RESOU RCE OPTIONS • RESOU RCE OPTIONS • RESOU RCE OPTIONS • RESOU RCE
Trigonometric Graphs and Identities
Section 14A
Section 14B
Exploring Trigonometric Graphs
Trigonometric Identities
14-1 Graphs of Sine and Cosine
14-3 Technology Lab Graph Trigonometric Identities
14-2 Graphs of Other Trigonometric Functions
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 14
DAY 1
DAY 2
DAY 3
DAY 4
DAY 5
14-1 Lesson
14-2 Lesson
14-2 Lesson
Multi-Step Test Prep
Ready to Go On?
14-3 Technology Lab
Connecting Algebra
to Geometry
14-3 Lesson
DAY 6
DAY 7
DAY 8
DAY 9
DAY 10
14-3 Lesson
14-4 Lesson
14-4 Lesson
14-5 Lesson
14-5 Lesson
14-6 Lesson
14-6 Lesson
Multi-Step Test Prep
Ready to Go On?
Chapter 14 Test
Pacing Guide for 90-Minute Classes
Calendar Planner®
Chapter 14
DAY 1
DAY 2
DAY 3
DAY 4
DAY 5
14-1 Lesson
14-2 Lesson
14-2 Lesson
Multi-Step Test Prep
Ready to Go On?
14-3 Technology Lab
Connecting Algebra
to Geometry
14-3 Lesson
14-4 Lesson
14-4 Lesson
14-5 Lesson
14-6 Lesson
14-6 Lesson
Multi-Step Test Prep
Ready to Go On?
Chapter 14 Test
986A
Chapter 14
E OPTIONS • RESOU RCE OPTIONS • RESOU RCE OPTIONS • RESOU RCE OPTIONS
DIAGNOSE
Assess
Prior
Knowledge
PRESCRIBE
Before Chapter 14
Diagnose readiness for the chapter.
Prescribe intervention.
Are You Ready? SE p. 987
Are You Ready? Intervention
Before Every Lesson
Diagnose readiness for the lesson.
Prescribe intervention.
Warm Up TE
Reteach CRB
During Every Lesson
Diagnose understanding of lesson concepts.
Prescribe intervention.
Check It Out! SE
Questioning Strategies TE
Think and Discuss SE
Write About It SE
Journal TE
Reading Strategies CRB
Success for Every Learner
Lesson Tutorial Videos
After Every Lesson
Formative
Assessment
Diagnose mastery of lesson concepts.
Prescribe intervention.
Lesson Quiz TE
Test Prep SE
Test and Practice Generator
Reteach CRB
Test Prep Doctor TE
Homework Help Online
Before Chapter 14 Testing
Diagnose mastery of concepts in chapter.
Prescribe intervention.
Ready to Go On? SE pp. 1005, 1041
Multi-Step Test Prep SE pp. 1004, 1034
Section Quizzes AR
Test and Practice Generator
Ready to Go On? Intervention
Scaffolding Questions TE pp. 1004, 1034
Reteach CRB
Lesson Tutorial Videos
Before High Stakes Testing
Diagnose mastery of benchmark concepts.
Prescribe intervention.
College Entrance Exam Practice SE p. 1041
Standardized Test Prep SE pp. 1044–1045
College Entrance Exam Practice
After Chapter 14
Summative
Assessment
KEY:
SE = Student Edition
Check mastery of chapter concepts.
Prescribe intervention.
Multiple-Choice Tests (Forms A, B, C)
Free-Response Tests (Forms A, B, C)
Performance Assessment AR
Cumulative Test AR
Test and Practice Generator
Reteach CRB
Lesson Tutorial Videos
TE = Teacher’s Edition
CRB = Chapter Resource Book AR = Assessment Resources
Available online
Available on CD- or DVD-ROM
986B
RESOU RCE OPTIONS • RESOU RCE OPTIONS • RESOU RCE OPTIONS • RESOU RCE
E
Lesson Resources
Before the Lesson
Practice the Lesson
Prepare Teacher One Stop JG8E@J?
Practice Chapter Resources
• Editable lesson plans
• Calendar Planner
• Easy access to all chapter resources
• Practice A, B, C
Practice and Problem Solving Workbook
IDEA Works!® Modified Worksheets and Tests
ExamView Test and Practice Generator
Lesson Transparencies
• Teacher Tools
Homework Help Online
JG8E@J?
