundredth if necessary.

Example

A regular octagon is inscribed in a circle with a radius of 5 cm. What is the length of each side of the octagon?

Make a sketch of the problem.

s = 2R sin (180°_n ) Choose a formula relating the radius of

the circumscribed circle to the side length of the polygon.

s = 2 (5) sin (180°_8 ) Substitute 5 for R and 8 for n.

s = 10 sin 22.5° ≈ 3.83 cm

The figures show regular polygons. A regular polygon has sides of equal length and equal interior angles. Here are some useful relationships for regular polygons.

R bisects θ. θ = (n - 2_n )180° r = R cos (180°_

n ) s = 2r tan (180°_n )= 2R sin (180°_

n )

Geometry

Connecting Algebra to Geometry 1007

≈2.59 in.

4. pink triangle: 5 cm, 4.10 cm, 2.87 cm; yellow triangle: 5 cm, 3.83 cm, 3.21 cm; blue triangle: 3.30 cm, 7.07 cm, and 7.80 cm

A211NLS_c14cn1_1007.indd 1007 8/7/09 9:32:02 AM

Connecting Algebra to Geometry 1007

Geometry

Pacing: Traditional 1 __ 2 dayBlock 1 __ 4 day

Objective: Apply trigonometric functions to solving problems involving inscribed and circumscribed polygons.

Online Edition

TeachRememberStudents review and apply the Pythagorean Theorem and angle relationships in circles and polygons.

Visual Point out that there are two radii on the top right figure. R is the radius

of the circumscribed circle. r is the radius of the inscribed circle and is called the apothem.

CloseAssessAsk students to find the value of θ and the side length for a regular hexagon circumscribing a circle of radius 3 cm. θ = 12 0 ◦ ; s ≈ 3.46 cm


Organizer

Answers 1. 4.62 in.; 6.93 in.

2. 28.28 cm, 28.28 cm, and 40 cm

A211NLT_c14_labc_1007.indd 1007A211NLT_c14_labc_1007.indd 1007 8/31/09 3:08:30 PM8/31/09 3:08:30 PM


14-3 FundamentalTrigonometric Identities

ObjectiveUse fundamental trigonometric identities to simplify and rewrite expressions and to verify other identities.

You can use trigonometric identities to simplify trigonometric expressions. Recall that an identity is a mathematical statement that is true for all values of the variables for which the statement is defined.

A derivation for a Pythagorean identity is shown below.

x2 + y2 = r2 Pythagorean Theorem

x2_

r2 +

y2_r2

= 1 Divide both sides by r2.

co s 2 θ + si n 2 θ = 1 Substitute cos θ for x_r and sin θ for y_r .

Fundamental Trigonometric Identities

Reciprocal Identities

Tangent and Cotangent Ratio Identities

Pythagorean Identities

Negative-AngleIdentities

cscθ = 1_sinθ

secθ = 1_cosθ

cotθ = 1_tanθ

tanθ = sinθ

_cosθ

cotθ = cosθ

_sinθ

cos2θ + si n2

θ = 1

1 + ta n2θ = se c2

θ

cot2θ + 1 = cs c2

θ

sin(-θ) = -sinθ

cos(-θ) = cosθ

tan(-θ) = -tanθ

To prove that an equation is an identity, alter one side of the equation until it is the same as the other side. Justify your steps by using the fundamental identities.

1E X A M P L E Proving Trigonometric Identities

Prove each trigonometric identity.

A sec θ = csc θ tan θ

sec θ = csc θ tan θ Choose the right-hand side to modify.

= ( 1_sin θ

)( sin θ_cos θ

) Reciprocal and ratio identities

= 1_cos θ

Simplify.

= sec θ Reciprocal identity

You may start with either side of the given equation. It is often easier to begin with the more complicated side and simplify it to match the simpler side.

Who uses this?Ski supply manufacturers can use trigonometric identities to determine the type of wax to use on skis. (See Example 3.)

Introduce1




Block 1 __ 2 day

Objectives: Use fundamental trigonometric identities to simplify and rewrite expressions and to verify other identities.

Technology LabIn Technology Lab Activities

Online EditionGraphing Calculator, Tutorial Videos

Warm UpSimplify.

1. ( sin A _ cos A

) ( co s 2 A _ sin A

) cos A

2. tan A ( sin A _ tan A

) ( 1 _ sin A

) 1


Student: Superheroes would be in big trouble if villains knew trig.

Parent: Why is that?

Student: Because then the villains could figure out the superheroes’ secret identities.

MotivateRecall that the vertex form and the standard form of a quadratic function are equivalent but have different uses: the vertex form gives the vertex and the standard form gives the y-intercept. The same holds true for many trigonometric expres-sions; one form may be more useful than another for a given situation. Trigonometric identities can be used to rewrite expressions.

1008 Chapter 14


14-3 Fundamental Trigonometric Identities 1009


B csc (-θ) = -csc θ

csc (-θ) = -csc θ Choose the left-hand side to modify.

1_sin(-θ)

= Reciprocal identity

1_-sin θ

= Negative-angle identity

- ( 1_sin θ

) = -csc θ

-csc θ = -csc θ Reciprocal identity


1a. sin θ cot θ = cos θ 1b. 1 - sec(-θ) = 1 - sec θ

You can use the fundamental trigonometric identities to simplify expressions.

2E X A M P L E Using Trigonometric Identities to Rewrite Trigonometric Expressions

Rewrite each expression in terms of cos θ, and simplify.

A sin 2 θ

_1 - cos θ

B sec θ - tan θ sin θ

1_cos θ

- ( sin θ

_cos θ

) · sin θ Substitute.

1_cos θ

- si n 2 θ

_cos θ

Multiply.

1 - si n 2 θ

_cos θ

Subtract fractions.

co s 2 θ

_cos θ

Pythagorean identity

cos θ Simplify.

1 - co s 2 θ

_1 - cos θ


(1 + cos θ) (1 - cos θ)

___1 - cos θ

Factor the difference of two squares.

(1 + cos θ) (1 - cos θ) __1 - cos θ

Simplify.

1 + cos θ

Rewrite each expression in terms of sin θ, and simplify.

2a. co s 2 θ_1 - sin θ

2b. co t 2 θ

If you get stuck, try converting all of the trigonometric functions into sine and cosine functions.

I like to use a graphing calculator to check for equivalent expressions.

For Example 2A, enter y = sin2θ

_(1 - cosθ)

and

y = 1 + cosθ. Graph both functions in the same viewing window.

The graphs appear to coincide, so the expressions are most likely equivalent.

Graphing to Check for Equivalent Expressions

Julia ZaragozaOak Ridge High School

1 + sin θ 1_ sin 2 θ

- 1

Lesson 14-3 1009

Example 1


A. tan θ = sec θ

_ csc θ

=

(

1 _ cos θ

)

_ ( 1 _ sin θ

)

= 1 _ cos θ

· sin θ = tan θ

B. 1 - cot θ = 1 + cot (-θ)

= 1 + 1 _ tan (-θ)

= 1 + 1 _ -tan θ

= 1 + (-cot θ)

= 1 - cot θ

Example 2


A. sec θ (1- si n 2 θ) cos θ

B. sin θ cos θ (tan θ + cot θ) 1

Additional Examples


EXAMPLE 1

• How can the identity in Example 1B be demonstrated by using the unit circle?

EXAMPLE 2

• How do you know when you should use one of the Pythagorean identities?

Answers to Check It Out!

1a. sin θ cot θ = sin θ (

cos θ

_ sin θ

)

= cos θ

b. 1 - sec (-θ) = 1 -

1 _ cos (-θ)

= 1 -

1 _ cos θ

= 1 - sec θ

Guided InstructionGuide students through the table on p. 1008 containing the Fundamental Trigonometric Identities. Remind students that an identity is an equation that is true for all values of the variable. To prove an identity, students must make substitutions using other, previously proven identities.

Teach2

Through Cognitive Strategies

Have students work in groups to derive the Pythagorean identities 1 + co t 2 θ = cs c 2 θ and 1 + ta n 2 θ = se c 2 θ so that students can recall the identities by understanding.

For example, si n 2 θ

_ si n 2 θ

+ co s 2 θ

_ si n 2 θ

= 1 _ si n 2 θ

yields 1 + co t 2 θ = cs c 2 θ.



3E X A M P L E Sports Application

A ski supply company is testing the friction of a new ski wax by placing a waxed wood block on an inclined plane of wet snow. The incline plane is slowly raised until the wood block begins to slide.

At the instant the block starts to slide, the component of the weight of the block parallel to the incline, mg sin θ, and the resistive force of friction, μmg cos θ, are equal. μ is the coefficient of friction. At what angle will the block start to move if μ = 0.14?

Set the expression for the weight component equal to the expression for the force of friction.

mg sin θ = μmg cos θ

sin θ = μ cos θ Divide both sides by mg.

sin θ = 0.14 cos θ Substitute 0.14 for μ.

sin θ

_cos θ

= 0.14 Divide both sides by cosθ.

tan θ = 0.14 Ratio identity

θ ≈ 8° Evaluate inverse tangent.

The wood block will start to move when the wet snow incline is raised to an angle of about 8°.

3. Use the equation mg sin θ = μmg cos θ to determine the angle at which a waxed wood block on a wood incline with μ = 0.4 begins to slide.

THINK AND DISCUSS 1. DESCRIBE how you prove that an equation is an identity.

2. EXPLAIN which identity can be used to prove that (1 - cos θ) (1 + cos θ) = si n 2 θ.

3. GET ORGANIZED Copy and complete the graphic organizer by writing the three Pythagorean identities.

The symbol μ is read as “mu.”

θ ≈ 22°

Precalculus Many aspects of trigonometry become important when students

begin their study of calculus. In cal-culus, students will encounter situa-tions in which expressions must be modified in order to solve problems.

Example 3

At what angle will a wooden block on a concrete incline start to move if the coefficient of fric-tion is 0.62?

θ ≈ 3 2 ◦

Additional Examples


EXAMPLE 3

• If you knew the angle at which the block began to slide, how could you find the coefficient of friction?





and INTERVENTIONSummarizeTrigonometric identities can be proven true by substituting simpler trigonometric identities. Tell students that when they are proving an identity, it is good practice to work with only one side of an equation at a time. Converting all terms to sine and cosine may be a good strategy if students become stuck.

Close3 Answers to Think and DiscussPossible answers:

1. Use identities to modify one side of the equation until it is written in the same form as the other side of the equation.

2. After multiplying the left side of the equation and simplifying by combining like terms, use the Pythagorean identity si n 2 θ + co s 2 θ = 1.

3. See p. A14.

1010 Chapter 14



ExercisesExercises14-3KEYWORD: MB7 14-3

KEYWORD: MB7 Parent

GUIDED PRACTICE



1. sin θ sec θ = tan θ 2. co t (-θ) = -co t θ 3. co s 2 θ (se c 2 θ - 1) = si n 2 θ



4. csc θ tan θ 5. (1 + se c 2 θ) (1 - si n 2 θ) 6. si n 2 θ + co s 2 θ + ta n 2 θ


7. Physics Use the equation mg sin θ = μmg cos θ to determine the angle at which a glass-top table can be tilted before a glass plate on the table begins to slide. Assume μ = 0.94.



8–11 1 12–15 2 16 3

Independent Practice Prove each trigonometric identity.

8. sec θ cot θ = csc θ 9. sin θ - cos θ

__sin θ

= 1 - cot θ

10. tan θ sin θ = sec θ - cos θ 11. se c 2 θ (1 - co s 2 θ) = ta n 2 θ

Rewrite each expression in terms of sin θ, and simplify.

12. co s 2 θ

_1 + sin θ

13. tan θ

_cot θ

14. cos θ cot θ + sin θ 15. sec 2 θ - 1_1 + tan 2 θ

16. Physics Use the equation mg sin θ = μmg cos θ to determine the steepest slope of the street shown on which a car with rubber tires can park without sliding.

Multi-Step Rewrite each expression in terms of a single trigonometric function.

17. tan θ cot θ 18. sin θ cot θ tan θ 19. cos θ + sin θ tan θ

20. sin θ csc θ - co s 2 θ 21. co s 2 θ sec θ csc θ 22. cos θ(ta n 2 θ + 1) 23. csc θ(1 - co s 2 θ) 24. csc θ cos θ tan θ 25. sin θ

_1 - co s 2 θ

26. si n 2 θ

_1 - co s 2 θ

27. tan θ

_sin θ sec θ

28. cos θ

_sin θ cot θ

29. tan θ (tan θ + cot θ) 30. si n 2 θ + co s 2 θ + co t 2 θ 31. si n 2 θ sec θ csc θ

Verify each identity.

32. cos θ -1_co s 2 θ

= sec θ - se c 2 θ 33. si n 2 θ (cs c 2 θ -1) = co s 2 θ 34. tan θ + cot θ = sec θ csc θ

35. cos θ

_1 - si n 2 θ

= sec θ 36. 1 - co s 2 θ

_tan θ

= sin θ cos θ 37. cs c 2 θ

_1 + ta n 2 θ

= co t 2 θ

Prove each fundamental identity without using any of the other fundamental identities. (Hint: Use the trigonometric ratios with x, y, and r.)

38. tan θ = sin θ

_cos θ

39. cot θ = cos θ

_sin θ

40. 1 + co t 2 θ = cs c 2 θ

41. csc θ = 1_sin θ

42. sec θ = 1_cos θ

43. 1 + ta n 2 θ = se c 2 θ


Extra Practice

1_cos θ

1 + cos 2 θ

6. 1_ cos 2 θ

θ ≈ 43°

1 - sin θ sin 2 θ

_1 - sin 2 θ

θ ≈ 42°

21. cot θ

22. sec θ

23. sin θ1

csc θ

csc 2 θ

1_sin θ

sin θ1 sec θ

sin 2 θ

1 1

sec 2 θ

tan θ

1

sin 2 θ

Lesson 14-3 1011

ExercisesExercises


Assignment Guide


Basic 8–22, 32–34, 44–50, 56–61, 70–75

Average 8–16, 17–43 odd, 44–63, 70–75

Advanced 8–44 even, 45–75

Homework Quick CheckQuickly check key concepts.Exercises: 8, 10, 12, 14, 16

Answers 1. sin θ sec θ = sin θ

(

1 _ cos θ

)

= sin θ

_ cos θ

= tan θ

2. cot (-θ) = cos (-θ)

_ sin (-θ)

= cos θ

_ -sin θ

= -

(

cos θ

_ sin θ

)

= -cot θ

3. cos 2 θ ( sec 2 θ - 1) = cos 2 θ ( tan 2 θ)

= cos 2 θ (

sin θ

_ cos θ

)

2

= cos 2 θ (

sin 2 θ

_ cos 2 θ

)

= sin 2 θ

8. sec θ cot θ = (

1 _ cos θ

)

(

cos θ

_ sin θ

)

= 1 _ sin θ

= csc θ

9. sin θ - cos θ

__ sin θ

= sin θ

_ sin θ

-

cos θ

_ sin θ

= 1 - cot θ

14-3

10. tan θ sin θ = si n 2 θ

_

cos θ

=

1 - cos 2 θ

_ cos θ

= sec θ - cos θ

11. sec 2 θ (1 - cos 2 θ) = (

1 _ cos 2 θ

)

( sin 2 θ)

= sin 2 θ

_ cos 2 θ

= tan 2 θ

32. cos θ - 1 _ cos 2 θ

= cos θ

_ cos 2 θ

-

1 _ cos 2 θ

= 1 _ cos θ

-

1 _ cos 2 θ

= sec θ - sec 2 θ

33. sin 2 θ ( csc 2 θ - 1) = sin 2 θ cot 2 θ

= sin 2 θ (

cos 2 θ

_ sin 2 θ

)

= cos 2 θ

34. tan θ + cot θ = sin θ

_ cos θ

+ cos θ

_ sin θ

= sin 2 θ + cos 2 θ

__ sin θ cos θ

= 1 _ sin θ cos θ

= (

1 _ sin θ

)

(

1 _

cos θ )

= sec θ csc θ

35–43. See p. A52.



Graphing Calculator Use a graphing calculator to determine whether each of the following equations represents an identity. (Hint: You may need to rewrite the equations in terms of sine, cosine, and tangent.)

45. (csc θ - 1)(csc θ + 1) = ta n 2 θ 46. sec θ - cos θ = sin θ

47. cos θ(sec θ + cos θ cs c 2 θ) = cs c 2 θ 48. cot θ(cos θ + sin θ tan θ) = csc θ

49. cos θ = 0.99 cos θ 50. sin θ cos θ = tan θ - tan θ si n 2 θ

51. Physics A conical pendulum is created by a pendulum that travels in a circle rather than side to side and traces out the shape of a cone. The radius r of the base of the

cone is given by the formula r = g tan θ

_____

ω 2 , where g represents

the force of gravity and ω represents the angular velocity of the pendulum.

a. Use ω = √ �� g_____

� cos θ and fundamental trigonometric

identities to rewrite the formula for the radius.

b. Find a formula for � in terms of g, ω, and a single trigonometric function.

Critical Thinking A function is called odd if f (-x) = - f (x) and even if f (-x) = f (x).

52. Which of the six trigonometric functions are odd? Which are even?

53. What distinguishes the graph of an odd function from an even function or a function that is neither odd nor even?

54. Determine whether the following functions are odd, even, or neither.

a. b.

55. Critical Thinking In how many equivalent forms can tan θ = sin θ

____ cos θ

be expressed? Write at least three of its forms.

56. Write About It Use the fact that sin (-θ) = - sin θ and cos (-θ) = cos θ to explain why tan (-θ) = - tan θ.


The displacement y of a mass attached to a spring is modeled by y(t) = 5 sin t, where t is the time in seconds. The displacement z of another mass attached to a spring is modeled by z(t) = 2.6 cos t.

a. The two masses are set in motion at t = 0. When do the masses have the same displacement for the first time?

b. What is the displacement at this time?

c. At what other times will the masses have the same displacement?

≈ 0.48 s

≈ 0.48 + πn where n is an integer

≈ 2.31

no no

yes yes

no yes

odd even

55. an infinite number of equivalent forms;

tan θ = sin θ

_cos θ

,

cos θ = sin θ

_tan θ

,

sin θ = tan θ cos θ

r = � sin θ

� = g_

ω 2 sec θ

52. odd: sine, tangent, cotangent, cosecant; even: cosine, secant

53. The graphs of even functions show reflection symmetry across the y-axis.Odd functions show 180° rotational symmetry about the origin, or both a reflection across the x-axis and the y-axis.

1012 Chapter 14

Exercise 44 involves solving trigonometric equations. This exer-

cise prepares students for the Multi-Step Test Prep on page 1034.

Answers 56. Because tan θ =

sin θ

_ cos θ

,

tan (-θ) = sin (-θ)

_ cos (-θ)

. Use

sin (-θ) = -sinθ and

cos (-θ) = cos θ to get

tan (-θ) =

-sin θ

_ cos θ = -tanθ.


14-3 PRACTICE C

14-3 PRACTICE B

14-3 PRACTICE A


Answers


57. Which expression is equivalent to sec θ sin θ?

sin θ cos θ csc θ tan θ

58. Which expression is NOT equivalent to the other expressions?

sec θ csc θ 1_sinθ cos θ

tanθ

_sin2

θ

cos2

θ

_cotθ

59. Which trigonometric statement is NOT an identity?

1 + co s2θ = si n2

θ 1 + ta n2θ = se c2

θ

cs c2θ - 1 = co t2

θ 1 - sin2θ = co s2

θ

60. Which is equivalent to 1 - sec2θ?

ta n2θ -tan2

θ co t2θ -cot2

θ

61. Short Response Verify that sin θ + cot θ cos θ = csc θ is an identity. Write the justification for each step.

CHALLENGE AND EXTENDWrite each expression as a single fraction.

62. 1_cos θ

+ 1_co s 2 θ

63. cos θ

_sin θ

+ sin θ

_cos θ

64. 1 - cos θ

_sin θ

65. 1_1 - cos θ

- cos θ

_1 - co s 2 θ

Simplify.

66.

1____si n 2 θ

- 1_co s 2 θ

_____si n 2 θ

67. 1____

sin θ + 1____

cos θ

_

1________sin θ cos θ

68. 1____

sin θ - 1____

cos θ

__sin θ

____cos θ

- cos θ

____sin θ

69. 1 - 1____

sin θ

_1 - 1____

si n 2 θ

SPIRAL REVIEW 70. Travel A statistician kept a record of the number of tourists in Hawaii for six

months. Match each situation to its corresponding graph. (Lesson 9-1)

A B

a. There were predictions of hurricanes in March and April.

b. High airfares and high temperatures cause tourism to drop off in the summer.

Find each probability. (Lesson 11-3)

71. rolling a 4 on a number cube 72. getting heads on both tosses and a 4 on another number cube when a coin is tossed 2 times

Find four values for which each function is undefined. (Lesson 14-2)

73. y = - tan θ 74. y = sec (0.5 θ) 75. y = - csc θ

1_sin θ cos θ

1_1 - cos 2 θ

1

sin θ + cos θ - 1__sin θ + cos θ

sin θ

_sin θ + 1

graph B

π

_2 , 3π

_

2 , -

π

_2 , -

3π

_2 π, 3π, -π, -3π 0,π, -π, 2π

1_36

1_4

sin θ - cos θ

__sin θ

cos θ + 1_ cos 2 θ

graph A

Lesson 14-3 1013

Students having difficulty with Exercise 58 should

remember to change everything to sine and cosine. Students having dif-ficulty with Exercise 60 should begin with the identity 1 + ta n 2 θ = se c 2 θ and try to solve from there.

JournalHave students use an example to explain how to choose which side of an identity to begin working with and how to choose which identity to use.

Have students create three of their own trigonometric identities includ-ing all six trigonometric functions by working backwards from a true statement such as sin θ = sin θ and substituting for each side, e.g., from

sin θ = sin θ to 1 _ csc θ

= tan θ cos θ.

14-3


1. sin θ sec θ = 1 - co s 2 θ __ sin θ cos θ

= (

si n 2 θ _ sin θ cos θ

)

= sin θ

_ cos θ

= sin θ sec θ

2. se c 2 θ = 1 + si n 2 θ se c 2 θ

= 1 + si n 2 θ _ co s 2 θ

= 1 + ta n 2 θ

= se c 2 θ


3. si n 2 θ co t 2 θ sec θ cos θ

4. 2 (cs c 2 θ - co t 2 θ)

__

sec θ 2 cos θ


14-3 PROBLEM SOLVING 14-3 CHALLENGE

61. sin θ + cot θ cos θ

= sin θ + (

cos θ

_ sin θ

)

cos θ Given,

ratio identity

= sin 2 θ

_ sin θ

+ cos 2 θ

_ sin θ

Common

denominators

= sin 2 θ + cos 2 θ

__

sin θ Add fractions.

= 1 _ sin θ


= csc θ Ratio identity



Matrix multiplication and sum and difference identities are tools to find the coordinates of points rotated about the origin on a plane.

Sum and Difference Identities

Sum Identities Difference Identities

sin (A + B) = sin A cos B + cos A sin B sin (A - B) = sin A cos B - cosA sin B

cos (A + B) = cos A cos B - sin A sin B cos (A - B) = cos A cos B + sin A sin B

tan (A + B) = tanA + tan B__1 - tan A tan B

tan (A - B) = tanA - tan B__1 + tan A tan B

1E X A M P L E Evaluating Expressions with Sum and Difference Identities

Find the exact value of each expression.

A sin 75°

sin 75° = sin (30° + 45°) Write 75° as the sum 30° + 45° because

trigonometric values of 30° and 45° are known.

= sin 30° cos 45° + cos 30° sin 45° Apply identity for sin (A + B).

= 1_2

· √ � 2_2

+ √ � 3_2

· √ � 2_2

Evaluate.

= √ � 2_4

+ √ � 6_4

= √ � 2 + √ � 6_

4 Simplify.

B cos (- π _12)

cos(- π _12) = cos ( π

_6

- π _4 ) Write - π

_12

as the difference π _6

- π _4

.

= cos π _6

cos π _4

+ sin π _6

sin π _4

Apply the identity for cos (A - B).

= √ � 3_2

· √ � 2_2

+ 1_2

· √ � 2_2

Evaluate.

= √ � 6_4

+ √ � 2_4

= √ � 2 + √ � 6_

4 Simplify.


1a. tan 105° 1b. sin (- 11π

_12 )


ObjectivesEvaluate trigonometric expressions by using sum and difference identities.

Use matrix multiplication with sum and difference identities to perform rotations.

Vocabularyrotation matrix

Why learn this?You can use sum and difference identities and matrices to form images made from rotations. (See Example 4.)

In Example 1B, there is more than one way to get - π__

12. For

example, (π

__6

- π __4 ) or

(π

__4 - π

__3 ).

-2 - √

�

3√

�

2 - √

�

6_4

Introduce1




Block 1 __ 2 day

Objectives: Evaluate trigonometric expressions by using sum and difference identities.

Use matrix multiplication with sum and difference identities to perform rotations.

Online EditionTutorial Videos

Warm UpFind each product, if possible.

A =

⎡

⎢

⎣

1 _ 2

√ � 3

_ 2

-

√ � 3 _ 2

1 _ 2

⎤

�

⎦

B = ⎡

⎢

⎣

√ � 3

1

1

√ � 3 ⎤

�

⎦

1. AB ⎡

⎢

⎣

0

2

-1

√ �

3 ⎤

�

⎦

2. BA ⎡

⎢

⎣

√

� 3

2

-1

0 ⎤

�

⎦


“A good mathematical joke is bet-ter, and better mathematics, than a dozen mediocre papers.”

J. E. Littlewood

MotivateHave students consider the parabolas y = x 2 and x = y 2 . Have students brainstorm the type of transformation that would be necessary to trans-form one of these parabolas into the other. The transformation is a rotation, and its methods and techniques will be investigated in this lesson.

1014 Chapter 14


14-4 Sum and Difference Identities 1015

Shifting the cosine function right π radians is equivalent to reflecting it across the x-axis. A proof of this is shown in Example 2 by using a difference identity.

Phase Shift Right π Radians Reflection Across x-axis

2E X A M P L E Proving Identities with Sum and Difference Identities

Prove the identity cos (x - π) = -cos x.

cos (x - π) = -cos x Choose the left-hand side to modify.

cos x cos π + sin x sin π = Apply the identity for cos (A - B).

-1 · cos x + 0 · sin x = Evaluate.

-cos x = -cos x Simplify.

2. Prove the identity cos (x + π _2 ) = -sin x.

3E X A M P L E Using the Pythagorean Theorem with Sum and Difference Identities

Find tan (A + B) if sin A = -

7_25

with 180° < A < 270° and if cos B = 8_17

with 0° < B < 180°.

Step 1 Find tan A and tan B.

Use reference angles and the ratio definitions sin A = y__r and cos B = x__

r .Draw a triangle in the appropriate quadrant and label x, y, and r for each angle.

In Quadrant III (QIII), In Quadrant I (QI), 180° < A < 270° 0° < B < 180°and sin A = - 7_

25 . and cos B = 8_

17 .

x2 + (-7) 2

= 25 2 8 2 + y2 = 17 2

x = - √ �� 625 - 49 = -24 y = √ �� 289 - 64 = 15

Thus, tan A = y_x

= 7_24

. Thus, tan B = y_x

= 15_8

.

Refer to Lessons 13-2 and 13-3 to review reference angles.

Lesson 14-4 1015

Example 1


A. cos 1 5 ◦ √

� 6 + √

� 2 _

4

B. tan ( 11π

_ 12

) √ �

3 - 2

Example 2

Prove the identity.

tan (θ + π _ 4 ) = 1 + tan θ _

1 - tan θ

tan (θ + π _ 4 ) =

tan θ + tan π __ 4 __

1 - tan θ tan π __ 4

= 1 + tan θ

_ 1 - tan θ

Example 3

Find cos (A - B) if sin A = 1 _ 3

with 0 < A < π _ 2 and if tan B = 3 _

4

with 0 < B < π _ 2 .

8 √ �

2 + 3 _

15

Additional Examples


EXAMPLE 1

• How can you decide whether to add or subtract values?

EXAMPLE 2

• How can you decide what values to use for A and B in the sum and difference formulas?

EXAMPLE 3

• How can you check to see if your answer is reasonable?

Answers to Check It Out! 2. cos

(

π _ 2 + x

)

= cos (

π _ 2 )

cos x - sin (

π _ 2 )

sin x

= (0) cos x - (1) sin x

= -sin xGuided InstructionIntroduce the sum and difference identi-ties for sine, cosine, and tangent. Practice with students adding and subtracting angle measures from the unit circle to help iden-tify what angle measures can be evaluated with the sum and difference identities.

Teach2

Through Auditory Cues

To help students remember the sum and difference formulas, show students the following:

For sines, the formulas are the same:

sin (A + B) = sin A cos B + sin B cos A

sin (A - B) = sin A cos B - sin B cos A

For cosines, the formulas contradict:

cos (A + B) = cos A cos B - sin A sin B

cos (A - B) = cos A cos B + sin A sin B



Step 2 Use the angle-sum identity to find tan (A + B).

tan (A + B) = tan A + tan B__1 - tan A tan B

Apply identity for tan (A + B).

= ( 7__

24) + (15__8 )__

1 - ( 7__24)(15__

8 ) Substitute 7__

24 for tan A and 15__

8 for tan B.

tan (A + B) = 52__24_

1 - 35__64

, or 416_87

Simplify.

3. Find sin (A - B) if sin A = 4__5 with 90° < A < 180° and if

cos B = 3__5 with 0° < B < 90°.

To rotate a point P(x,y) through an angle θ, use a rotation matrix .

The sum identities for sine and cosine are used to derive the system of equations that yields the rotation matrix.

Using a Rotation Matrix

If P (x, y) is any point in a plane, then the coordinates P ′ (x′, y′) of the image after a rotation of θ degrees counterclockwise about the origin can be found by using the rotation matrix:

⎡

⎢

⎣ cosθ

sinθ

-sinθ

cosθ

⎤ �

⎦ ⎡ ⎢

⎣ x

y

⎤ �

⎦ =

⎡

⎢

⎣ x′

y′

⎤

�

⎦

4E X A M P L E Using a Rotation Matrix

Find the coordinates, to the nearest hundredth, of the points in the figure shown after a 30° rotation about the origin.

Step 1 Write matrices for a 30° rotation and for the points in the figure.

R 30° =⎡

⎢

⎣ cos 30°

sin 30°

-sin 30°

cos 30°

⎤ �

⎦ Rotation matrix

S =⎡ ⎢

⎣ 02

04

√ 3 1

-√ 3 1

⎤ �

⎦ Matrix of point coordinates

Step 2 Find the matrix product.

R 30° × S = ⎡ ⎢

⎣ cos 30°

sin 30°

-sin 30°

cos 30°

⎤

�

⎦ ⎡ ⎢

⎣ 02

04

√ 3 1

-√ 3 1

⎤ �

⎦

= ⎡

⎢

⎣ -1√ 3

-22√ 3

1 √ 3

-2

0 ⎤ �

⎦

Step 3 The approximate coordinates of the points after a 30° rotation

are A′ (-1, √ 3) , B′ (-2, 2 √ 3) ,

C ′ (1, √ 3) , and D ′ (-2, 0) .

4. Find the coordinates, to the nearest hundredth, of the points in the original figure after a 60° rotation about the origin.

24_25

A′ (- √

�

3 , 1) , B′ (-2√

�

3 , 2) , C′ (0, 2) , D′ (- √

�

3 , -1)

1016 Chapter 14

Example 4

Find the coordinates to the near-est hundredth of the points (1, 1) and (2, 0) after a 4 0 ◦ rotation about the origin. (0.12, 1.41) , (1.53, 1.29)

Additional Examples


EXAMPLE 4

• What are the coordinates of the point (1, 0) after a rotation of θ?

Critical Thinking Students may wonder how to show a clockwise rotation. This

rotation can be shown by

multiplying by ⎡

⎢

⎣

cos θ

-sin θ

sin θ

cos θ

⎤

⎦

.

Tell students that the clockwise rota-tion is equivalent to a (360 - θ)

◦

counterclockwise rotation.





and INTERVENTION SummarizeBy using angle sum and difference formu-las, it is possible to evaluate trigonometric functions for additional angles without resorting to a calculator. By using rotation matrices, you can rotate a point or a figure about the origin.

Close3




GUIDED PRACTICE 1. Vocabulary A geometric rotation requires that a center point of rotation be

defined. Which point and which direction does a rotation matrix such as Rθ assume?



2. cos 105° 3. sin 11π

_12

4. tan π _12

5. cos (-75°)


Prove each identity.

6. sin(π

_2

+ x) = cos x 7. tan(π + x) = tan x 8. cos(3π

_2

- x) = -sin x


Find each value if sin A = -

12_13

with 180° < A < 270° and if sin B = 4_5

with 90° < B < 180°.

9. sin (A + B) 10. cos (A - B) 11. tan (A + B) 12. tan (A - B)


13. Find the coordinates, to the nearest hundredth, of the vertices of triangle ABC with A(0, 2) , B (0, -1) , and C (3, 0) after a 120° rotation about the origin.



14–17 1 18–20 2 21–24 3 25 4

Independent Practice Find the exact value of each expression.

14. sin 7π

_12

15. tan 165° 16. sin 195° 17. cos 11π

_12


18. cos (3π

_2

+ x) = sin x 19. sin (3π

_2

+ x) = -cos x 20. tan (x - 2π) = tan x

Find each value if cos A = -

12___13

with 90° < A < 180° and if sin B = -

4__5 with

270° < B < 360°.

21. sin (A + B) 22. tan (A - B) 23. cos (A + B) 24. cos (A - B)

KEYWORD: MB7 Parent

ExercisesExercises

THINK AND DISCUSS 1. DESCRIBE three different ways that you can use the difference identity

to find the exact value of sin 15°.

2. EXPLAIN the similarities and differences between the identity formulas for sine and cosine. How do the signs of the terms relate to whether the identity is a sum or a difference?

3. GET ORGANIZED Copy and complete the graphic organizer. For each type of function, give the sum and difference identity and an example.


Extra Practice

A rotation matrix assumes a counterclockwise rotation about the origin.

√

�

2 - √

�

6_4

√

�

6 - √

�

2_4

2 - √

�

3√

�

6 - √

�

2_4

16_65

-33_65

16_63

-

56_33

A′ (-1.73, -1) , B′ (0.87, 0.5) , C′ (-1.5, 2.60)

√

�

6 + √

�

2_4 √

�

2 - √

�

6_4

- √

�

2 - √

�

6__4

√

�

3 - 2

63_65

33_56

-

16_65

-

56_65

Answers 6. sin

(

π _ 2 + x

)

= sin π _ 2 cos x + cos π

_ 2 sin x

= (1) cos x - (0) sin x

= cos x

7. tan (π + x)

= tan π + tan x

__ 1 - tan π tan x

= 0 + tan x

_ 1 - 0

= tan x

8. cos ( 3π

_ 2 - x

)

= cos 3π

_ 2 cos x + sin 3π

_ 2 sin x

= (0) cos x + (-1) sin x

= -sin x

18. cos ( 3π

_ 2 + x

)

= cos 3π

_ 2 cos x - sin 3π

_ 2 sin x

= (0) cos x - (

-1) sin x

= sin x

19, 20. See p. A53.