Online Interactivities
Interactive Online Edition
Teach the Lesson
• Homework Help
Introduce Alternate Openers: Explorations
Lesson Transparencies
Apply Chapter Resources
• Warm Up
• Problem of the Day
• Problem Solving
Practice and Problem Solving Workbook
Interactive Online Edition
Teach Lesson Transparencies
• Chapter Project
• Teaching Transparencies
Project Teacher Support
Know-It Notebook™
• Vocabulary
• Key Concepts
Power Presentations
Lesson Tutorial Videos
Interactive Online Edition
After the Lesson
JG8E@J?
Reteach Chapter Resources
• Reteach
• Reading Strategies %,,
• Lesson Activities
• Lesson Tutorial Videos
Lab Activities
Lab Resources Online
Online Interactivities
TechKeys
Success for Every Learner
Review Interactive Answers and Solutions
Solutions Key
Know-It Notebook™
JG8E@J?
• Big Ideas
• Chapter Review
Extend Chapter Resources
• Challenge
Technology Highlights for the Teacher
Power Presentations
Teacher One Stop
Dynamic presentations to engage students.
Complete PowerPoint® presentations for
every lesson in Chapter 14.
KEY:
986C
SE = Student Edition
Chapter 14
TE = Teacher’s Edition
%,,
JG8E@J?
Easy access to Chapter 14 resources and
assessments. Includes lesson planning, test
generation, and puzzle creation software.
English Language Learners
JG8E@J? Spanish version available
Premier Online Edition
JG8E@J?
Chapter 14 includes Tutorial Videos, Lesson
Activities, Lesson Quizzes, Homework Help,
and Chapter Project.
Available online
Available on CD- or DVD-ROM
E OPTIONS • RESOU RCE OPTIONS • RESOU RCE OPTIONS • RESOU RCE OPTIONS
Reaching All Learners
Teaching tips to help all learners appear throughout the chapter. A few that target specific students are included in the lists below.
All Learners
On-Level Learners
Lab Activities
Practice and Problem Solving Workbook
Know-It Notebook
Practice B ............................................................................. CRB
Problem Solving .................................................................. CRB
Vocabulary Connections ..............................................SE p. 988
Questioning Strategies ...........................................................TE
Ready to Go On? Intervention JG8E@J?
Know-It Notebook
Homework Help Online
JG8E@J?
Online Interactivities
Special Needs Students
Practice A ............................................................................. CRB
Reteach................................................................................. CRB
Reading Strategies ............................................................... CRB
Are You Ready?............................................................SE p. 987
Inclusion .......................................................................TE p. 996
IDEA Works!® Modified Worksheets and Tests
Ready to Go On? Intervention JG8E@J?
Know-It Notebook
JG8E@J?
Online Interactivities
JG8E@J?
Lesson Tutorial Videos
Advanced Learners
Practice C ............................................................................. CRB
Challenge ............................................................................. CRB
Challenge Exercises ...............................................................SE
Reading and Writing Math Extend ..............................TE p. 989
Critical Thinking ........................................................TE p. 1016
Are You Ready? Enrichment JG8E@J?
Developing Learners
Practice 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 JG8E@J?
Know-It Notebook
Homework Help Online
JG8E@J?
Online Interactivities
JG8E@J?
Lesson Tutorial Videos
Ready To Go On? Enrichment JG8E@J?
English Language Learners
ENGLISH
LANGUAGE
LEARNERS
Reading Strategies ............................................................... CRB
Are You Ready? Vocabulary ........................................SE p. 987
Vocabulary Connections ..............................................SE p. 988
Vocabulary Review ....................................................SE p. 1037
English Language Learners..TE pp. 989, 993, 1018, 1046, 1047
Success for Every Learner
Know-It Notebook
Spanish Study Guide (Resumen y Repaso)
Multilingual Glossary
JG8E@J?
Lesson Tutorial Videos
Technology Highlights for Reaching All Learners
Lesson Tutorial Videos
JG8E@J?
Starring Holt authors Ed Burger and Freddie
Renfro! Live tutorials to support every
lesson in Chapter 14.
KEY:
SE = Student Edition TE = Teacher’s Edition
Multilingual Glossary
Searchable glossary includes definitions
in English, Spanish, Vietnamese, Chinese,
Hmong, Korean, and 4 other languages.
CRB = Chapter Resource Book
Online Interactivities
Interactive tutorials provide visually engaging
alternative opportunities to learn concepts and
master skills.