Lesson 14-4 1017


Assignment Guide


If you finished Examples 1–2 Basic 14–20 Average 14–20, 26–28 Advanced 14–20, 26–31, 54

If you finished Examples 1–4 Basic 14–25, 35–37, 41,

43–52, 60–67 Average 14–28, 35–53, 58–67 Advanced 14–67


14-4 ExercisesExercises

Answers to Think and DiscussPossible answers:

1. Evaluate sin (6 0 ◦ - 4 5 ◦ ) ,

sin (4 5 ◦ - 3 0 ◦ ) , or

sin (13 5 ◦ - 12 0 ◦ ) .

2. Both the sine and cosine identi-ties are in the form of a sum or difference of 2 products. The products each involve the 2 dif-ferent angles given in the same order (A then B) .

In sine identities, a cosine is multiplied by a sine, and the sign of the second term matches the sign between A and B. In cosine identities, cosines are multiplied by each other, sines are mul-tiplied by each other, and the signs are opposites.

3. See p. A14.



25. Find the coordinates, to the nearest hundredth, of the vertices of figure ABC with A(0, 2) , B(1, 2) , and C (0, 1) after a 45° rotation about the origin.


26. sin 165° 27. tan (-105°) 28. cos 195°

29. sin (-15°) 30. cos 19π

_12

31. tan 5π

_12

32. sin 255° 33. tan 195° 34. cos π _12

Find the value for each unknown angle given that 0° ≤ θ ≤ 180°.

35. cos (θ - 30°) = 1_2

36. cos (20° + θ) = √ � 2_2

37. sin (180° - θ) = 1_2

38. Physics Light enters glass of thickness t at an angle θi and leaves the glass at the same angle θi.However, the exiting ray of light is offset from the

initial ray by a distance Δ = ( sin(θ i - θ r)_________sin θ i cos θ r) t, indicated

in the figure shown.

a. Write the formula for Δ in terms of tangent and cotangent by using the difference identities and other trigonometric identities.

b. Use the figure to write a ratio for sin (θ i - θ r).

Multi-Step Find tan (A + B), cos (A + B), and sin (A - B) for each situation.

39. sin A = - 7_25

with 180° < A < 270° and cos B = 12_13

with 0° < B < 90°

40. sin A = - 1_3

with 270° < A < 360° and sin B = 4_5

with 0° < B < 90°

41. The figure PQRS will be rotated about the origin repeatedly to create the logo for a new product.

a. Write the rotation matrices for 90°, 180°, and 270° rotations.

b. Use your answers to part a to find the coordinates of the vertices of the figure after each of the three rotations.

c. Graph the three rotations on the same graph as PQRSto create the logo.

42. Critical Thinking Is it possible to find the exact value of sin (11π

_24 ) by using sum or

difference identities? Explain.


The displacement y of a mass attached to a spring is modeled by

y(t) = 4.2 sin (2π

___3

t - π __2 ) , where t is the time in seconds.

a. What are the amplitude and period of the function?

b. Use a trigonometric identity to write the displacement, using only the cosine function.

c. What is the displacement of the mass when t = 8 s?

A′(-1.41,1.41) , B′(-0.71, 2.12) , C′(-0.71, 0.71)

θ = 90° θ = 25° θ = 30° or 150°

2.1

y(t) = -4.2 cos 2π

_3

t

4.2; 3

38a.

Δ =

(1 - cot θ i tan θ r) t

26.√

�

6 - √

�

2_4

28.

- √

�

2 - √

�

6__4

29.√

�

2 - √

�

6_4

30.√

�

6 - √

�

2_4

32.

- √

�

6 - √

�

2__4

34.√

�

2 + √

�

6_4

2 + √

�

3

2 + √

�

3

2 -√

�3

39. 204_253

; -

253_325

;

36_325

40. 54 - 25√

�

2_28

;

4 + 6 √

�

2_15

;

-3 - 8√

�

2_15

sin(

θ i - θ r) = �

_h

1018 Chapter 14

Language For Exercise 38, ensure that students know that,

in this context, to offset is to displace or move out of position.

Multiple Representations Matrices like the one in Exercise 41 can be used to

represent other transformations. For example, a 18 0 ◦ rotation is equiva-lent to a reflection about the x- and y-axes.

Exercise 43 involves applying trigonomet-ric identities. This

exercise prepares students for the Multi-Step Test Prep on page 1034.

Answers

41a.⎡

⎢

⎣

0 1

-1

0 ⎤

�

⎦

; ⎡

⎢

⎣

-1 0

0

-1 ⎤

�

⎦

; ⎡

⎢

⎣

0 -1

1

0 ⎤

�

⎦

b. P ′ (0, 0) , Q ′ (-1, 1) , R ′ (0, 4) , S ′ (1, 1) ;

P ′′ (0, 0) , Q ′′ (-1, -1) , R ′′ (-4, 0) , S ′′ (-1, 1) ;

P ′′′ (0, 0) , Q ′′′ (1, -1) , R ′′′ (0, -4) , S ′′′ (-1, -1)

c.

42. Possible answer: No; 11π

_ 24

cannot be expressed as a sum or difference of values from the unit circle.


14-4 PRACTICE C

14-4 PRACTICE B

14-4 PRACTICE A




Geometry Find the coordinates, to the nearest hundredth, of the vertices of figure ABCD with A

(0, 3

) , B

(1, 4

) , C

(2, 3

) ,

and D(

2, 0)

after each rotation about the origin.

44. 45° 45. 60°

46. 120° 47. -30°

48. Write About It In general, does sin(A + B) = sin A + sin B? Give an example to support your response.

49. Which is the value of cos 15° cos 45° - sin 15° sin 45°?

1_2

√ � 2_2

- √ � 2_2

2 + √ � 2_2

50. Which gives the value for x if sin (π

_2

+ x) = 1_2

?π

_6

π

_4

π

_3

π

_2

51. Given sin A = 1__2 with 0° < A < 90° and cos B = 3__

5 with 0° < B < 90°, which expression gives the value of cos (A - B)?

3√ � 3 + 4_10

3√ � 3 - 4_10

3 + 4 √ � 3_10

3 - 4 √ � 3_10

52. Short Response Find the exact value for sin (-15°) . Show your work.

CHALLENGE AND EXTEND 53. Verify that the rotation matrix for θ is the inverse of the

rotation matrix for -θ.

54. Derive the identity for tan (A + B).

55. Derive the rotation matrix by using the sum identities for sine and cosine and recalling from Lesson 13-2 that any point P(x, y) can be represented as (r cos α, r sin α) by using a reference angle.

Find the angle by which a figure ABC with vertices A(

1, 0)

, B(

0, 2)

, and C(

-1, 0)

was rotated to get A′B′C ′.

56. A′ (0, 1) , B ′ (-2, 0) , C ′ (0, -1) 57. A′ ( √ � 2_2

, √ � 2_2 ) , B ′ (-√ � 2 , 2) , C ′ (-

√ � 2_2

, - √ � 2_2 )

58. A′ (-1, 0) , B ′ (0, -2) , C ′ (1, 0) 59. A′ ( √ � 3_

2 , 1_

2) , B ′ (-1, √ � 3) , C ′ (- √ � 3_2

, - 1_2)

SPIRAL REVIEWDivide. Assume that all expressions are defined. (Lesson 8-2)

60. 3x2_ 7y3

÷ 6x_21y

61. x2 + x - 2__x2 - 2x - 8

÷ x2 + 3x + 2__

x2 - 3x - 4 62.

9x3y2_ 15xy 4

÷ 6x4y_ 3x2y5

Identify the conic section that each equation represents. (Lesson 10-6)

63. x2 + 2xy + y2 + 12x - 25 = 0 64. 5x2 + 5y2 + 20x - 15y = 0

Rewrite each expression in terms of a single trigonometric function. (Lesson 14-3)

65. cot θ sec θ

_sin θ cos θ

66. cot θ tan θ csc θ 67. tan θ

_sec θ

sin θcsc θ sin 2 θ

90° 45°

3x_2y2

x - 1_x + 2

3y2_10

parabola circle

65. 1__cos θ - cos3

θ

180° 30°

Lesson 14-4 1019

Students having difficulty with Exercise 49 should

condense the expression into cos (1 5 ◦ + 4 5 ◦ ) rather than evaluating cos1 5 ◦ and sin1 5 ◦ .

Students having difficulty with Exercise 50 may wish to refer to the unit circle as a guide. If

sin ( π _ 2 + x) = 1 _

2 , then π

_ 2 + x = 5π

_ 6 .

Answers44–48, 52–55. See p. A53.

JournalHave students explain how the sum formulas can be used to show that adding 36 0 ◦ has no effect on the values of sine, cosine, or tangent.

Have students create their own quiz modeled after the one below, and have them supply complete answers. The quiz should use sine, cosine, and tangent formulas.

14-4

1. Find the exact value of cos 7 5 ◦ .

√

� 6 - √

� 2 _

4

2. Prove the identity

sin ( π _ 2 - θ) = cos θ.

= sin π _ 2 cos θ - cos π

_ 2 sin θ

= cos θ - 0 = cos θ

3. Find tan (A - B) for sin A = 12 _ 13

with 0 < A < π _ 2 and

cos B = 8 _ 17

with 0 < B < π _ 2 .

21 _ 220

4. Find the coordinates to the nearest hundredth of the point (3, 4) after a 6 0 ◦ rotation about the origin.

≈ ⎡

⎢

⎣

-1.96 4.60

⎤

⎦






Who uses this?Double-angle formulas can be used to find the horizontal distance for a projectile such as a golf ball. (See Exercise 49.)

ObjectiveEvaluate and simplify expressions by using double-angle and half-angle identities.

You can use sum identities to derive the double-angle identities.

sin 2θ = sin (θ + θ)

= sin θ cos θ + cos θ sin θ

= 2 sin θ cos θ

You can derive the double-angle identities for cosine and tangent in the same way. There are three forms of the identity for cos 2θ, which are derived by using sin2

θ + co s 2θ = 1. It is common to rewrite expressions as functions of θ only.

Double-Angle Identities

sin 2θ = 2 sin θ cos θ

cos 2θ = co s2θ - sin2

θ

cos 2θ = 2 co s2θ - 1

cos 2θ = 1 - 2 si n2θ

tan 2θ = 2 tan θ

_1 - tan2

θ

1E X A M P L E Evaluating Expressions with Double-Angle Identities

Find sin 2θ and cos 2θ if cos θ = -

3__4 and 90° < θ < 180°.

Step 1 Find sin θ to evaluate sin 2θ = 2 sin θ cos θ.

Method 1 Use the reference angle.

In QII, 90°< θ < 180°, and cos θ = - 3__4 .

(-3)2 + y2 = 4 2 Use the Pythagorean Theorem.

y = √ �� 16 - 9 = √ � 7 Solve for y.

sin θ = √

�

7_4

Method 2 Solve si n 2 θ = 1 - co s 2 θ.

si n 2 θ = 1- co s 2θ

sin θ = √ �� 1- (

-

3__4 )

2 Substitute - 3__

4 for cosθ.

= √ �� 1 - 9__16

= √ � 7___4 Simplify.

sin θ = √

�

7_4

The signs of x and y depend on the quadrant for angle θ. sin cos

QI + +QII + -QIII - -QIV - +

A211NLS_c14l05_1020_1026.indd 1020 2/9/10 8:46:34 AM

1020 Chapter 14

Introduce1




Block 1 __ 2 day

Objectives: Evaluate and simplify expressions by using double-angle and half-angle identities.

Online EditionTutorial Videos

Warm UpFind tan θ for 0 ≤ θ ≤ 9 0 ◦ , if

1. sin θ = 3 _ 5 . tan θ = 3 _

4

2. sin θ = 1 _ 3 . tan θ =

√ �

2 _

4

3. cos θ = x. tan θ = √ �� 1 - x 2

_ x


Teacher: Why is your answer sin θ _ 2

when the problem asks for sin 2θ?

Student: I guess it’s a case of mistaken identity.

MotivateAsk students how they might find the exact value of sin 22. 5 ◦ . Giving exact trigonometric function values for angles other than those on the unit circle may be necessary sometimes. In addition to using sum and difference identities, you can use the double-angle and half-angle identities.


14-5 Double-Angle and Half-Angle Identities 1021

Step 2 Find sin 2θ.

sin 2θ = 2 sin θ cos θ Apply the identity for sin 2θ.

= 2 (

√ �

7_4 )(

-

3_4 )

Substitute√ � 7_2

for sin θ and - 3_4

for cos θ.

= - 3√ � 7_

8 Simplify.

Step 3 Find cos 2θ.

cos 2θ = 2 co s 2 θ - 1 Select a double-angle identity.

= 2(

-

3_4 )

2

-1 Substitute - 3_4

for cos θ.

= 2 ( 9_16) -1 Simplify.

= 1_8

1. Find tan 2θ and cos 2θ if cos θ = 1__3 and 270° < θ < 360°.

You can use double-angle identities to prove trigonometric identities.

2E X A M P L E Proving Identities with Double-Angle Identities


A si n 2 θ = 1_2

(1 - cos 2θ)

sin 2 θ = 1_2

(1 - cos 2θ) Choose the right-hand side to modify.

= 1_2

(1- (1 - 2 sin 2 θ) ) Apply the identity for cos 2θ.

= 1_2

(2 sin 2 θ) Simplify.

sin 2 θ = sin 2 θ

B (cos θ + sin θ) 2

= 1 + sin 2θ

(cos θ + sin θ) 2 = 1 + sin 2θ

Choose the left-hand side to modify.

co s 2 θ + 2 cos θ sin θ + si n 2 θ = Expand the square.

(co s 2 θ + si n 2 θ) + (2 cos θ sin θ) = Regroup.

1 + sin 2θ = Rewrite using 1 = co s2θ + sin2

θ and sin 2θ = 2 sin θ cos θ.

1 + sin 2θ = 1 + sin 2θ


2a. co s 4 θ - sin4θ = cos 2θ 2b. sin 2θ = 2 tan θ

_1 + ta n 2 θ

You can use double-angle identities for cosine to derive the half-angle identitiesby substituting θ

__2 for θ. For example, cos 2θ = 2 cos 2 θ - 1 can be rewritten as

cos θ = 2 cos 2 θ __2 - 1. Then solve for cos θ

__2 .

Choose to modify either the left side or the right side of an identity. Do not work on both sides at once.

4 √

�

2_7 ; -

7_9

2b. Possible answer:

2 tan θ

_1 + tan 2 θ

= 2

(

sin θ

____cos θ

)

_ sec 2 θ

= 2

(

sin θ

____cos θ

)

_

1_____ cos 2 θ

· (

cos 2 θ

_____1 )_

(

cos 2 θ

_____1 )

= 2 (

sin θ

_cos θ

)

(

cos 2 θ

_1 )

= 2 sin θ cos θ = sin 2θ

Lesson 14-5 1021

Students may consider sin 2θ to be a function of only θ. Although θ is the only variable, the intent of the rewriting is to move the constants away from θ.

Example 1

Find sin 2θ and tan 2θ if

sin θ = 2 _ 5 and 0 ◦ < θ < 9 0 ◦ .

4 √

�� 21 _

25 ;

4 √ ��

21 _

17

Example 2


A. sin 2θ = 2 tan θ - 2 tan θ si n 2 θ

= 2 tan θ (1 - si n 2 θ)

= 2 tan θ co s 2 θ

= 2 (tan θ cos θ) cos θ

= 2 sin θ cos θ

= sin 2θ

B. cos 2θ = (2 - se c 2 θ) (1 - si n 2 θ)

= (2 - se c 2 θ) (co s 2 θ)

= 2 co s 2 θ - 1

= cos 2θ

Additional Examples


EXAMPLE 1

• When will the signs for sine, cosine, and tangent of 2θ be the same as the signs for sine, cosine, and tangent of θ?

EXAMPLE 2

• How do you decide what to do first when proving an identity?

Answers to Check It Out! 2a. Possible answer:

cos 4 θ - sin 4 θ

= ( cos 2 θ + sin 2 θ)

( cos 2 θ - sin 2 θ)

= (1) (cos 2θ)

= cos 2θ

Guided InstructionIntroduce the half-angle and double-angle identities. To help students identify the value of θ, encourage them to set up simple equations for each problem, such

as θ _ 2 = 12 0 ◦ , or 2θ = 12 0 ◦ .

Teach2

Through Critical Thinking

Have students show that the double-angle identities are special cases of the angle sum and difference identities from Lesson 14-4. For example, begin with sin (2θ) = sin (θ + θ) to derive the double-angle identity for sine.



Half-Angle Identities

sin θ

_2

= ± √ �� 1 - cosθ

_2

cos θ _2

= ±√ �� 1 + cosθ

_2

tan θ _2

= ± √ �� 1 - cosθ

_1 + cosθ

Choose + or - depending on the location of θ

__2.

Half-angle identities are useful in calculating exact values for trigonometric expressions.

3E X A M P L E Evaluating Expressions with Half-Angle Identities

Use half-angle identities to find the exact value of each trigonometric expression.

A cos 165° B sin π

_8

sin 1_2

(π

_4 )

+

√ ��

1 - cos

(

π

__4 )__

2

Positivein QI

√ ��

1 -

√ �

2___2_

2 cos π

_4

=√ � 2_2

√ ��

(2 - √ � 2_2 )(1_

2) Simplify.

√ �� 2 - √ � 2 _

2

cos 330° _2

- √ ��

1 + cos 330° __2

Negativein QII

- √

��

1 +

(

√ �

3___2

)

_2

cos 330 ° =√ � 3_2

- √ ��

(2 + √ � 3_2 )(1_

2) Simplify.

- √ �� 2 + √ � 3 _

2

Check Use your calculator. Check Use your calculator.


3a. tan 75° 3b. cos 5π

_8

4E X A M P L E Using the Pythagorean Theorem with Half-Angle Identities

Find sin θ __2 and tan θ

__2 if sin θ = -

5___13

and 180° < θ < 270°.

Step 1 Find cos θ to evaluate the half-angle identities.

Use the reference angle.

In QIII, 180° < θ < 270°, and sin θ = - 5__13

.

x 2 + (-5) 2 = 1 3 2 Pythagorean Theorem

x = - √ �� 169 - 25 = -12 Solve for the missing side x.

Thus, cos θ = -

12_13

.

In Example 3, the

expressions - √ �� 2 + √ � 3 ________

2

and√ �� 2 - √ � 2_______

2 are in

reduced form and cannot be simplified further.

3a. √ �� 7 + 4 √ � 3

b. -

√

��

2 - √

�

2 _2

a211se_c14l05_1020_1026.indd 1022 7/21/09 2:32:00 PM

1022 Chapter 14

Example 3


A. cos 1 5 ◦ √

��

√ �

3 + 2 _

2

B. tan 7π

_ 8 -

√

��

3 - 2 √ �

2

Example 4

Find cos θ _ 2 and tan θ

_ 2 if

tan θ = 7 _ 24

and 0 < θ < π _ 2 .

7 √

� 2 _

10 ; 1 _

7

Additional Examples


EXAMPLE 3

• How do you decide which sign to select when evaluating an expression by using the half-angle identity?

EXAMPLE 4

• What is the connection between the half-angle formulas and the corresponding difference formulas from Lesson 14-4?



Step 2 Evaluate sin θ_2

.

sin θ_2

+ √��1 - cos θ_2

Choose + for sin θ_2

where 90° < θ_2

< 135°.

√��1 -

(

-

12___13 )_

2Evaluate.

√��(25_13)(1_

2) Simplify.

√��25_26

5√ � 26 _26

Step 3 Evaluate tan θ_2

.

tan θ_2

- √��1 - cos θ_1 + cos θ

Choose - for tan θ_2

where 90° < θ_2

< 135°.

-√��1 - (

-

12___13 )_

1 + (

-

12___13 )

Evaluate.

-√��(25_13)(13_

1 ) Simplify.

-√�25

-5

4. Find sin θ__2 and cos θ__

2 if tan θ = 4__

3 and 0° < θ < 90°.

THINK AND DISCUSS 1. EXPLAIN which double-angle identity you would use to

simplify cos 2θ

_________sin θ + cos θ

.

2. DESCRIBE how to determine the sign of the value for sin θ __2 and

for cos θ __2 .

3. GET ORGANIZED Copy and complete the graphic organizer. In each box, write one of the identities.

Be careful to choose the correct sign for sin θ

__2 and cos θ

__2. If

180° < θ < 270°, then 90° < θ

__2

< 135°.

√

�

5_5 ; 2

√

�

5_5

Lesson 14-5 1023





and INTERVENTIONSummarizeExpressions in terms of twice an angle or half an angle can be rewritten as expres-sions in terms of the angle by using dou-ble-angle and half-angle identities.


1. cos 2θ = co s 2 θ - si n 2 θ, because it can be factored into (cos θ + sin θ) (cos θ - sin θ) , and the factor (cos θ + sin θ) can be divided out to eliminate the denominator

2. Determine the quadrant that θ __ 2 lies in

based on the quadrant that θ lies in, and identify the sign of sine (or cosine) in that quadrant.

3. See p. A14.

In problems such as Example 3, students may drop or combine the radicals. The nature of the calcula-tions is such that the answers will often contain radicals inside of other radicals. Remind students that these expressions are in simplest form.

Math Background To avoid confusion, you may wish to encourage stu-

dents to work on only one side of an equation at a time when proving identities. However, for more orga-nized students, it is legitimate to manipulate both sides of the identity until they are equal.



ExercisesExercises

GUIDED PRACTICE


Find sin 2θ, cos 2θ, and tan 2θ for each set of conditions.

1. cos θ = - 5_13

and π _2

< θ < π 2. sin θ = 4_5

and 0° < θ < 90°



3. 2 cos 2θ = 4 cos 2 θ - 2 4. si n 2 θ = 1 - cos 2θ + 1_2

5. 1 + cos 2θ

_sin 2θ

= cot θ 6. sin 2θ = 2 tan θ

_1 + ta n 2θ



7. cos 67.5° 8. cos π _12

9. tan 3π

_8

10. sin 112.5°


Find sin θ _2

, cos θ _2

, and tan θ _2

for each set of conditions.

11. sin θ = - 24_25

and 180° < θ < 270° 12. cos θ = 1_4

and 270° < θ < 360°



13–14 1 15–18 2 19–22 3 23–24 4

Independent Practice Find sin 2θ, cos 2θ, and tan 2θ for each set of conditions.

13. cos θ = - 7_25

and 90° < θ < 180° 14. tan θ = 20_21

and 0 ≤ θ ≤ π _2


15. sin 2θ

_sin θ

= 2 cos θ 16. cos 2 θ = 1_2

(1 + cos 2θ)

17. tan θ = 1 - cos 2θ

_sin 2θ

18. tan θ = sin 2θ

_1 + cos 2θ


19. sin 7π

_12

20. cos 5π

_12

21. sin 22.5° 22. tan 15°

Find sin θ _2

, cos θ _2

, and tan θ _2


23. tan θ = - 12_35

and 3π

_2

< θ < 2π 24. sin θ = - 3_5

and 180° < θ < 270°

Multi-Step Rewrite each expression in terms of trigonometric functions of θrather than multiples of θ. Then simplify.

25. sin 3θ 26. sin 4θ

27. cos 3θ 28. cos 4θ

29. cos 2θ + 2 sin 2 θ 30. cos 2θ + 1

31. tan 2θ(2 - sec 2 θ) 32. cos 2θ__cos θ + sin θ

33. cos θ sin 2θ

_1 + cos 2θ

34. cos 2θ - 1_ sin 2 θ


KEYWORD: MB7 Parent


Extra Practice

-

120_169

; -

119_169

; 120_119

24_25

; -

7_25

; -

24_7

4_5 ; -

3_5 ; -

4_3

√

�

6_4

; -

√

�

10_4

; -

√

�

15_5

-

336_625

; -

527_625

; 336_527

840_841

; 41_841

; 840_41

19.√

��

2 + √

�

3 _2

20.√

��

2 - √

�

3 _2

21.√

��

2 - √

�

2 _2

22.

√

��

2 - √

�

3_2 + √

�

3 1

sin θ

2 cos 2 θ

2 tan θ cos θ - sin θ

-2

25. 3 sin θ cos 2 θ - sin 3 θ 26. 4 sin θ cos 3 θ - 4 cos θ sin 3 θ

23.√

�

37_37

; -

6√

�

37_37

; -

1_6

24. 3√

�

10_10

;-√

�

10_10

; -3

7.√

��

2 - √

�

2 _2

√

��

2 + √

�

3 _2

√

��

2 + √

�

2_2 - √

�

2

√

��

2 + √

�

2 _2

1024 Chapter 14



14-5

Assignment Guide


If you finished Examples 1–2 Basic 13–18, 29–30 Average 13–18, 29–32 Advanced 13–18, 29–32, 57

If you finished Examples 1–4 Basic 13–24, 29–37, 48, 49,

51–56, 65–74 Average 13–37, 44–57, 65–74 Advanced 14–34 even, 35–74


Answers 3. 2 cos 2 θ = 2 (2 cos 2 θ - 1)

= 4 cos 2 θ - 2

4. sin 2 θ = 1 - cos 2 θ

= 1 -

2 cos 2 θ

_ 2

= 1 -

(2 cos 2 θ - 1) + 1

__ 2

= 1 -

cos 2θ + 1 _ 2

5. 1 + cos 2θ

_ sin 2θ

= 1 + (2 cos 2 θ - 1)

__ (2 sin θ cos θ)

= 2 cos 2 θ

__ 2 sin θ cos θ

= cos θ

_ sin θ

= cot θ

6, 15–18. See p. A53. 27. cos θ (1 - 4 sin 2 θ)

28. sin 4 θ + cos 4 θ - 6 sin 2 θ cos 2 θ

ExercisesExercises



The Tevatron at Fermi National Accelerator Lab in Batavia, Illinois, uses superconducting magnets to study subatomic particles by colliding matter and antimatter inside of a ring with a diameter of 6.3 km.

Physics

Multi-Step Find sin 2θ, cos 2θ, tan 2θ, sin θ _2

, cos θ _2

, and tan θ _2


36. cos θ = 3_8

and π _2

< θ < π 37. cos θ = - √ � 5_3

and 180° < θ < 270°

38. sin θ = 2_5

and 0° < θ < 90° 39. tan θ = - 1 _ 2

and 3π

_ 2

< θ < 2π


40. cos 7π

_8

41. sin 11π

_12

42. cos 105° 43. sin (-15°)

44. Physics The change in momentum of a scattered nuclear particle is given by ΔP = Pf - Pi , where Pf is the final momentum, and Pi is the initial momentum.

a. Use the diagram and the Pythagorean Theorem to write a formula for ΔP in terms of Pi . Then write a formula for ΔP in terms of Pf .

b. Compare your two answers to part a. What does this tell you about the magnitude, or size, of the momentum before and after the “collision”?

c. Write the formula for ΔP in terms of cos θ.


45. co s 2 θ _2

= sin 2 θ

__2(1 - cos θ)

46. cos 2θ = 1 - tan2θ

_1 + tan 2 θ

47. tan θ + sin θ

__2 tan θ

= cos 2 θ _2

48. Graphing Calculator Graph y = (cos x)(1 - cos 2x)_____________

sin 2x to discover an identity.

Then prove the identity.

49. Multi-Step A golf ball is hit with an initial velocity of v0 in feet per second at

an angle of elevation θ. The function d (θ) = v0 2 sin θ cos θ

__________

16 gives the horizontal distance

d in feet that the ball travels.

a. Rewrite the function in terms of the double angle 2θ.

b. Calculate the horizontal distance for an initial velocity of 80 ft/s for angles of 15°, 30°, 45°, 60°, and 75°.

c. For a given velocity, what angle gives the maximum horizontal distance?

d. What if...? If the initial velocity is 80 ft/s, through what approximate range of angles will the ball travel horizontally at least 175 ft?

50. Critical Thinking Explain how to find the exact value for sin 7.5°.

51. Write About It How do you know when to use a double-angle or a half-angle identity?


The displacement y of a mass attached to a spring is modeled by y(t) = 3.1 sin 2t,where t is the time in seconds.

a. Rewrite the function by using a double-angle identity.

b. The displacement w of another mass attached to a spring is given by w(t) = 3.8 cos t. The two masses are set in motion at t = 0. When do the masses have the same displacement for the first time?

c. What is the displacement at this time?

ΔP = 2 Pf √

��

1 - cos θ

_2

100 ft; ≈ 173 ft; 200 ft; ≈ 173 ft; 100 ft45°

30.52° < θ < 59.48°

d(θ) = v0

2 sin 2θ

_

32

Possible answer: If θ is multiplied by 2, use a double-angle identity. If θ is divided by 2, use a half-angle identity.

about 3.00 mabout 0.66 s

y(t) = 6.2 sin t cos t

42. -

√

��

2 - √

�

3 _2

43. -

√

��

2 - √

�

3 _2

-

√

��

2 + √

�

2 _2

√

��

2 - √

�

3 _2

It does not change.

Lesson 14-5 1025

14-5 PRACTICE C

14-5 PRACTICE B

14-5 PRACTICE A


Exercise 35 involves applying double-angle identities. This

exercise prepares students for the Multi-Step Test Prep on page 1034.

Answers

36. -

3 √ ��

55 _ 32

; -

23 _ 32

; 3 √

�� 55 _

23 ;

√

�� 11 _

4 ;

√ �

5 _

4 ;

√ ��

55 _ 5

37. 4 √ �

5 _ 9 ; 1 _

9 ; 4 √

� 5 ;

√

��

18 + 6 √ �

5 __

6 ;

-

√

��

18 - 6 √ �

5 __

6 ; -

√

��

3 + √ �

5 _

√

��

3 - √ �

5

38. 4 √ ��

21 _ 25

; 17 _ 25

; 4 √

�� 21 _

17 ;

√

��

50 - 10 √ ��

21 __

10 ;

√

��

50 + 10 √ ��

21 __

10 ;

√

��

5 - √ ��

21 __

√

��

5 + √ ��

21

39. -

4 _ 5 ; 3 _

5 ; -

4 _ 3 ;

√

��

5 - 2 √

�

5 _

10 ;

-

√

��

5 + 2 √

�

5 _

10 ; -

√

��

5 - 2 √

�

5 _

5 + 2 √ �

5

44a. ΔP = 2 P i sin (

θ _ 2 )

;

ΔP = 2 P f sin (

θ _ 2 )

45–48. See p. A53.

50. First find sin 15° by using

sin 30°

_ 2 , and then find

sin 7.5° by using sin 15°

_ 2 .



52. What is the value of sin 2 θ if cos θ = - √ � 2_2

and 90° < θ < 180°?

1_2

√ � 2_2

1 -1

53. What is the value for cos 2 θ if sin θ = cos θ?

0 1 2 sin 2θ 2 cos2θ

54. What is the value for sin θ _2

if cos θ = - 12_13

and 90° < θ < 180°?

√ � 26_26

-√ � 26_26

5√ � 26_26

-5√ � 26_

26

55. What is the exact value for sin 157.5°?

- √ �� 2 - √ � 2_

2 √ �� 2 - √ � 2_

2-

√ �� 2 + √ � 2_2

√ �� 2 + √ � 2_

2

56. Short Response Verify that cos 2 θ

_________sinθ + cos θ

= cos θ - sinθ for 0 ≤ θ ≤ π

__2. Show each

step in your justification process.

CHALLENGE AND EXTEND 57. Derive the double-angle formula for tan 2θ by using the ratio identity for tangent and

the double-angle identities for sine and cosine.

58. Derive the half-angle formula for tan θ _2

by using the ratio identity for tangent.

Use half-angle identities to find the exact value of each expression.

59. tan 7.5° 60. tan π _16

61. sin π _24

62. cos 11.25°

63. Write About It For what values of θ is sin 2θ = 2 sin θ true? Explain first by using graphs and then by solving the equation.

64. Derive the product-to-sum formulas sin A sin B = 1__2 ⎡ ⎣ cos(A - B) - cos(A + B)⎤ ⎦ and

cos A cos B = 1__2 ⎡ ⎣ cos(A + B) + cos (A - B)⎤ ⎦ by using the angle sum and difference

formulas.

SPIRAL REVIEWUse the vertical-line test to determine whether each relation is a function. (Lesson 1-6)

65. 66.

Add or subtract. Identify any x-values for which the expression is undefined. (Lesson 8-3)

67. 3x - 2_x + 7

+ 2x + 14_x + 7

68. 4x - 1_x + 6x - 2_

2x

69. 7x + 4_x + 1

- 5x + 8_x - 3

70. x + 9_x2

- x_x + 2

Find the exact value of each expression. (Lesson 14-4)

71. sin (- π _12) 72. sin 105° 73. cos 7π

_12

74. cos 255°

no yes

67. 5x + 12 _x + 7

; x ≠ -7

68. 7x - 2_x ; x ≠ 0

69. 2x2- 30x - 20__

(x + 1)(x - 3)

;

x ≠ -1, 3

71. √

�

2 - √

�

6_4

- x3+ x2

+ 11x + 18__x2

(x + 2)

;

x ≠ -2, 0

√

�

2 + √

�

6_4

√

�

2 - √

�

6_4

√

�

2 - √

�

6_4

a207se_c14l05_1020_1026.indd 1026 2/16/07 12:48:44 PM

60. tan ⎡

⎢

⎣

1 _ 2 (

1 _ 2 · π

_ 4 )

⎤

⎦

= √

��

2 -

√

��

2 + √ �

2 __

2 + √

��

2 + √ �

2

61. sin ⎡

⎢

⎣

1 _ 2 (

1 _ 2 · π

_ 6 )

⎤

⎦

= 1 _ 2 √

��

2 - √

��

2 + √ �

3

62. cos ⎡

⎢

⎣

1 _ 2 (

45°

_ 2

)

⎤

⎦

= 1 _ 2 √

��

2 + √

��

2 + √ �

2

63. π n, where n is an integer; possible answer:

Solve the equation:

sin 2 θ = 2 sin θ

2 sin θ cos θ - 2 sin θ = 0

2 sin θ (cos θ - 1) = 0

So cos θ = 1 or sin θ = 0, which are both true when θ = πn, where n is an integer.

57, 58, 64. See p. A53.

1026 Chapter 14

In Exercise 55, stu-dents who have dif-ficulty should refer to

the unit circle. The answer must be a positive number because 157. 5 ◦ is in quadrant 2. The value must also be closer to zero than it is to one. These characteristics apply to only choice G.

Answers 56. Possible answer:

cos 2θ

__ sin θ + cos θ

= ( cos 2 θ - sin 2 θ)

__

sin θ + cos θ

= (cos θ - sin θ) (cos θ + sin θ)

___

sin θ + cos θ

= cos θ - sin θ

59. tan ⎡

⎢

⎣

1 _ 2 (

30°

_ 2

)

⎤

⎦

= √

��

2 -

√

��

2 + √ �

3 __

2 + √

��

2 + √ �

3

JournalHave students show that, given the first double-angle identity for cosines, cos 2θ = co s 2 θ - si n 2 θ, they can derive the other two double-angle identities by using the Pythagorean identities.

Have students present, preferably to the class, a derivation of one double-angle identity. They should also include a numerical example of the identity in use.

14-5

1. Find cos θ _ 2 and cos 2θ if

sin θ = 5 _ 8 and 0 < θ < π

_ 2 .