JG8E@J? Spanish version available
Available online
Available on CD- or DVD-ROM
986D
RESOU RCE OPTIONS • RESOU RCE OPTIONS • RESOU RCE OPTIONS • RESOU RCE
E
Ongoing Assessment
Assessing Prior Knowledge
Lesson Assessment
Determine whether students have the prerequisite concepts
and skills for success in Chapter 14.
Provide formative assessment for each lesson of Chapter 14.
Are You Ready? JG8E@J?
............................SE p. 987
Warm Up
.................................................................TE
Test Preparation
Provide 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! Modified Worksheets and Tests
Questioning Strategies ...........................................................TE
Think and Discuss ...................................................................SE
Check It Out! Exercises ...........................................................SE
Write About It .........................................................................SE
Journal ....................................................................................TE
Lesson Quiz
.............................................................TE
Alternative Assessment ..........................................................TE
IDEA Works!® Modified Worksheets and Tests
Weekly Assessment
Provide formative assessment for each section of Chapter 14.
Multi-Step Test Prep ......................................SE pp. 1004, 1034
JG8E@J? .............SE pp. 1005, 1035
Ready to Go On?
Section Quizzes ..................................................................... AR
Test and Practice Generator
................. Teacher One Stop
Alternative Assessment
Assess students’ understanding of Chapter 14 concepts
and combined problem-solving skills.
Chapter Assessment
Chapter 14 Project .......................................................SE p. 986
Alternative Assessment ..........................................................TE
Performance Assessment ..................................................... AR
Portfolio Assessment ............................................................ AR
Provide 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
®
IDEA Works! Modified Worksheets and Tests
Technology Highlights for Assessment
Are You Ready?
Ready to Go On?
JG8E@J?
Automatically assess readiness and prescribe
intervention for Chapter 14 prerequisite skills.
KEY:
986E
SE = Student Edition
Chapter 14
TE = Teacher’s Edition
Automatically assess understanding
of and prescribe intervention for
Sections 14A and 14B.
AR = Assessment Resources
JG8E@J? Spanish version available
Test and Practice Generator
Use Chapter 14 problem banks to create
assessments and worksheets to print out
or deliver online. Includes dynamic problems.
Available online
Available on CD- or DVD-ROM
E OPTIONS • RESOU RCE OPTIONS • RESOU RCE OPTIONS • RESOU RCE 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
A Chapter 14 Test
C Chapter 14 Test
C Chapter 14 Test
MULTIPLE CHOICE
FREE RESPONSE
PERFORMANCE ASSESSMENT
B Chapter 14 Test
B Chapter 14 Test
Chapter 14 Test
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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.
(continued)
DSINV
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______ Calculates the period of sunspot cycles given frequency.
Chapter 14 Test
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OR 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.)
Level 1: Student is not able to solve any of the problems.
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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.
Level 2: Student solves some problems but gives no calculations to support work.
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Review the characteristics of the graphs of sine and cosine functions,
including frequency, period, amplitude, phase shift, and vertical shift.
Level 3: Student solves problems but gives inadequate calculations.
B Chapter 14 Test
(continued)
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This performance task assesses the student’s ability to model real-world
periodic behavior with a transformed trigonometric function.
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CHAPTER
14
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.
This is one of 47 sunspot
drawings made by the Italian
scholar Galileo Galilei in 1612.
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.
____________________
Modified chapter tests that address special
learning needs are available in IDEA Works!®
Modified Worksheets and Tests.
Create and customize Chapter 14 Tests. Instantly
generate multiple test versions, answer keys, and
practice versions of test items.
986F
Trigonometric Graphs
and Identities
SECTION
14A
Exploring Trigonometric
Graphs
14-1 Graphs of Sine and Cosine
14-2 Graphs of Other Trigonometric
Functions
On page 1004,
students write and
graph functions to
model real-world
movement of tides.
14B Trigonometric
Identities
Lab
Exercises designed to
prepare students for
success on the MultiStep Test Prep can be
found on pages 996
and 1002.
SECTION
14A Exploring Trigonometric
Graphs
14-3
Graph Trigonometric
Identities
Fundamental Trigonometric
Identities
14-4 Sum and Difference Identities
14-5
Double-Angle and Half-Angle
Identities
14-6
Solving Trigonometric
Equations
14B
Trigonometric Identities
On page 1034, students apply trigonometric identities and
solve trigonometric
equations to model real-world
motion of springs.
Exercises designed to
prepare students for
success on the MultiStep Test Prep can be
found on pages 1012,
1018, 1025, and 1032.