√

��

√ ��

39 + 8 __

4 ; 7 _

32

2. Prove the following identity:

tan θ = sin 2θ

__ 2 - 2 si n 2 θ

= 2 sin θ cos θ

__ 2 (1-si n 2 θ)

= sin θ cos θ

_ co s 2 θ

= tan θ

3. Find the exact value of cos 22. 5 ◦ .

√

��

√ �

2 + 2 _

2



14-6 Solving Trigonometric Equations 1027

Unlike trigonometric identities, most trigonometric equations are true only for certain values of the variable, called solutions. To solve trigonometric equations, apply the same methods used for solving algebraic equations.

1E X A M P L E Solving Trigonometric Equations with Infinitely Many Solutions

Find all of the solutions of 3 tan θ = tan θ + 2.

Method 1 Use algebra.

Solve for θ over one cycle of the tangent, -90° < θ < 90°.

3 tan θ = tan θ + 2

3 tan θ - tan θ = 2 Subtract tan θ from both sides.

2 tan θ = 2 Combine like terms.

tan θ = 1 Divide by 2.

θ = tan -1 1 Apply the inverse tangent.

θ = 45° Find θ when tan θ = 1.

Find all real number values of θ, where n is an integer.

θ = 45° + 180°n Use the period of the tangent function.

Method 2 Use a graph.

Graph y = 3 tan θ and y = tan θ + 2 in the same viewing window for -90° ≤ θ ≤ 90°.

Use the intersect feature of your graphing calculator to find the points of intersection.

The graphs intersect at θ = 45°. Thus, θ = 45° + 180°n, where n is an integer.

1. Find all of the solutions of 2 cos θ + √ � 3 = 0.

Some trigonometric equations can be solved by applying the same methods used for quadratic equations.


ObjectivesSolve equations involving trigonometric functions.

Why learn this?You can use trigonometric equations to determine the day of the year that the sun will rise at a given time. (See Example 4.)

Compare Example 1 with this solution: 3x = x + 2 3x-x = 2 2x = 2 x = 1

150° + 360°n, 210° + 360°n

Lesson 14-6 1027



Block 1 __ 2 day

Objectives: Solve equations involving trigonometric functions.

Online EditionGraphing Calculator, Tutorial Videos, Interactivity, TechKeys

Warm UpSolve.

1. x 2 + 3x - 4 = 0 x = 1 or -4

2. 3 x 2 + 7x = 6 x = 2 _ 3 or -3

Evaluate each inverse trigonometric function.

3. Ta n -1 1 4 5 ◦

4. Si n -1 - √

� 3 _

2 -6 0 ◦


“Do not worry about your difficulties in mathematics. I can assure you that mine are still greater.”

Albert Einstein

Introduce1


MotivateRemind students of properties of the real num-bers such as the Distributive Property. Point out that these properties are in fact algebraic identi-ties and were the basic tools used for solving algebraic equations. In this lesson, the trigono-metric identities that have been studied will be used to solve trigonometric equations. Also, some methods used to solve algebraic equations will be used to solve trigonometric equations.



2E X A M P L E Solving Trigonometric Equations in Quadratic Form

Solve each equation for the given domain.

A sin 2 θ - 2 sin θ = 3 for 0 ≤ θ < 2π

sin 2 θ - 2 sin θ - 3 = 0 Subtract 3 from both sides.

(sin θ + 1)(sin θ - 3) = 0 Factor the quadratic expression by

comparing it with x2 - 2x - 3 = 0.sin θ = -1 or sin θ = 3 Apply the Zero Product Property.

sin θ = 3 has no solution because -1 ≤ sin θ ≤ 1.

θ = 3π

_2

The only solution will come from sinθ = -1.

B cos 2 θ + 2 cos θ - 1 = 0 for 0° ≤ θ < 360°

The equation is in quadratic form but cannot easily be factored. Use the Quadratic Formula.

cos θ = - (2) ± √ �� (2)2 - 4(1)(-1) ___

2(1)

Substitute 1 for a, 2 for b, and -1 for c.

cos θ = -1 ± √ � 2 Simplify.

-1 - √ � 2 < -1 so cos θ = -1 - √ � 2 has no solution.

θ = cos -1(-1 + √ � 2) Apply the inverse cosine.

≈ 65.5° or 294.5° Use a calculator. Find both angles for 0° ≤ θ < 360°.

Solve each equation for 0 ≤ θ < 2π.

2a. cos 2 θ + 2 cos θ = 3 2b. sin 2 θ + 5 sin θ - 2 = 0

You can often write trigonometric equations involving more than one function as equations of only one function by using trigonometric identities.

3E X A M P L E Solving Trigonometric Equations with Trigonometric Identities

Use trigonometric identities to solve each equation for 0 ≤ θ < 2π.

A 2 cos 2 θ = sin θ + 1

2 (1 - sin2θ ) - sin θ - 1 = 0

Substitute 1 - sin2θ for cos2

θ by the Pythagorean identity.

-2 sin 2 θ - sin θ + 2 - 1 = 0 Simplify.

2 sin 2 θ + sin θ - 1 = 0 Multiply by -1.

(2 sin θ - 1)(sin θ + 1) = 0 Factor.

sin θ = 1_2

or sin θ = -1 Apply the Zero Product Property.

θ = π _6

or 5π

_6

or θ = 3π

_2

Check Use the intersect feature of your graphing calculator. A graph supports your answer.

A trigonometric equation may have zero, one, two, or an infinite number of solutions, depending on the equation and domain of θ.

≈ 21.9°, ≈ 158.1°

0

a207se_c14l06_1027_1033.indd 1029 2/15/07 2:44:54 PM

1028 Chapter 14

Guided InstructionBefore solving quadratic trigonometric equations, some students may need to review solving basic quadratic equations. Make the connection between a trigono-metric equation and an algebraic equa-tion by replacing sine or cosine with x. Demonstrate that the process of factoring and isolating the variable will remain the same.

Teach2

Through Cooperative Learning

When solving equations or systems of equations, students may use different approaches to arrive at the correct answer. Have students work in small groups to discuss and share the techniques that they used on specific problems, and the way that they decided which method to use.

Example 1

Find all solutions for

sin θ = 1 _ 2 sin θ + 1 _

4 .

θ = {30° + 360°n, 150° + 360°n}

Example 2


A. 4 ta n 2 θ - 7 tan θ + 3 = 0 for 0 ≤ θ ≤ 36 0 ◦ .

(tan θ - 1) (4 tan θ - 3) = 0θ = 4 5 ◦ or 22 5 ◦ , or θ ≈ 36.9 or 216.9

B. 2 co s 2 θ - cos θ = 1 for 0 ≤ θ ≤ π.

(2 cos θ + 1) (cos θ - 1) = 0

θ = 2π

_ 3 or θ = 0

Additional Examples


EXAMPLE 1

• If a trigonometric equation with no domain restriction has at least one solution, will it always have infinitely many solutions?

EXAMPLE 2

• What are possible numbers of solu-tions for a quadratic trigonometric equation with one variable 0 < θ < 36 0 ◦ ?


Augusta Mt. DesertIsland

N

S

W E

M A I N E

A t l a n t i cO c e a n


Use trigonometric identities to solve each equation for 0° ≤ θ < 360°.

B cos 2θ + 3 cos θ + 2 = 0

2 cos 2 θ - 1 + 3 cos θ + 2 = 0 Substitute 2 cos2

θ - 1 for cos 2θ by the double-angle identity.

2 cos 2 θ + 3 cos θ + 1 = 0 Combine like terms.

(2 cos θ + 1)(cos θ + 1) = 0 Factor.

cos θ = - 1_2

Apply the Zero Product Property.

or

cos θ = -1

θ = 120° or 240° or θ = 180°

Check Use the intersect feature of your graphing calculator. A graph supports your answer.

Use trigonometric identities to solve each equation for the given domain.

3a. 4 sin 2 θ + 4 cos θ = 5 for 0° ≤ θ < 360°

3b. sin 2θ = -cos θ for 0 ≤ θ < 2π

4E X A M P L E Problem-Solving Application

The first sunrise in the United States each day is observed from Cadillac Mountain on Mount Desert Island in Maine. The time of the sunrise can be modeled by t(m) = 1.665 sin π

__6 (m + 3) + 5.485,

where t is hours after midnight and m is the number of months after January 1. When does the sun rise at 7 A.M.?

1 Understand the Problem

The answer will be months of the year.List the important information:• The function model is

t(m) = 1.665 sin π __6 (m + 3) + 5.485.

• Sunrise is at 7 A.M., which is represented by t = 7.• m represents the number of months after January 1.

2 Make a Plan

Substitute 7 for t in the model. Then solve the equation for m by using algebra.

60°, 300°

90°, 210°, 270°, 330°

a207se_c14l06_1027_1033.indd 1029 2/15/07 2:44:54 PM

Lesson 14-6 1029

Example 3

Use trigonometric identities to solve each equation.

A. ta n 2 θ + se c 2 θ = 3 for 0 ≤ θ ≤ 2π.

θ = ⎧

⎨

⎩

π _ 4 , 3π

_ 4 , 5π

_ 4 , 7π

_ 4

⎫

⎬

⎭

B. co s 2 θ = 1 + si n 2 θ for 0 ≤ θ ≤ 36 0 ◦ .

θ = 0 ◦ or 18 0 ◦ or 36 0 ◦

Additional Examples


EXAMPLE 3

• How do you determine which identity will help solve an equation or whether one is even necessary?

• Is it possible to solve an equation containing more than one trigono-metric function without converting everything to the same function?



3 Solve

7 = 1.665 sin π_6

(m + 3) + 5.485 Substitute 7 for t.

7 - 5.485_1.665

= sin π_6

(m + 3) Isolate the sine term.

sin -1(0.9−−−099) = π_

6(m + 3) Apply the inverse sine.

Sine is positive in Quadrants I and II. Compute both values.

QI: sin -1(0.9−−−099) = π_

6(m + 3) QII: π - sin -1(0.9

−−−099) = π_

6(m + 3)

1.143 ≈ π_6

(m + 3) π - 1.143 ≈ π_6

(m + 3)

(

6_π

)1.143 ≈ m + 3

(

6_π

)

(π - 1.143) ≈ m + 3

-0.817 ≈ m 0.817 ≈ mThe value m = 0.817 corresponds to late January and the valuem = -0.817 corresponds to early December.

4 Look Back

Check your answer by using a graphing calculator. Enter y = 1.665 sin π__

6(x + 3) + 5.485

and y = 7. Graph the functions on the same viewing window, and find the points of intersection.The graphs intersect at about 0.817 and -0.817.

4. The number of hours h of sunlight in a day at Cadillac Mountain can be modeled by

h(d) = 3.31 sin π____182.5

(d - 85.25) + 12.22, where

d is the number of days after January 1. When are there 12 hours of sunlight?

THINK AND DISCUSS 1. DESCRIBE the general procedure for finding all real-number solutions

of a trigonometric equation.

2. GET ORGANIZED Copy and complete the graphic organizer. Write when each method is most useful, and give an example.

Be sure to have your calculator in radian mode when working with angles expressed in radians.

late March and late September

1030 Chapter 14





and INTERVENTIONSummarizeReview with students the different meth-ods for solving trigonometric equations, such as graphing, factoring, and substitut-ing by using trigonometric identities.


1. First solve the equation for a restricted domain equal to the period of the given function. Then use an under-standing of the periodicity of the given function to find all solutions.

2. See p. A14.

Example 4

When does the sun rise at 4 a.m. on Cadillac Mountain? Use the equation from Example 4.

early June and late July

Additional Examples


EXAMPLE 4

• How do you know what month the value of m corresponds to?



ExercisesExercises

GUIDED PRACTICE


Find all of the solutions of each equation.

1. 6 cos θ - 1 = 2 2. 2 sin θ - √ � 3 = 0 3. cos θ = √ � 3 - cos θ



4. 2 sin 2 θ + 3 sin θ = -1 for 0 ≤ θ < 2π 5. cos 2 θ - 4 cos θ + 1 = 0 for 0° ≤ θ < 360°


Multi-Step Use trigonometric identities to solve each equation for the given domain.

6. 2 sin 2 θ - cos 2θ = 0 for 0° ≤ θ < 360° 7. sin 2 θ + cos θ = -1 for 0 ≤ θ < 2π


8. Heating The amount of energy from natural gas used for heating a manufacturing plant is modeled by E(m) = 350 sin π

__6 (m + 1.5) + 650, where E is the energy used in

dekatherms, and m is the month where m = 0 represents January 1. When is the gas usage 825 dekatherms? Assume an average of 30 days per month.



9–12 1 13–14 2 15–16 3 17 4

Independent Practice Find all of the solutions of each equation.

9. 1 - 2 cos θ = 0 10. √ � 3 tan θ - 3 = 0

11. 2 cos θ + √ � 3 = 0 12. 2 sin θ + 1 = 2 + sin θ


13. 2 cos 2 θ + cos θ - 1 = 0 for 0 ≤ θ < 2π 14. sin 2 θ + 2 sin θ - 2 = 0 for 0° ≤ θ < 360°

Multi-Step Use trigonometric identities to solve each equation for the given domain.

15. cos 2θ + cos θ + 1 = 0 for 0° ≤ θ < 360° 16. cos 2θ = sin θ for 0 ≤ θ < 2π

17. Multi-Step The amount of energy used by a large office building is modeled by E(t) = 100 sin π

__12

(t - 8) + 800, where E is the energy in kilowatt-hours, and t is the time in hours after midnight.

a. During what time in the day is the electricity use 850 kilowatt-hours?

b. When are the least and greatest amounts of electricity used? Are your answers reasonable? Explain.

Solve each equation algebraically for 0° ≤ θ < 360°.

18. 2 sin 2 θ = sin θ 19. 2 cos 2 θ = sin θ + 1

20. cos 2θ - 2 sin θ + 2 = 0 21. 2 cos 2 θ + 3 sin θ = 3

22. cos 2 θ + sin θ - 1 = 0 23. 2 sin 2 θ + sin θ = 0

Solve each equation algebraically for 0 ≤ θ < 2π.

24. sin 2 θ - sin θ = 0 25. cos 2 θ - 3 cos θ = 4

26. cos θ (0.5 + cos θ) = 0 27. 2 sin 2 θ - 3 sin θ = 2

28. cos 2 θ + 1_2

cos θ = 5 29. sin 2 θ + 3 sin θ + 3 = 0

30. cos 2 θ + 4 cos θ - 3 = 0 31. tan 2 θ = √ � 3 tan θ


KEYWORD: MB7 Parent


Extra Practice

60° + 360°n, 300° + 360°n 60° + 360°n, 120° + 360°n

30° + 360°n, 330° + 360°n

≈ 74.5° or 285.5°

30°, 150°, 210°, 330°π

mid-April and mid-December

60° + 180°n

150° + 360°n, 210° + 360°n90° + 360°n

60° + 360°n, 300° + 360°n

13. π _3

, π, or 5π

_3

≈ 47.1°, ≈ 132.9°

π

_6 , 5π

_

6 , 3π

_

290°, 120°, 240°, 270°

17a. 10:00 A.M. and 6:00 P.M.

0°, 180°, 210°, 330°

0°, 30°, 150°, 180° 30°, 150°, 270°

≈ 55.4°, ≈ 124.6°30°, 90°, 150°

0°, 90°, 180°

24. 0, π _2

, π

27. 7π

_6

, 11π

_6

28. no solutionπ

π

_2 , 2π

_

3 , 4π

_

3 , 3π

_

2no solution

≈ 0.869, ≈ 5.4140,

π

_3 , π, 4π

_

3

Lesson 14-6 1031

ExercisesExercises


Assignment Guide


If you finished Examples 1–2 Basic 9–14, 18–23 Average 9–14, 18–23, 52 Advanced 9–14, 18–23, 52, 55

If you finished Examples 1–4 Basic 9–27, 33–51, 58–64 Average 9–51, 52, 56, 58–64 Advanced 9–34, 36–64


Science Link For Exercise 8, note that the metric prefix deka- is

sometimes written deca-. A therm is equivalent to 100,000 Btu. One Btu (British thermal unit) is the amount of heat required to raise the temper-ature of 1 pound of water 1 degree Fahrenheit.

Answers 4. 3π

_ 2 , 7π

_ 6 , 11π

_

6

17b. Possible answer: The minimum electricity use is at 2:00 a.m., when no one is at work. The maximum electricity use is at 2:00 p.m., when the building is fully occupied. 2:00 p.m. is also near the hottest time of the day, so the building’s electricity use may be very high because of air conditioning. The answers are reasonable.

14-6



Traditional Japanese kabuki theaters were round and were able to be rotated to change scenes. The stages were also equipped with trapdoors and bridges that led through the audience.

Performing Arts

32. Sports A baseball is thrown with an initial velocity of 96 feet per second at an angle θ degrees with a horizontal.

a. The horizontal range R in feet that the ball travels can be modeled by

R(θ) = v2 sin 2θ

_

32 . At what angle(s) with the horizontal will the ball travel 250 feet?

b. The maximum vertical height Hmax in feet that the ball travels upward can be

modeled by Hmax(θ) = v 2 sin 2 θ

_

64 . At what angle(s) with the horizontal will the ball

travel 50 feet?

33. Performing Arts A theater has a rotating stage that can be turned for different scenes. The stage has a radius of 18 feet, and the area in square feet of the segment of the circle formed by connecting two radii

as shown is A = r2_

2 (θ - sin θ) , with θ in radians.

a. What angle gives a segment area of 92 square feet? How many such sets can simultaneously fit on the full rotating stage?

b. What angle gives a segment area of 50 square feet? About how many such sets can simultaneously fit on the full rotating stage?

34. Oceanography The height of the water on a certain day at a pier in Cape Cod, Massachusetts, can be modeled by h(t) = 4.5 sin π

_6.25

(t + 4) + 7.5, where h is the

height in feet and t is the time in hours after midnight.

a. On this particular day, when is the height of the water 5 feet?

b. How much time is there between high and low tides?

c. What is the period for the tide?

d. Does the cycle of tides fit evenly in a 24-hour day? Explain.

35. /////ERROR ANALYSIS///// Below are two solution procedures for solving sin 2 θ - 1__

2 sin θ = 0 for 0° ≤ θ < 360°. Which is incorrect? Explain the error.

36. Critical Thinking What is the difference between a trigonometric equation and a trigonometric identity? Explain by using examples.

37. Graphing Calculator Use your graphing calculator to find all solutions of the equation 2 cos x = 0.25x.


The displacement in centimeters of a mass attached to a spring is modeled by y (t) = 2.9 cos (2π

___3

t + π __4 ) + 3, where t is the time in seconds.

a. What are the maximum and minimum displacements of the mass?

b. The mass is set in motion at t = 0. When is the displacement of the mass equal to 1 cm for the first time?

c. At what other times will the displacement be 1 cm?

32a. ≈ 30° and ≈ 60°

a. ≈ π _2

; 4 sets b. ≈ 2π

_5

; 5 sets

34a. 3:25 A.M.,7:20 A.M., 3:55 P.M.,and 7:50 P.M. 6.25 h

12.5 hd. No; the period of the model is 12.5 h.

35. B is incorrect; Possible answer: Division by sin θeliminates the solution when sin θ = 0.

5.9 cm; 0.1 cm

0.74 s

0.74 + 3n and 1.51 + 3n where n is an integer

x ≈ -4.165, -1.797, 1.395, 5.464, 6.831

1032 Chapter 14

14-6 PRACTICE C

14-6 PRACTICE B

14-6 PRACTICE A

Science Link Exercise 34 points out that the length of a tidal

day, a period containing two com-plete cycles of high and low tides, is not 24 hours. The actual factors contributing to tidal patterns include the gravitational pulls of both the Sun and the Moon, centrifugal force that causes a bulge on the dark side of Earth, frictional forces that delay tides, and irregularities in the paths of all of the objects involved. The average length of a tidal day is actu-ally about 24 hours and 50 minutes.

Exercise 38 involves solving and interpret-ing a trigonometric

equation. This exercise prepares stu-dents for the Multi-Step Test Prep on page 1034.

Answers 32b. ≈ 36°; The other answer of

144° is not reasonable because it implies that the ball is thrown in the opposite direction.

36. Possible answer: An equation that includes trigonometric func-tions is a trigonometric equation, such as sin 2 θ - sin θ = 0, and may be true only for some values. A trigonometric identity is a trigonometric equation that is true for all values, such as tan θ = sin θ

____ cos θ.




Estimation Use a graphing calculator to approximate the solution to each equation to the nearest tenth of a degree for 0° ≤ θ < 360°.

39. tan θ - 12 = -1 40. sin θ + cos θ + 1.25 = 0

41. 4 sin 2(2θ - 30) = 4 42. tan 2 θ + tan θ = 3

43. sin 2 θ + 5 sin θ = 3.5 44. cos 2 θ - cos 2 θ + 1 = 0

45. Write About It How many solutions can a trigonometric equation have? Explain by using examples.

46. Which values are solutions of 2 cos θ + √ � 3 = 2 √ � 3 for 0° ≤ θ < 360°?

30° or 150° 60° or 120°

30° or 330° 60° or 320°

47. Which gives an approximate solution to 5 tan θ - √ � 3 = tan θ for -90° ≤ θ ≤ 90°?

-23.4° -19.1° 19.1° 23.4°

48. Which value for θ is NOT a solution to sin2θ = sin θ?

0° 90° 180° 270°

49. Which gives all of the solutions of cos θ - 1 = - 1__2 for 0 ≤ θ < 2π?

2π

_3

or 5π

_3

2π

_3

or 4π

_3

π

_3

or 2π

_3

π

_3

or 5π

_3

50. Which gives the solution to sin2θ - sin θ - 2 = 0 for 0° ≤ θ < 360°?

90° 90° or 270°

270° No solution

51. Short Response Solve 2 cos2θ + cos θ - 2 = 0 algebraically. Show the steps in the

solution process.

CHALLENGE AND EXTENDSolve each equation algebraically for 0° ≤ θ < 360°.

52. 9 cos 3 θ - cos θ = 0 53. 4 cos 3 θ - cos θ = 0 54. 16 sin 4 θ - 16 sin 2 θ + 3 = 0

55. sin 2 θ - 4.5 sin θ = 2.5 56. ⎪sin θ⎥ = 1_2

57. ⎪cos θ⎥ = √ � 3_2

SPIRAL REVIEWOrder the given numbers from least to greatest. (Lesson 1-1)

58. √ � 3_2

, -1, 0.8 −

6 , 1, 5_6

59. 2 √ � 5 , 19_4

, 4. −− 47 , √ � 21 , π

_0.65

60. Technology An e-commerce company constructed a Web site for a local business. Each time a customer purchases a product on the Web site, the e-commerce company receives 5% of the sale. Write a function to represent the e-commerce company’s revenue based on total website sales per day. What is the value of the function for an input of 259, and what does it represent? (Lesson 1-7)

Simplify each expression by writing it only in terms of θ. (Lesson 14-5)

61. cos 2θ - 2 cos 2 θ 62. sin 2θ

_2 sin θ

63. cos 2θ + sin 2 θ 64. cos 2θ + 1_2

≈ 84.8°, ≈ 264.8°≈ 197.1°, ≈ 252.9°

60°, 150°, 240°, 330° 52.5°, 113.5°, 232.5°, 293.5°

no solution38.5°, 141.5°

Possible answer: A trigonometric equation may have no solution, such as sin x = 2, or an infinite number of solutions, such as sin x = 1, or any given number of solutions if there are restrictions on the domain.

52. 90°, 270°, 109.5°, 250.5°, 70.5°, 289.5°

53. 90°, 270°, 120°, 240°, 60°, 300°

54. 60°, 150°, 240°, 330°, 30°, 120°, 210°, 300°

210°, 330° 30°, 150°, 210°, 330° 30°, 150°, 210°, 330°

-1, 5_6 ,

√

�

3_2 , 0.8

−

6 , 12√

�5 , 4. −−

47 , √�21 , 19_4 , π_

0.65

60. f (x) = 0.05x;$12.95; 5% of $259, the e-commerce company’s revenue from web sales of $259

-1cos θ

cos 2 θ

cos 2 θ

Lesson 14-6 1033


In Exercises 46 and 50, students can substitute the answer

choices into the equations to elimi-nate choices.

JournalHave students describe how they choose which method to use when solving a trigonometric equation, including when to use a graphing calculator and when an exact solu-tion is possible.

Have students create three differ-ent trigonometric equations with solutions worked out. At least one equation should be quadratic, and at least one should involve a Pythagorean or double-angle iden-tity. At least two of them should be solvable without a calculator.

14-6

1. Find all solutions for cos θ = √ � 2 - cos θ.

θ = 4 5 ◦ + n · 36 0 ◦ or 31 5 ◦ + n · 36 0 ◦

2. Solve 3 si n 2 θ - 4 sin θ - 4 = 0 for 0 ≤ θ ≤ 36 0 ◦ .

θ ≈ 221. 8 ◦ or 318. 2 ◦

3. Solve cos 2θ = 3 sin θ + 2 for 0 ≤ θ ≤ 2π.

θ = ⎧

⎨

⎩

7π

_ 6 , 3π

_ 2 , 11π

_

6

⎫

⎬

⎭


Answers 51. cos θ =

-1 ± √

��

(1) 2 - 4 (2) (-2) ___

2 (2)

=

-1 ± √ ��

17 __ 4

θ = cos -1 ( -1 + √

�

17 _ 4

) or

θ = cos -1 ( -1 -

√ ��

17 __

4

)

θ ≈ 38.7° or 321.3° or no solu-tion. Thus, θ ≈ 38.7° or 321.3°.


Trigonometric IdentitiesSpring into Action Simple harmonic motion refers to motion that repeats in a regular pattern. The bouncing motion of a mass attached to a spring is a good example of simple harmonic motion. As shown in the figure, the displacement y of the mass as a function of time t in seconds is a sine or cosine function. The amplitude is the distance from the center of the motion to either extreme. The period is the time that it takes to complete one full cycle of the motion.

1. The displacement in inches of a mass attached to a spring is modeled by y1(t) = 3 \sin(2π

___5 t + π

__2 ) ,

where t is the time in seconds. What is the amplitude of the motion? What is the period?

2. What is the initial displacement when t = 0 s? How long does it take until the displacement is 1.8 in.?

3. At what other times will the displacement be 1.8 in.?

4. Use trigonometric identities to write the displacement by using only the cosine function.

5. The displacement of a second mass attached to a spring is modeled by y2(t) = sin 2π

___5 t. Both masses are set in motion at t = 0 s. How long does it

take until both masses have the same displacement?

6. The displacement of a third mass attached to a spring is modeled by y3(t) = cos π

__5 t. The second and third masses are set in motion at

t = 0 s. How long does it take until both masses have the same displacement?

Amplitude

Period

0

y

t

1034

SECTION 14B

3 in.; 5 s

3 in.; 0.74 s

5n ± 0.74 s where n is an integer

y(t) = 3 cos 2π

_5

t

about 0.99 s

about 0.83 s

OrganizerObjective: Assess students’ ability to apply concepts and skills in Lessons 14-3 through 14-6 in a real-world format.

Online Edition

ResourcesAlgebra II Assessments

www.mathtekstoolkit.org

Problem Text Reference

1 Lesson 14-1

2–3 Lesson 14-6

4 Lesson 14-3

5–6 Lesson 14-6

INTERVENTION

Scaffolding Questions 1. What are the maximum and minimum

values of the function? 3, -3

2. What equation should you solve in order to determine when the displacement is

1.8 in.? 1.8 = 3 sin ( 2π

___ 5 t + π

__ 2 ) 3. How can you use the period to find when

the displacement will be 1.8 in. again? Add 5 s to the answer from Problem 2.

4. What type of identity should you use to rewrite the function? sum identity

5. What equation do you need to solve?

3 cos 2π

___ 5 t = sin 2π

___ 5 t

6. What equation do you need to solve?

sin 2π

___ 5 t = cos π

__ 5 t

ExtensionAt what times will the second and third masses have the same displacement? t = 2.5 + 5n, t = 0.83 + 10n, and t = 4.16 + 10n, for n ∈ �


S E C T I O N

14B

1034 Chapter 14

a207te_c14_m&r_1034-1035.indd 1034a207te_c14_m&r_1034-1035.indd 1034 12/20/05 12:09:11 PM12/20/05 12:09:11 PM


Quiz for Lessons 14-3 Through 14-6

14-3 Fundamental Trigonometric IdentitiesProve each trigonometric identity.

1. si n 2θ sec θ csc θ = tan θ 2. sin (-θ) sec θ cot θ = -1 3. co t 2θ - 1_co t 2θ + 1

= 1 - 2 si n 2 θ

Rewrite each expression in terms of a single trigonometric function.

4. cot θ sec θ 5. 1_cos(-θ)

6. cs c 2 θ

__tan θ + cot θ

14-4 Sum and Difference IdentitiesFind the exact value of each expression.

7. cos 5π

_12

8. sin (-75°) 9. tan 75°

Find each value if sin A = 1__4 with 90° < A < 180° and if cos B = 12___

13 with

270° < B < 360°.

10. sin (A + B) 11. cos (A + B) 12. cos (A - B)

13. Find the coordinates, to the nearest hundredth, of the vertices of figure ABCD with A(0, 0) , B(4, 1) , C (0, 2) , and D (-1, 1) after a 120° rotation about the origin.

14-5 Double-Angle and Half-Angle IdentitiesFind each expression if cos θ = -

4__5 and 180° < θ < 270°.

14. sin 2θ 15. cos 2θ 16. tan 2θ

17. sin θ _2

18. cos θ _2

19. tan θ _2

20. Use half-angle identities to find the exact value of cos 22.5°.


21. Find all solutions of 1 + 2 sin θ = 0 where θ is in radians.

Solve each equation for 0° ≤ θ < 360°.

22. cos 2θ + 2 cos θ = 3 23. 8 si n 2 θ - 2 sin θ = 1


24. cos 2θ = 3 cos θ + 1 25. si n 2 θ + cos θ + 1 = 0

26. The average daily minimum temperature for Houston, Texas, can be modeled by T(x) = -15.85 cos π

__6 (x - 1) + 76.85, where T is the temperature in degrees

Fahrenheit, x is the time in months, and x = 0 is January 1. When is the temperature 65°F? 85°F?

SECTION 14B

csc θ sec θ cot θ

√

�

6 - √

�

2_4

-√

�

6 -√

�

2__4

2 +√

�

3

12 + 5 √

�

15__52

-12√

�

15 + 5__52

-12√

�

15 - 5__52

(0, 0), (-2.87, 2.96), (-1.73, -1), (-0.37, -1.37)

24_25

7_25

24_7

3√

�

10_10

-

√

�

10_10

-3

√

��

2 +√

�

2 _2

7π

_6 + 2πn, 11π

_

6 + 2πn

0°

2π

_3 , 4π

_

3

mid-March and mid-December; early June and late September

30°, 150°, ≈ 194.5°, ≈ 345.5°

π


OrganizerObjective: Assess students’ mastery of concepts and skills in Lessons 14-3 through 14-6.

Online Edition


Section 14B Quiz


INTERVENTION

ResourcesReady to Go On? Intervention and Enrichment Worksheets

Ready to Go On? CD-ROM

Ready to Go On? Online


NOINTERVENE

YESENRICH

READY TO GO ON? Intervention, Section 14BReady to Go On?

Intervention Worksheets CD-ROM Online

Lesson 14-3 14-3 Intervention Activity 14-3

Diagnose and Prescribe Online




Diagnose and Prescribe

READY TO GO ON? Enrichment, Section 14B


S E C T I O N

14B

A211NLT_c14_m&r_1034-1035.indd 1035A211NLT_c14_m&r_1034-1035.indd 1035 9/1/09 1:43:30 AM9/1/09 1:43:30 AM


amplitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 991

cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 990

frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 992

period . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 990

periodic function . . . . . . . . . . . . . . . . . . . . . . . . . 990

phase shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 993

rotation matrix . . . . . . . . . . . . . . . . . . . . . . . . . . 1016

Complete the sentences below with vocabulary words from the list above.

1. The shortest repeating portion of a periodic function is known as a(n) −−−−−− ? .

2. The number of cycles in a given unit of time is called −−−−−− ? .

3. The −−−−−− ? gives the length of a complete cycle for a periodic function.

4. A horizontal translation of a periodic function is known as a(n) −−−−−− ? .


5. f (x) = cos 3x 6. g(x) = cos 1_2

x

7. h(x) = - 1_3

sin x 8. j(x) = 2 sin πx

9. f (x) = 1_2

cos 2x 10. g(x) = π _2

sin πx


11. f (x) = cos (x + π) 12. g(x) = sin (x + π _4 )

13. h(x) = sin (x - 3π

_2 ) 14. j(x) = cos (x + 3π

_2 )

Biology In photosynthesis, a plant converts carbon dioxide and water to sugar and oxygen. This process is studied by measuring a plant’s carbon assimilation C (in micromoles of C O 2 per square meter per second). For a bean plant, C(t) = 1.2 sin π

__12

(t - 6) + 7, where t is time in hours starting at midnight.

15. Graph the function for two complete cycles.

16. What is the period of the function?

17. What is the maximum and at what time does it occur?

■ Using f (x) = cos x as a guide, graphg(x) = -2 cos π

__2

x. Identify the amplitude and period.

Step 1 Identify the period and amplitude.

Because a = -2, amplitude is ⎪a⎥ = ⎪-2⎥ = 2.

Because b = π _2

, the period is 2π

_⎪b⎥

= 2π

_

⎪ π __2 ⎥

= 4.

Step 2 Graph.

The curve is reflected over the x-axis.

■ Using f (x) = sin x as a guide, graph g(x) = sin

(

x - 5π

___4 )

. Identify the x-intercepts and phase shift.

The amplitude is 1. The period is 2π.

- 5π

___4 indicates a shift 5π

___4 units right.

The first x-intercept occurs at π

__4 . Thus,

the intercepts occur at π

__4 + nπ,

where n is an integer.

14-1 Graphs of Sine and Cosine (pp. 990–997)

EXERCISESE X A M P L E S

Vocabulary

1036 Chapter 14

Study Guide: Review

OrganizerObjective: Help students organize and review key concepts and skills in Chapter 14.

Online EditionMultilingual Glossary

Resources

PuzzleViewTest & Practice Generator

Multilingual Glossary Online

KEYWORD: MB7 Glossary

Lesson Tutorial Videos CD-ROM

Answers 1. cycle

3. period

2. frequency

4. phase shift


_ 3


7. amplitude: 1 _ 3 ; period: 2π

8. amplitude: 2; period: 2

C H A P T E R

14

9. amplitude: 1 _ 2 ; period: π

10. amplitude: π _ 2 ; period: 2

11. x-intercepts: π _ 2 + nπ;

phase shift: π left

12. x-intercepts: 3π

_ 4 + πn;

phase shift: π _ 4 left

13. x-intercepts: π _ 2 + πn; phase

shift: 3π

_ 2 right

14. x-intercepts: πn;

phase shift: 3π

_ 2 left

A211NLT_c14_em_1036-1043.indd 1036A211NLT_c14_em_1036-1043.indd 1036 8/31/09 3:21:08 PM8/31/09 3:21:08 PM

Study Guide: Review 1037

Using f (x) = tan x or f (x) = cot x as a guide, graph each function. Identify the period, x-intercepts, and asymptotes.