You can use graphs of trigonometric
functions and trigonometric identities to model the motion of a circle
or a wheel in a variety of situations.
▼
Interactivities Online
• Make connections among representations of trigonometric functions.
• Use reasoning to solve problems
involving trigonometric ratios.
KEYWORD: MB7 ChProj
986
Lessons 14-1 and 14-6
Chapter 14
Spinning Wheels
Lesson Tutorials Online
About the Project
Project Resources
In the Chapter Project, students write and
graph functions to model the circular motion
of a wheel. Students then investigate an
unusual trigonometric identity.
All project resources for teachers and
students are provided online.
A211NLS_c14opn_0986-0987.indd 986
Lesson Tutorial Videos are
available for EVERY example.
Materials:
• ruler
• graphing calculator
KEYWORD: MB7 ProjectTS
986
Chapter 14
8/10/09 10:32:35 AM
Vocabulary
Match each term on the left with a definition on the right.
A. the ratio of the length of the leg adjacent the
1. cosecant D
angle to the length of the opposite leg
2. cosine B
B. the ratio of the length of the leg adjacent the
3. hypotenuse E
angle to the length of the hypotenuse
4. tangent of an angle C
C. the ratio of the length of the leg opposite the
Organizer
Objective: Assess students’
understanding of prerequisite skills.
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
Prerequisite Skills
E. the side opposite the right angle
Multiply and Divide Fractions
Simplify Radical Expressions
Divide Fractions
Multiply Binomials
Divide.
_
_
5 6
5. _
5
4 3
6. _
1
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7. _
1
2
2
8
3
__
__
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__
__
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__
2
2
__
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_
3
- 8
8. _
21
-__74
Assessing Prior
Knowledge
Simplify Radical Expressions
Simplify each expression.
9. √
6 · √
2
10. √
100 - 64
3
2 √
_
√
9 1
11. _
√
36 2
6
INTERVENTION
_
4 2
12. _
25 5
Diagnose and Prescribe
Use this page to determine
whether intervention is necessary
or whether enrichment is
appropriate.
Multiply Binomials
Multiply.
13. (x + 11)(x + 7) x 2 + 18x + 77
14.
(y - 4)(y - 9) y 2 - 13y + 36
15. (2x - 3)(x + 5) 2x 2 + 7x - 15
16.
(k + 3)(3k - 3) 3k 2 + 6k - 9
17. (4z - 4)(z + 1) 4z 2 - 4
18.
(y + 0.5)(y - 1) y 2 - 0.5y - 0.5
Resources
Are You Ready?
Intervention and
Enrichment Worksheets
Are You Ready? CD-ROM
Special Products of Binomials
Are You Ready? Online
Multiply.
(3y - 2) 2 9y 2 - 12y + 4
19. (2x + 5) 2 4x 2 + 20x + 25
20.
21. (4x - 6)(4x + 6) 16x 2 - 36
22. (2m + 1)(2m - 1) 4m 2- 1
23. (s + 7) s + 14s + 49
24.
2
2
(-p + 4)(-p - 4) p 2 - 16
Trigonometric Graphs and Identities
NO
INTERVENE
987
YES
Diagnose and Prescribe
ENRICH
ARE YOU READY? Intervention, Chapter 14
Prerequisite Skill
Worksheets
CD-ROM
Multiply and Divide Fractions
Skill 47
Activity 47
Simplify Radical Expressions
Skill 53
Activity 53
Multiply Binomials
Skill 64
Activity 64
Special Products of Binomials
Skill 65
Activity 65
Online
Diagnose and
Prescribe Online
ARE YOU READY?
Enrichment, Chapter 14
Worksheets
CD-ROM
Online
Are You Ready?
987
CHAPTER
Study Guide:
Preview
14
Organizer
Key
Vocabulary/Vocabulario
Objective: Help students
In previous chapters, you
GI
organize the new concepts they
will learn in Chapter 14.
<D
@<I
• solved problems involving
triangles and trigonometric
ratios.
Online Edition
• factored to solve quadratic
Multilingual Glossary
•
equations.
applied function models to
solve real-world problems.
• solved equations by using
Resources
algebra and graphs.
PuzzleView
You will study
• problems involving
trigonometric functions.
• factoring to solve
KEYWORD: MB7 Glossary
•
Answers to
Vocabulary Connections
trigonometric equations.
trigonometric function
models of real-world
problems.
• solving trigonometric
Possible answers:
equations by using algebra
and graphs.
1. to increase in amount or importance; 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 movement 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.