18. f (x) = 1_4

tan x 19. g(x) = tan πx

20. h(x) = tan 1_2

πx 21. g(x) = 5 cot x

22. j(x) = -0.5 cot x 23. j(x) = cot πx


24. f (x) = 2 sec x 25. g(x) = csc 2x

26. h(x) = 4 csc x 27. j(x) = 0.2 sec x

28. h(x) = sec (-x) 29. j(x) = -2 csc x

■ Using f (x) = cot x as a guide, graph g(x) = cot π

__2

x. Identify the period, x-intercepts, and asymptotes.


Because b = π _2

, the period is π

_⎪b⎥

= π

_

⎪ π __2 ⎥

= 2.


The first x-intercept occurs at 1. Thus, the x-intercepts occur at 1 + 2n, where n is an integer.


The asymptotes occur at x = πn_⎪b⎥

= πn_

⎪ π __2 ⎥

= 2n.

Step 4 Graph.

14-2 Graphs of Other Trigonometric Functions (pp. 998–1003)

EXERCISESE X A M P L E


30. sec θ sin θ cot θ = 1

31. si n 2(-θ)_

tan θ

= sin θ cos θ

32. (sec θ + 1)(sec θ - 1) = ta n 2 θ

33. cos θ sec θ + co s 2 θ cs c 2 θ = cs c 2 θ

34. (tan θ + cot θ)2 = se c 2 θ + cs c 2 θ

35. tan θ + cot θ = sec θ csc θ

36. si n 2 θ tan θ = tan θ - sin θ cos θ

37. tan θ

_1 - co s 2 θ

= sec θ csc θ

Rewrite each expression in terms of a single trigonometric function, and simplify.

38. cot θ sec θ 39. sec θ sin θ

_cot θ

40. tan(-θ)_

cot θ

41. cos θ cot θ

_cs c 2 θ - 1

■ Prove tan θ

_______1 - co s 2θ

= sec θ csc θ.

( sin θ

____cos θ

)_(si n 2 θ)

=Modify the left side. Apply

the ratio and Pythagorean identities.

( sin θ

_cos θ

)( 1_si n 2 θ

) = Multiply by the reciprocal.

( 1_cos θ

)( 1_sin θ

) = Simplify.

sec θ csc θ Reciprocal identities

■ Rewrite cot θ + tan θ

_________

csc θ in terms of a single

trigonometric function, and simplify.

(cot θ + tan θ) sin θ Given.

(cos θ

_sin θ

+ sin θ

_cos θ

) sin θ Ratio identities

co s 2 θ + si n 2 θ

__cos θ

Add fractions and

simplify.

1_cos θ

= sec θ Pythagorean and

reciprocal identities

14-3 Fundamental Trigonometric Identities (pp. 1008–1013)



Answers 15.

16. 24 h

17. 8.2; noon

18. period: π; x-intercepts: πn; asymptotes: π

_ 2 + πn

19. period: 1; x-intercepts: n;

asymptotes: 1 _ 2 + n

20. period: 2; x-intercepts: 2n; asymptotes: 1 + 2n

21. period: π; x-intercepts: π _ 2 + πn;

asymptotes: πn

22. period: π; x-intercepts: π _ 2 + πn;

asymptotes: πn

23. period: 1; x-intercepts: 1 _ 2 + n;

asymptotes: n

24. period: 2π; asymptotes: π _ 2

+ πn

25. period: π; asymptotes: π _ 2 n

26. period: 2π; asymptotes: π + πn


+ πn


+ πn

29. period: 2π; asymptotes: π + πn

a207TE_c14_em_1036-1045.indd 1037a207TE_c14_em_1036-1045.indd 1037 12/20/05 12:08:08 PM12/20/05 12:08:08 PM



42. sin 19π

_12

43. cos 165°

44. cos 15° 45. tan π _12

Find each value if tan A = 3__4 with

0° < A < 90° and if tan B = -

5___12

with 90° < B < 180°.

46. sin (A + B) 47. cos (A + B)

48. tan (A - B) 49. tan (A + B)

50. sin (A - B) 51. cos (A - B)

Find each value if sin A = √

�

7___4 with

0° < A < 90° and if cos B = -

5___13

with 90° < B < 180°.

52. sin (A + B) 53. cos (A + B)

54. tan (A - B) 55. tan (A + B)

56. sin (A - B) 57. cos (A - B)

Find the coordinates, to the nearest hundredth, of the vertices of figure ABCDwith A

(0, 0

) , B

(3, 0

) , C

(4, 2

) , and D

(1, 2

)


58. 30° rotation 59. 45° rotation


Find the coordinates, to the nearest hundredth, of the vertices of figure ABCDwith A

(0, 0

) , B

(5, 2

) , C

(0, 4

) , and D

(-5, 2

)




■ Find sin (A + B) if cos A = -

1__3 with

180° < A < 270° and if sin B = 4__5 with

90° < B < 180°.

Step 1 Find sin A and cos B by using the Pythagorean Theorem with reference triangles.

180° < A < 270° 90° < B < 180°

cos A = - 1_3

sin B = 4_5

y = - √ � 8 , sin A = - √ � 8_

3 x = -3, cos B = -3_

5Step 2 Use the angle-sum identity.

sin (A + B) = sin A cos B + cos A sin B

= ( - √ � 8_

3 )(-3_5 ) + (- 1_

3)(4_5)

= 3√ � 8 - 4_

15

■ Find the coordinates to the nearest hundredth of the vertices of figure ABCwith A

(0, 2

) , B

(1, 2

) , and

C(

0, 1)

after a 60° rotation about the origin.

Step 1 Write matrices for a 60° rotation and for the points in the figure.

R60° =⎡ ⎢

⎣ cos 60°

sin 60°

-sin 60°

cos 60°

⎤

�

⎦ Rotation matrix

S =⎡ ⎢

⎣ 02

12

01

⎤ �

⎦ Matrix of points

Step 2 Find the matrix product.

R60° × S =⎡

⎢

⎣ cos 60°

sin 60°

-sin 60°

cos 60°

⎤

�

⎦ ⎡ ⎢

⎣ 02

12

01

⎤

�

⎦

≈⎡

⎢

⎣ -1.73

1

-1.231.87

-0.87

0.5

⎤

�

⎦

Step 3 The approximate coordinates of the points after a 60° rotation are A'(-1.73, 1) , B ' (-1.23, 1.87) , and C ' (-0.87, 0.5) .

14-4 Sum and Difference Identities (pp. 1014–1019)


1038 Chapter 14

Answers30.

sec θ sin θ cot θ = ( 1 _ cos θ

) sin θ ( cos θ

_ sin θ

)

= ( cos θ

_ cos θ

) ( sin θ

_ sin θ

) = 1

31. si n 2 (-θ)

_

tan θ =

(-sin θ) (-sin θ)

__

sin θ

____ cos θ

= (sin θ) (sin θ) ( cos θ

_ sin θ

) = sin θ cos θ

32. (sec θ + 1) (sec θ - 1) = se c 2 θ - 1

= ta n 2 θ

33. cos θ sec θ + co s 2 θ cs c 2 θ

= 1 + co s 2 θ ( 1 _ si n 2 θ

)

= 1 + co t 2 θ

= cs c 2 θ

34. (tan θ + cot θ) 2

= ta n 2 θ + 2 tan θ cot θ + co t 2 θ

= ta n 2 θ + 2 + co t 2 θ

= (ta n 2 θ + 1) + (1 + co t 2 θ)

= se c 2 θ + cs c 2 θ

35. tan θ + cot θ = sin θ

_ cos θ

+ cos θ

_ sin θ

= si n 2 θ + co s 2 θ

__ sin θ cos θ

= 1 _ sin θ cos θ

= sec θ csc θ

36. si n 2 θ tan θ = (1 - co s 2 θ) tan θ

= tan θ - co s 2 tan θ

= tan θ - co s 2 θ ( sin θ

_ cos θ

)

= tan θ - sin θ cos θ

37. tan θ

_ 1 - co s 2 θ

=

(

sin θ

____ cos θ )

_ (si n 2 θ)

= ( sin θ

_ cos θ

) ( 1 _

si n 2 θ )

= ( 1 _ cos θ

) ( 1 _

sin θ )

= sec θ csc θ

38. csc θ 39. ta n 2 θ

40. -ta n 2 θ 41. sin θ

42. - √

� 2 - √

� 6 __

4 43.

- √

�

2 - √ �

6 __

4

44. √

� 6 + √

� 2 _

4 45. 2 - √

� 3

46. -

16 _ 65

47. -

63 _ 65

48. 56 _ 33

49. 16 _ 63

50. -

56 _ 65

51. -

33 _ 65

52. 36 - 5 √

� 7 _

52

53. -15 - 12 √

� 7 __

52

54. 5 √

� 7 + 36

__ 15 - 12 √

� 7

55. 5 √

� 7 - 36

__ 15 + 12 √

� 7

a207TE_c14_em_1036-1045.indd 1038a207TE_c14_em_1036-1045.indd 1038 6/13/06 10:56:25 AM6/13/06 10:56:25 AM


Find all of the solutions of each equation.

76. √ � 2 cos θ + 1 = 0 77. cos θ = 2 + 3 cos θ

78. ta n 2 θ + tan θ = 0 79. si n 2 θ - co s 2 θ = 1_2

Solve each equation for 0 ≤ θ < 2π.

80. 2 co s 2 θ - 3 cos θ = 2 81. co s 2 θ + 5 cos θ - 6 = 0

82. si n 2 θ - 1 = 0 83. 2 si n 2 θ - sin θ = 3


84. cos 2θ = cos θ 85. sin 2θ + cos θ = 0

86. Earth Science The number of minutes of daylight for each day of the year can be modeled with a trigonometric function. For Washington, D.C., S is the number of minutes of daylight in the model S(d) = 180 sin (0.0172d - 1.376) + 720, where d is the number of days since January 1.

a. What is the maximum number of daylight minutes, and when does it occur?

b. What is the minimum number of daylight minutes, and when does it occur?

■ Find all of the solutions of 3 cos θ - √

� 3 = cos θ.

3 cos θ - √ � 3 = cos θ

3 cos θ - cos θ = √ � 3 Subtract tan θ.

2 cos θ = √ � 3 Combine like terms.

cos θ = √ � 3_2

Divide by 2.

θ = co s -1( √ � 3_2 ) Apply the inverse

cosine.

θ = 30° or 330° Find θ for0° ≤ θ < 360°.

θ = 30° + 360°n

or 330° + 360°n

■ Solve 6 si n 2 θ + 5 sin θ = -1 for 0° ≤ θ < 360°.

6 si n 2 θ + 5 sin θ + 1 = 0 Set equal to 0.

(2 sin θ + 1)(3 sin θ + 1) = 0 Factor.

sin θ = -1 or sin θ = 3 Zero Product Property

θ = 210°, 330°

or ≈ 199.5°, 340.5°

sinθ = 3 has no solution since -1 ≤ sin θ ≤ 1.

14-6 Solving Trigonometric Equations (pp. 1027–1033)


Find each expression if tan θ =

4__3

and 0° < θ < 90°.

66. sin 2θ 67. cos 2θ

68. tan θ_2

69. sin θ_2

Find each expression if cos θ =

3__4

and 3π___2

< θ < 2π.

70. tan 2θ 71. cos 2θ

72. cos θ_2

73. sin θ_2


74. sin π_12

75. cos 75°

Find each expression if sin θ = 1__4

and 270° < θ < 360°.

■ sin 2θ

For sin θ = 1_4

in QIV, cos θ = -

√

��

15 _4

.

sin 2θ = 2 sin θ cos θ Identity for sin 2θ

= 2 ( 1_4 )(-

√

��

15_4 ) = -

√ � 15_8

Substitute.

■ cos θ_2

cos θ_2

= ± √��1 + cos θ_2

Identity for cos θ_2

= -

√��1 +

(-

√

��

15 ____4 )

__2

Negative for cos θ

_2

in QII

= -√��(4 - √ � 15 _4 )(1_

2) = - √ �� 4 - √ � 15 _

√ � 8

14-5 Double-Angle and Half-Angle Identities (pp. 1020–1026)



Answers

56. -36 - 5 √

� 7 __

52

57. -15 + 12 √

� 7 __

52

58. ≈

⎡

⎢

⎣

0 0

2.60

1.50

2.46

3.73

-0.13

2.23 ⎤

⎦

59. ≈

⎡

⎢

⎣

0 0

2.12

2.12

1.41

4.24

-0.71

2.12 ⎤

⎦

60. ≈

⎡

⎢

⎣

0 0

1.5

2.60

0.27

4.46

-1.23

1.87 ⎤

⎦

61. ≈

⎡

⎢

⎣

0 0

0

3

-2

4

-2

1 ⎤

⎦

62. ≈

⎡

⎢

⎣

0 0

-4.23

3.33

-3.46

-2

0.77

-5.33 ⎤

⎦

63. ≈

⎡

⎢

⎣

0 0

-5

-2

0

-4

5

-2 ⎤

⎦

64. ≈

⎡

⎢

⎣

0 0

-0.77

-5.33

3.46

-2

4.23

3.33 ⎤

⎦

65. ≈

⎡

⎢

⎣

0 0

2

-5

4

0

2

5 ⎤

⎦

66. 24 _ 25

67. -

7 _ 25

68. 1 _ 2 69.

√ �

5 _

5

70. -3 √ �

7 71. 1 _ 8

72. -

√

��

14 _ 4 73.

√ �

2 _

4

74. √

��

2 - √

�

3 _

2 75.

√

��

2- √

�

3 _

2

76. 13 5 ◦ + 36 0 ◦ n, 22 5 ◦ + 36 0 ◦ n

77. 18 0 ◦ + 36 0 ◦ n

78. 0 ◦ + 18 0 ◦ n, 13 5 ◦ + 18 0 ◦ n

79. 6 0 ◦ + 18 0 ◦ n, 12 0 ◦ + 18 0 ◦ n

80. 2π

_ 3 , 4π

_ 3

81. 0

82. π _ 2 , 3π

_ 2

83. 3π

_ 2

84. 0, 2π

_ 3 , 4π

_ 3

85. π _ 2 , 7π

_ 6 , 3π

_ 2 , 11π

_

6

86a. 900 min; late June

b. 540 min; late December



1. Using f (x) = cos x as a guide, graph g (x) = 1__2 cos 2x. Identify the amplitude and period.

2. Using f (x) = sin x as a guide, graph g (x) = sin (x + π __3 ) . Identify the x-intercepts

and phase shift.

3. A torque τ in newton meters (N·m) applied to an object is given by τ (θ) = Fr sin θ,where r is the length of the lever arm in meters, F is the applied force in newtons, and θ is the angle between F and r in degrees. Find the amount and angle for the maximum torque and the minimum torque for a lever arm of 0.5 m and a force of 500 newtons, where 0° ≤ θ ≤ 90°.

4. Using f (x) = tan x as a guide, graph g(x) = 2 tan πx. Identify the period, x-intercepts, and asymptotes.

5. Using f (x) = cot x as a guide, graph g(x) = cot 4x. Identify the period, x-intercepts, and asymptotes.

6. Using f (x) = sin x as a guide, graph g(x) = 1_4

csc x. Identify the period and asymptotes.

7. Prove the trigonometric identity cot θ = co s 2 θ sec θ csc θ.

Rewrite each expression in terms of a single trigonometric function.

8. (sec θ + 1)(sec θ - 1) 9.sin(-θ)_cos(-θ)

Find each value if tan A = 3__4 with 0° < A < 90° and if sin B = -

12___13

with 180° < B < 270°.

10. sin (A + B) 11. cos (A - B)

12. Find the coordinates, to the nearest hundredth, of the vertices of figure ABCD with A(0, 1) , B(2, 1) , C(3, 3) , and D(-1, 3) after a 30° rotation about the origin.

Find each expression if tan θ = -

12___5 and 90° < θ < 180°.

13. sin 2θ 14. cos 2θ 15. cos θ _2

16. Use half-angle identities to find the exact value of sin 3π

_8

.

17. Find all of the solutions of tan θ + √ � 3 = 0.

18. Solve 2 si n 2 θ = sin θ for 0° ≤ θ < 360°.

19. Use trigonometric identities to solve 2 co s 2 θ + 3 sin θ = 0 for 0 ≤ θ < 2π.

20. The voltage at a wall plug in a home can be modeled by V (t) = 156 sin 2π(60t),where V is the voltage in volts and t is time in seconds. At what times is the voltage equal to 110 volts?

co s 2 θ sec θ csc θ = co s 2 θ(

1_cos θ

)(

1_sin θ

)

= cos θ

_sin θ

= cot θ

ta n 2 θ -tan θ

-

63_65

-

56_65

(-0.50, 0.87), (1.23, 1.87), (1.10, 4.10), (-2.37, 2.10)

-

120_169

-

119_169

2√

�

13_13

√

��

2 + √

�

2 _2 17. 120° + 180°n, 2π

_

3 + πn

0°, 30°, 150°, 180°

7π

_6

, 11π

_6

≈ 0.0021 + n 1_60

s, ≈ 0.0063 + n 1_60

s

max.: 250 N·m at 90°; min.: 0 N·m at 0°

1040 Chapter 14

C H A P T E R

14

OrganizerObjective: Assess students’ mastery of concepts and skills in Chapter 14.

Online Edition


Chapter 14 Tests

• Free Response (Levels A, B, C)

• Multiple Choice (Levels A, B, C)

• Performance Assessment


Answers 1.

amplitude: 1 _ 2 ; period: π


Answers

2.

x-intercepts: 2π

_ 3 + πn;

phase shift: π _ 3 left

4.

period: 1; intercepts: n;

asymptotes: 1 _ 2 + n

5.

period: π _ 4 ;

intercepts: π _ 8 +

π

_ 4 n;

asymptotes: π _ 4 n

6.

period: 2π; asymptotes: πn


College Entrance Exam Practice 1041

FOCUS ON SAT MATHEMATICS SUBJECT TESTSTo help decide which standardized tests you should take, make a list of colleges that you might like to attend. Find out the admission requirements for each school. Make sure that you register for and take the appropriate tests early enough for colleges to receive your scores.