You can use the skills
in this chapter
• in your future math classes,
particularly Calculus.
• in other classes, such
as Physics, Biology, and
Economics.
3. blinking; 15 blinks per minute
4. a length of time; a function that
repeats at regular intervals
• outside of school to observe
cyclical patterns and make
conjectures.
5. a rotation of whatever is operated on by the rotation matrix
988
Chapter 14
amplitud
cycle
ciclo
frequency
frecuencia
period
periodo
periodic function
función periódica
phase shift
cambio de fase
rotation matrix
matriz de rotación
Vocabulary Connections
Multilingual Glossary Online
988
amplitude
Chapter 14
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 ?
CHAPTER
14
Organizer
Study Strategy: Prepare for Your Final Exam
Objective: Help students apply
Math 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.
GI
strategies to understand and retain
key concepts.
<D
@<I
Online Edition
ENGLISH
LANGUAGE
LEARNERS
2 weeks before the final:
• Look at previous exams and homework to
determine areas I need to focus on; rework
problems that were incorrect or incomplete.
• Make a list of all formulas and theorems that
I need to know for the final.
• Create a practice exam using problems from
the book that are similar to problems from
each exam.
Study Strategy:
Prepare for Your Final
Exam
Discuss 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.
1 week before the final:
• Take the practice exam and check it.
For each problem I miss, find two or
three similar ones and work those.
• Work with a friend in the class to quiz
each other on formulas, postulates,
and theorems from my list.
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.
1 day before the final:
• Make sure I have pencils and a
calculator (check batteries!).
batteries!)
Answers to Try This
1. Check students’ work.
Try This
Reading
Connection
1. Create a timeline that you will use to study for your final exam.
Trigonometric Graphs and Identities
989
Trig or Treat
by Y.E.O. Adrian
This 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 doubleangle and half-angle identities.
Reading and Writing Math
989
SECTION
14A Exploring Trigonometric Graphs
One-Minute Section Planner
Lesson
Lab Resources
Lesson 14-1 Graphs of Sine and Cosine
•
□
Recognize and graph periodic and trigonometric functions.
SAT-10
NAEP ✔ ACT
SAT
SAT Subject Tests
□
□
□
□
Lesson 14-2 Graphs of Other Trigonometric Functions
• Recognize and graph trigonometric functions.
✔ ACT □ SAT □ SAT Subject Tests
□ SAT-10 □ NAEP □
Materials
Algebra Lab Activities
14-1 Algebra Lab
Required
Algebra Lab Activities
14-2 Algebra Lab
Required
graphing calculator
graphing calculator
MK = Manipulatives Kit
990A
Chapter 14
Math Background
PERIODIC FUNCTIONS
Lesson 14-1
1
A 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 period
of 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.
Þ
*
v­Ýä®
Ýä
ÝäÊÊ*
Ý
ÝäÊÊ* ÝäÊÊÓ*
ÝäÊÊÎ*
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
FUNCTIONS
Lesson 14-1
The 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
π
shapes. The graph of y = cos x is a translation __
units
2
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 + _π2_ . This relationship 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.
(
)
-2P -P 1
y = sin x
P
y = cos x = sin (x +
2P
P
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 functions. For example, students sometimes have difficulty
π
remembering whether sin _π2_ = 0 or cos __
= 0. A quick
2
sketch of either graph should help them see that the
latter equation is correct.
THE GRAPHS OF THE TANGENT
AND COTANGENT FUNCTIONS
Lesson 14-2
The 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
π
function y = tan x is undefined at x = __
+ nπ, where
2
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 _π2_ from the left
π
), tan x approaches infinity; as x approaches
(i.e., x < __
2
π
π
__ from the right (i.e., x > __
), tan x approaches negative
2
2
infinity. The same is true at each vertical asymptote.
Once students are familiar with the graph of the tangent function, they can sketch the graph of the cotangent 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.
990B
14-1 Graphs of Sine
14-1 Organizer
and Cosine
Pacing: Traditional 1 day
1
Block __
day
2
Objectives: Recognize and
graph periodic and trigonometric
functions.
Algebra Lab
GI
In Algebra Lab Activities
<D
Online Edition
@<I
Graphing Calculator, Tutorial
Videos, Interactivity
Why learn this?
Periodic phenomena such as sound
waves can be modeled with trigonometric
functions. (See Example 3.)
Objective
Recognize and
graph periodic and
trigonometric functions.