You may want to time yourself as you take this practice test. It should take you about 6 minutes to complete.

1. Identify the range of f (x) = 3 sin x.

(A) -1 ≤ f (x) ≤ 1

(B) -3 < f (x) < 3

(C) 0 ≤ f (x) ≤ 3

(D) -3 ≤ f (x) ≤ 3

(E) -∞ < f (x) < ∞

2. If 2 si n 2 θ + 5 sin θ = 3, what could the value of θ be?

(A) π

_6

(B) π

_3

(C) 2π

_3

(D) 7π

_6

(E) 11π

_6

3. If sec θ = 4, what is ta n 2 θ?

(A) 1_16

(B) 3

(C) 5

(D) 15

(E) 17

4. Given the figure, what is the value of cos (A - B)?

(A) 0

(B) 7_25

(C) 24_25

(D) 1

(E) 28_25

5. If sin θ = 7__9 , what is cos 2θ?

(A) -8√ � 2_9

(B) -17_81

(C) 17_81

(D) 56√ � 2_81

(E) 8√ � 2_9

If your calculator malfunctions while you are taking an SAT Mathematics Subject Test, you may be able to have your score for that test canceled. To do so, you must inform a supervisor at the test center immediately when the malfunction occurs.

Not to scale

College Entrance Exam Practice 1041

OrganizerObjective: Provide practice for college entrance exams such as the SAT Mathematics Subject Tests.

Online Edition

ResourcesCollege Entrance Exam Practice

Questions on the SAT Mathematics Subject Tests Levels 1 and 2 rep-resent the following math content areas:

Level

1 2

Number and Operations

10—14% 10—14%

Algebra and Functions

38—42% 48—52%

Geometry and Measurement

38—42% 28—32%

Plane Euclidean 18—22% 0%

Coordinate 8—12% 10—14%

Three-dimensional

4—6% 4—6%

Trigonometry 6—8% 12—16%

Data Analysis, Statistics, and Probability

6—10% 6—10%

Items on this page focus on:

• Algebra and Functions

• Trigonometry

Text References:

Item 1 2 3 4 5

Lesson 1 6 3 4 5

C H A P T E R

14

1. Students may choose answer C because they found the amplitude of the function and then selected only half of the range.

2. Students may choose answers D or E because they found the correct refer-ence angle but forgot that the sine func-tion is negative in the third and fourth quadrants.

3. Students may choose answer E because they have added 1 to se c 2 θ instead of subtracted 1.

4. Students may choose answer A because they found the value of cos (A + B) .

5. Students may choose answer C because they have reversed the order of the terms in one of the equivalent forms of the double-angle formula for cosine.



Multiple Choice: Choose Answer CombinationsYou may be given a test item in which you are asked to choose from a combination of statements. To answer these types of test items, try comparing each given statement with the question and determining whether the statement is true or false. If you determine that more than one of the statements is correct, choose the combination that contains each correct statement.

Which exact solution makes the equation 2 co s 2 θ - 3 cos θ = 2 true?

I. θ = 2° Look at each statement separately, and determine if it is true or false.II. θ = 120°

III. θ = 240°

I only II only

II and III I, II, and III

As you consider each statement, mark it true or false.

Consider statement I: Substitute 2° for θ in the equation.2 co s 2(2°

) - 3 cos (2°

) ≈ -1.0006≠ 2

Statement I is false. So, the answer is not choice A or D.

Consider statement II: Substitute 120° for θ in the equation.2 co s 2(120°

) - 3 cos (120°

) = 2

Statement II is true. The answer could be choice B or C.

Consider statement III: Substitute 240° for θ in the equation.2 co s 2(240°

) - 3 cos (240°

) = 2

Statement III is true.

Because both statements II and III are true, choice B is the correct response.

You can also use a table to keep track of whether the statements are true or false.

Statement True/False

IIIIII

FalseTrueTrue

1042 Chapter 14

C H A P T E R

14

OrganizerObjective: Provide opportunities to learn and practice common test-taking strategies.

Online Edition

This Test Tackler focuses on choosing the best answer when

there are multiple correct answers or combinations of answers. Reinforce students to read the problem state-ment and answer choices thorough-ly. Encourage students to investigate each possibility.

As students practice this strategy, remind them to choose the best, most complete answer, not just the first correct answer that they dis-cover. Review the elimination strat-egy, and guide students to use logic to eliminate any obviously incorrect answer choice. If students are strug-gling with organization, show them how a table can be used to organize their process.


Test Tackler 1043

As you eliminate a statement, cross out the corresponding answer choice(s).

Read each test item and answer the questions that follow.

Item AWhich expression is equivalent to tan2

θ?

I. sec2θ - 1 III. 1_

cs c 2θ - 1

II. sec2θ + 1 IV. 1 - co s 2 θ

_1 - si n 2 θ

I and II I and III

II and III I, III, and IV

1. What are some of the identities that involve the tangent function?

2. Determine whether statements I, II, III, and IV are true or false. Explain your reasoning.

3. Sally realized that statement III was true and selected choice B as her response. Do you agree? If not, what would you have done differently?

Item BFor the graph of f (x) = 3 sin x + 2, which of the statements are true?

I. The function has a period of 2π

_3

.

II. The function has an amplitude of 3.

III. The function has a period of 2π.

I only II only

III only II and III

4. How do you determine the period of a trigonometric function?

5. How do you determine the amplitude of a trigonometric function?

6. Using your response to Problems 4 and 5, which of the three statements are true? Explain.

Item CWhich identities do you need to use to prove that tan θ csc θ = sec θ?

I. tan θ = sin θ

_cos θ

II. se c 2 θ = ta n 2 θ + 1

III. csc θ = 1_sin θ

I only I and II

II only I and III

7. Is statement I true or false? Can any answer choice be eliminated? Explain.

8. Is statement II true or false? Should you select the answer choice yet? Explain.

9. Is statement III true or false? Explain.

10. Which combination of statements is correct? How do you know?

Item DFor the graph of the function f (x) = sec 4x,which are equations of some of the asymptotes?

I. x =

π

_8

II. x = π _2

III. x = -

3π

_4

I only I, II, and III

II and III I and III

11. Create a table, and determine whether each statement is true or false.

12. Using your table, which choice is the most accurate?

Test Tackler 1043


AnswersPossible answers:

1. Some of the identities that involve the tangent function are

tan θ = sin θ

_ cos θ

,

se c 2 θ = ta n 2 θ + 1, and

cot θ = 1 _ tan θ

.

2. Statement I is true because it is a fundamental identity. Statement II is false because

ta n 2 θ = se c 2 θ - 1. Statement III is true because

1 _ cs c 2 θ - 1

= 1 _ co t 2 θ

= ta n 2 θ.

Statement IV is true because

1 - co s 2 θ

_ 1 - si n 2 θ

= si n 2 θ

_ co s 2 θ

= tan 2 θ.

3. No; check if Statements I and IV are true before selecting an answer choice.

4. The period of a trigonometric function in the form

f (x) = a sin bx is equal to 2π

_ ⎪b⎥

.

5. The amplitude of a trigonometric function in the form

f (x) = a sin bx is equal to ⎪a⎥ .

6. Statements II and III are true because the given values are true for the period and for the amplitude.

7. Statement I is true because you can substitute sin θ

____ cos θ for tan θ.

Statement II can be eliminated because there are no squared terms.

8. Statement II is false because you cannot use an expression for ta n 2 θ in this identity. No; It is necessary to determine if Statement III is true or false first.

Answers to Test Items A. D

B. J

C. D

D. F

Possible answers:

9. Statement III is true because you can

substitute 1 _ sin θ

for csc θ.

10. Statements I and III are true because you need both of these identities in order to prove that tan θ csc θ = sec θ.

11.

Statement True/False

I True

II False

III False

12. F


A211NLS_c14stp_1044-1045.indd 1045 8/13/09 2:05:33 AM

KEYWORD: MB7 TestPrep

CUMULATIVE ASSESSMENT, CHAPTERS 1–14

Multiple Choice 1. What is the exact value of tan 15°?

√ � 6 - √ � 2_4

√ � 6 + √ � 2_4

2 + √�3

2 - √�3

2. Where do the asymptotes occur in the given equation?

y = 1_3

cot 2x

2πn

πn_2

3πn

πn_3

3. What is the period of the given equation?

y = 5 cos 1_3

x

2π_5

5_3

2π_3

6π

4. A movie has 14 dialogue scenes and 10 action scenes. If these are the only two types of scenes, what is the probability that a randomly selected scene will be an action scene?

5_127_

125_77_5

5. What is the value of f(x) = 3x3 + 4x2 + 7x + 10 for x = -2?

-44

-12

0

36

6. Which graph shows an inverse variation function for which y = 2 when x = -1?

84

8

4

-8 -4

-8

x

y

0 84

8

-8 -4

-8

-4

x

y

0

42

4

2

0

-4

-4

x

y

0

-4

-2-4 -2

x

y

4

4

7. What is the exact value of cos157.5° using half-angle identities?

- √ �� 2 - √ � 2_

2 √ �� 2 - √ � 2_

2

- √ �� 2 + √ � 2_

2 √ �� 2 + √ � 2_

2

8. What type of function is f(x) = - 2x3 - x + 10?

cubic

exponential

quadratic

rational


A211NLS_c14stp_1044-1045.indd 1044 2/8/10 11:33:51 AM

C H A P T E R

14

OrganizerObjective: Provide review and practice for Chapters 1–14 and standardized tests.

Online Edition


Chapter 14 Cumulative Test

KEYWORD: MB7 Testprep


Answers 1. D

2. G

3. D

4. F

5. B

6. J

7. C

8. F

9. H

10. A

11. 4

12. 0.894

13. 0

14. 245

1044 Chapter 14

A211NLT_c14_stp_1044-1045.indd 1044A211NLT_c14_stp_1044-1045.indd 1044 2/8/10 12:00:57 PM2/8/10 12:00:57 PM

9. Which is a solution of 2 cos θ = 2 sin θ for π ≤ θ ≤ 3π?

π

_4

π

5π

_4

3π

10. Which is the equation of a circle with center (3, 2)and radius 5?

25 = (x - 3)2 + (y - 2)2

5 = (x - 3)2 + (y - 2)2

25 = (x + 3)2 + (y + 2)2

5 = (x + 3)2 + (y + 2)2

Gridded Response11. What is the value of x?

5 √��2x - 7 + 4 = 9

12. What is the value of cos θ? Round to the nearest thousandth.

7.2

3.68.05

θ

13. What is the y-value of the solution of the following system of nonlinear equations?

⎧ � ⎨ �

⎩ x - 4 = 1_

4y2

(x + 1)2_

25+

y2_36

= 1

14. Find the sum of the arithmetric series ∑ k=1

14(3k - 5).

Short Response15. The chart below shows the names of the students

on the academic bowl team.

Robin Drew Jim

Greg Sarah Mindy

Ashley Tina Justin

David Amy Kevin

a. Only 2 students can be chosen for the final academic bowl. How many different ways can the students be selected?

b. Explain why you solved the problem the way that you did.

16. Given the sequence:

4, 12, 36, 108, 324, . . .

a. Write the explicit rule for the nth term.

b. Find the 10th term.

Extended Response17. The chart below shows the grades in

Mr. Bradshaw’s class.

90 85 72 86 94 96

85 95 94 68 71 85

93 98 84 83 80 89

Round each answer to the nearest tenth.

a. Find the mean.

b. Find the median.

c. Find the mode.

d. Find the variance.

e. Find the standard deviation.

f. Find the range.

In Item 13, the answer will be a y -value only. It will be quickest and most efficient to isolate x in one equation and substitute for x in the second equation because then the first variable for which you obtain a value will be y.

Cumulative Assessment Chapters 1–14 1045

A211NLS_c14stp_1044-1045.indd 1045 8/13/09 2:05:33 AMA211NLS_c14stp_1044-1045.indd 1044 8/13/09 10:35:26 AM

Short-Response RubricItems 15–16

2 Points = The student’s answer is an accurate and complete execu-tion of the task or tasks.

1 Point = The student’s answer con-tains attributes of an appropriate response but is flawed.

0 Points = The student’s answer contains no attributes of an appro-priate response.

Extended-Response RubricItem 17

4 Points = The student correctly finds the value of each measure in parts a–f and rounds to the nearest tenth correctly.

3 Points = The student correctly finds the value of at least 5 of the values in parts a–f or makes errors in rounding.

2 Points = The student correctly finds the value of at least 3 of the values in parts a–f.

1 Point = The student correctly finds the value of 1 or 2 of the values in parts a–f.

0 Points = The student does not answer correctly and does not attempt all parts of the problem.

Answers 15a. 66

b. Possible answer: I used the combination formula with the values n = 12 and r = 2 because the order in which the students are selected does not matter.

16a. 4(3n - 1)

b. 78,732

17a. 86

b. 85.5

c. 85

d. 73.6

e. 8.6

f. 30

Cumulative Assessment 1045

A211NLT_c14_stp_1044-1045.indd 1045A211NLT_c14_stp_1044-1045.indd 1045 8/31/09 3:24:33 PM8/31/09 3:24:33 PM

Sandusky Bay

Cleveland

The Rock and Roll Hall of FameThe Rock and Roll Hall of Fame in downtown Cleveland traces the history of rock music through live performances and interactive exhibits. Designed by renowned architect I. M. Pei, the 50,000-square-foot exhibition space houses everything from vintage posters to handwritten lyrics to John Lennon’s report card.

Choose one or more strategies to solve each problem. For 1 and 2, use the diagram.

1. Visitors enter the museum through an enormous glass entryway in the shape of a tetrahedron. The figure shows the dimensions of the tetrahedron. What is the pitch of the tetrahedron’s slanted facade? (Hint: The pitch is shown in the figure by angle θ.)

2. What is the area of the triangular floor space enclosed by the glass tetrahedron?

3. The Hall of Fame exhibits are displayed in an eight-story, 162-foot tower. Pei originally designed a 200-foot tower but had to reduce its height in order to meet the requirements of a nearby airport. From the top of the existing tower, an observer sights the entrance to the museum’s plaza with an angle of depression of 18°. What would be the angle of depression to the entrance of the plaza from Pei’s original tower?

OHIO

≈ 35°

about 9964 f t 2

≈ 21.9°

A211NLS_c14psl_1046_1047.indd 1046 8/10/09 6:53:07 PM

1046 Chapter 14

OrganizerObjective: Choose appropriate problem-solving strategies, and use them with skills from Chapters 13 and 14 to solve real-world problems.

Online Edition

The Rock and Roll Hall of Fame

Reading StrategiesMake sure students understand the word façade in Problem 1. Ask ELL students to identify and define similar words in other languages (for example, the Spanish word fachada means “front”).

Using Data Be sure students understand that an angle of depression is measured down from the horizontal. Ask students whether the angle of depression from the 200 ft tower will be greater than or less than the angle of depression from the 162 ft tower. greater

Problem-Solving FocusEncourage students to use the four-step probelm-solving process for the problems. Focus on the first step: (1) Understand the Problem.

Discuss with students how they could organize the information about the building in Problem 3. Suggest that students draw a diagram that shows towers of both heights and the line of sight from the top of each tower to the entrance of the plaza in order to visualize the problem.



A211NLT_c14ncc_1046-1047.indd 1046A211NLT_c14ncc_1046-1047.indd 1046 9/2/09 4:55:02 AM9/2/09 4:55:02 AM

Real-World Connections 1047

Marblehead LighthouseSince its construction in 1821, Marblehead Lighthouse has stood at the entrance to Sandusky Bay, guiding sailors along Lake Erie’s rocky shores. The 65-foot tower is one of Ohio’s best-known landmarks and the oldest continuously operating lighthouse on the Great Lakes.

Choose one or more strategies to solve each problem.

1. The range of a lighthouse is the maximum distance at which its light is visible. In the figure, point A is the farthest point from which it is possible to see the light at the top of the lighthouse L. The distance along Earth s is the range. Assuming that the radius of Earth is 4000 miles, find the range of Marblehead Lighthouse.

2. In 1897, a new lighting system was installed in the lighthouse. A set of descending weights rotated the tower’s lantern to produce a flashing light. The rotation could be modeled by the function f (x) = sin π

__5 x,

where x is the time in seconds since the weights were released. The light briefly flashed on whenever f (x) = 1. How many times per minute did the light flash?

3. Today the flashing light of Marblehead Lighthouse can be modeled by g(x) = sin π

__3

x.How many seconds are there between each flash? Does the light flash more or less frequently than in 1897?

about 9.9 mi

6

6 s; more frequently

A211NLS_c14psl_1046_1047.indd 1047 8/7/09 9:30:19 AM

Real-World Connections 1047

Marblehead Lighthouse

Reading Strategies


Have students restate Problem 1 in their own words. Then ask them what values they will need to find in order to determine the lighthouse’s range. Students will need to find m∠LEA. Then they can use s = rθ to find the range.

Using Data Ask students what additional information they can add to the figure for Problem 1. For example, AE = 4000 mi and LE = 4000.0123 mi because the height of the lighthouse is 65 ft, which equals 0.0123 mi.

Problem-Solving FocusEncourage students to use the four-step problem-solving process for the problems. Focus on the second step: (2) Make a Plan.

Discuss with students the strategies that might be useful in solving the problem, such as making a table or graphing f (x) from x = 0 to x = 60.

A211NLT_c14ncc_1046-1047.indd 1047A211NLT_c14ncc_1046-1047.indd 1047 8/31/09 3:26:10 PM8/31/09 3:26:10 PM

Documents

Trigonometric Graphs and Identities - Spokane Public …swcontent.spokaneschools.org/.../Domain/3458/chapter14text.pdf · Chapter Test (Levels A, B, C) ... Trigonometric Graphs and