Vocabulary
periodic function
cycle
period
amplitude
frequency
phase 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
5. 145
◦
Ý
_ 0
√
2
_ _
35
◦
Þ
*iÀˆœ`
Ý
Evaluate.
π
π
0.5 2. cos
1. sin
6
2
π
π
3. cos
0.5 4. sin
3
4
Find the measure of the
reference angle for each
given angle.
_
_
Þ
*iÀˆœ`
Warm Up
Not Periodic
6. 317
◦
*iÀˆœ`
2
1
EXAMPLE
43
Identifying Periodic Functions
Identify whether each function is periodic. If the function is periodic, give
the period.
◦
Þ
A
£
£
£
£
Î
Î
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.
Student: I think I’ve done enough
problems about frequency.
Ý
Î
Ý
Î
Þ
B
*iÀˆœ`
Also available on transparency
Although there is some
symmetry, the pattern does
not repeat exactly.
This function is not periodic.
Teacher: How do you know?
Identify whether each function is periodic. If the function is
periodic, give the period.
Þ
Þ
1a.
1b.
Student: My brain hertz.
Ó
Ó
Ý
Ý
Ó
ä
not periodic
990
Ó
Ó
periodic; 3
£
Ó
Chapter 14 Trigonometric Graphs and Identities
1 Introduce
E X P L O R AT I O N
Motivate
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5SEYOURCALCULATORTOCOMPLETETHETABLEFORTHEFUNCTION
YSINX2OUNDEACHYVALUETOTHENEARESTHUNDREDTH
IFNECESSARY
XS D
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0LOTTHEPOINTSFROMYOUR
TABLEONACOORDINATEPLANE
LIKETHEONESHOWN
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KEYWORD: MB7 Resources
X
/ Ê Ê-
1--Ê
990
Chapter 14
Students 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 brainstorming examples of periodic behavior. Possible
answers: temperatures and weather, tides and
waves, and orbits of planets
$ESCRIBE THEMAXIMUMANDMINIMUMVALUESOFTHEFUNCTION
Explorations and answers are provided in
Alternate Openers: Explorations Transparencies.
x
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.
û
Ú
Ê
Ó
û
x(= θ)
y
_π
3
√3
2
5π
_
_1
6
2
ä
Example 1
Þ
_
4π
_
-
3
11π
_
Îû
Ú
Ê
£
v­Ý®ÊÊȘÊÝ
Ý
ä
ÚÚ
ûÊÊ
ÚÚÚ
ÊÎûÊ
û
Ó
Identify whether each function
is periodic. If the function is
periodic, give the period.
A.
Ó
_
√3
2
£
1
-_
6
Additional Examples
P
2
P
Ó
Similarly, the function f (x) = cos x can be graphed on the coordinate plane by
using x-values from points on the unit circle.
B.
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.
y = sin x
The graph of the
sine function passes
through the origin.
The graph of the
cosine function has
y-intercept 1.
DOMAIN
Þ
no
ä°x
ä
û
Ý
û
ä
ä°x
ä°x
⎧
⎫
⎨ x⎥ x ⎬
⎩
⎭
⎧
⎫
⎨ x⎥ x ⎬
⎩
⎭
û
RANGE
⎧
⎫
⎨ y⎥ -1 ≤ y ≤ 1 ⎬
⎩
⎭
⎧
⎫
⎨ y⎥ -1 ≤ y ≤ 1⎬
⎩
⎭
PERIOD
2π
2π
1
1
AMPLITUDE
Ý
û
y = cos x
ä°x
y
x
Þ
GRAPH
Characteristics of the Graphs of Sine and Cosine
FUNCTION
yes; π
INTERVENTION
Questioning Strategies
EX AM P LE
• How much of the curve do you
need to see before you can conclude that it is periodic?
• How can you determine the
period?
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.
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 connection with the y-coordinate. Repeat
for cosine.
Transformations of Sine and Cosine Graphs
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⎥.
2π .
• the period is _
⎪b⎥
14-1 Graphs of Sine and Cosine
1
991
2 Teach
Guided Instruction
Introduce 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.
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.
Lesson 14-1
991
EXAMPLE
2
Using f (x) = sin x as a guide, graph the function g (x) = 3 sin 2x. Identify the
amplitude and period.
Additional Examples
Step 1 Identify the amplitude and period.
Example 2
Because a = 3, the amplitude is ⎪a⎥ = ⎪3⎥ = 3.
2π = π.
2π = _
Because b = 2, the period is _
⎪b⎥ ⎪2⎥
Step 2 Graph.
Using f(x) = sin x as a
guide, graph the function
1
1
g(x) = _ sin _ x . Identify
2
2
the amplitude and period.
( )
x
û
_1 ; period: 4π
2.
x
û
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.
Sine and cosine functions can be used to model
real-world phenomena, such as sound waves.
Different sounds create different waves. One way
amplitude: 1 ; period: π to distinguish sounds is to measure frequency.
3
Frequency is the number of cycles in a given
unit of time, so it is the reciprocal of the period
of a function.
_
y
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.
x
EXAMPLE
Questioning Strategies
2
3.
y
x
frequency: 250 Hz
992
“«ˆÌÕ`i
*iÀˆœ`
ä
ä°ääÓ
ä°ää{
Ý
ä°ääÈ
ä°ään
Ó
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.
Chapter 14 Trigonometric Graphs and Identities
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 modulation) bands.
Chapter 14
Þ
Ó
3
• What are the frequencies of the
parent curves f(x) = sin x and
g(x) = cos x?
Sound Application
Use a horizontal scale where one
unit represents 0.001 second.
The period tells you that it takes
0.005 seconds to complete one
full cycle. The maximum and
minimum values are given by
the amplitude.
1
frequency = _
period
1 = 200 Hz
=_
0.005
INTERVENTION
• How do horizontal or vertical compressions of sine and cosine graphs
compare to these transformations
of other functions?
3
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.
frequency: 500 Hz
992
Ó
û
EX A M P L E
û
2. Using f (x) = cos x as a guide, graph the function
h(x) = __13 cos 2x. Identify the amplitude and period.
y
2
Example 3
EX A M P L E
ä
The maximum value of g is 3, and the minimum value is -3.
Ý
û
The parent function f has x-intercepts at
multiples of π and g has x-intercepts at
π
.
multiples of __
2
û
Þ
Ó
The curve is vertically stretched by a
factor of 3 and horizontally compressed
by a factor of __12 .
y
amplitude:
Stretching or Compressing Sine and Cosine Functions
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).
""
It is worth noting that an expression
π
such as sin _ (t - 7.5) is equivalent
15
⎧ π
⎫
to sin ⎨ _(t - 7.5)⎬ and not to
⎩ 15
⎭
π
_
sin
· (t - 7.5). The use of the
15
extra brackets may be cumbersome,
but it serves to eliminate confusion.
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).
4
EXAMPLE
Identifying Phase Shifts for Sine and Cosine Functions
π
Using f (x) = sin x as a guide, graph g (x) = sin x + __
. Identify the
2
x-intercepts and phase shift.
(
)
(
Step 1 Identify the amplitude and period.
Amplitude is ⎪a⎥ = ⎪1⎥ = 1.
2π = _
2π = 2π.
The period is _
⎪b⎥ ⎪1⎥
Step 2 Identify the phase shift.
π = x - -π
Identify h.
x+_
2
2
π
π radians to the left.
Because h = - , the phase shift is _
2
2
Example 4
_
π units
All x-intercepts, maxima, and minima of f (x) are shifted _
2
to the left.
Using f(x) = sin x as a guide,
π
graph g(x) = sin x – _ .
4
Identify the x-intercepts and
phase shift.
(
Step 3 Identify the x-intercepts.
π
. Because sin x has two x-intercepts in
The first x-intercept occurs at -__
2
π
each period of 2π, the x-intercepts occur at -__
+ nπ, where n is
2
an integer.
)
y
x
Step 4 Identify the maximum and minimum values.
û
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.
û
_
π
x-intercepts:
+ nπ; phase
4
π
units to the right
shift:
4
_
Step 5 Graph using all of the information about the function.
Þ
ä°x
)
Additional Examples
( _)
The repeating
pattern is maximum,
intercept, minimum,
intercept,…. So
intercepts occur
twice as often
as maximum or
minimum values.
Ê,,",
,/
v
Ý
û
}
ä
û
ä°x
INTERVENTION
Questioning Strategies
4.
4. Using f (x) = cos x as a guide, graph g (x) = cos(x - π). Identify
the x-intercepts and phase shift.
y
x
û
x-intercepts:
û
_π + πn;
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
Phase shift
2
phase shift: π right
Period
Vertical shift
14-1 Graphs of Sine and Cosine
993
EX AM P LE
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
ENGLISH
LANGUAGE
similar to a sine function,
LEARNERS
is sinusoidal.
Lesson 14-1
993
EXAMPLE
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
2π
(t - 1.75) + 80,
the six-person gondolas can be modeled by H (t) = 70 sin ___
7
where t is time in minutes.
Additional Examples
Example 5
a. Graph the height of a cabin for two complete periods.
2π (t - 1.75) + 80
2π
H(t) = 70 sin _
a = 70, b = ___
, h = 1.75, k = 80
7
7
The number of people, in
thousands, employed in a
resort town can be modeled by
π
g(x) = 1.5 sin _(x + 2) + 5.2,
6
where x is the month of the
year.
Step 1 Identify the important features of the graph.
2π = _
2π = 7
Period: _
2π
___
⎪b⎥
⎪7⎥
The period is equal to the time
required for one full rotation.
iˆ}…ÌÊ­vÌ®
Amplitude: 70
A. Graph the number of people
employed in the town for one
complete period.
0OPULATIONTHOUSANDS
5
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nä
{ä
Phase shift: 1.75 minutes right
y
ä
Ó
{
È
n
£ä
£Ó
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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
x
Step 2 Graph using all of the information about the function.
b. What is the maximum height of a cabin?
B. What is the maximum number
of people employed? 6700
The maximum height is 80 + 70 = 150 feet above the ground.
5a.
(EIGHTFT
INTERVENTION
b. What is the maximum height of a cabin? 40 ft
Questioning Strategies
s
EX A M P L E
5. What if...? Suppose that the height H of a Ferris wheel can
π
be modeled by H(t) = -16 cos __
t + 24, where t is the time
45
in seconds.
a. Graph the height of a cabin for two complete periods.
ft
5
4IMES
• How can you find the radius of the
Ferris wheel from the function that
describes the wheel’s height?
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.
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The graphs of f(x) = sin x and
g(x) = cos x are periodic, which is not surprising because they represent the movement 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.
994
Chapter 14
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Chapter 14 Trigonometric Graphs and Identities
Answers to Think and Discuss
3 Close
Summarize
œÀˆâœ˜Ì>
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and INTERVENTION
Diagnose Before the Lesson
14-1 Warm Up, TE p. 990
Monitor During the Lesson
Check It Out! Exercises, SE pp. 990–994
Questioning Strategies, TE pp. 991–994
Assess After the Lesson
14-1 Lesson Quiz, TE p. 997
Alternative Assessment, TE p. 997
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
1
is _.
2π
2. The period of sine and cosine functions tells how often the maxima and
minima occur. The amplitude affects
the value of the maxima and minima.
3. See p. A14.
14-1
Exercises
14–1 Exercises
KEYWORD: MB7 14-1
KEYWORD: MB7 Parent
Assignment Guide
GUIDED PRACTICE
Assign Guided Practice exercises
as necessary.
1. Vocabulary Periodic functions repeat in regular intervals called ? .
−−−
(cycles or periods) cycles
SEE EXAMPLE
1
p. 990
Identify whether each function is periodic. If the function is periodic, give
the period.
2.
3.
Þ
Þ
£
£
Ý
{
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û
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p. 992
SEE EXAMPLE
3
p. 992
SEE EXAMPLE 4
p. 993
SEE EXAMPLE
5
If you finished Examples 1–5
Basic 12–37, 39–43, 48–53
Average 12–44, 48–53
Advanced 12–53
û
not periodic
Using f (x) = sin x or f (x) = cos x as a guide, graph each function. Identify the
amplitude and the period.
1x
1 cos x
6. k(x) = sin πx
4. f (x) = 2 sin _
5. h(x) = _
4
2
Homework Quick Check
Quickly check key concepts.
Exercises: 12, 14, 20, 28
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. 100 Hz
Using f (x) = sin x or f (x) = cos x as a guide, graph each function. Identify the
x-intercepts and the phase shift.
π
π
3π
9. g(x) = cos x - _
10. h(x) = sin x - _
8. f (x) = sin x + _
4
2
2
(
p. 994
ä
£
periodic; 5
SEE EXAMPLE
If you finished Examples 1–2
Basic 12–17, 29–32
Average 12–17, 29–32
Advanced 12–17, 29–32, 44
)
(
)
(
Answers
4.
y
)
x
û
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°? 4 ft
û
amplitude: 2; period: 4π
5.
y
PRACTICE AND PROBLEM SOLVING
12–13
14–17
18
19–22
23
x
û
Identify whether each function is periodic. If the function is periodic,
give the period.
Independent Practice
For
See
Exercises Example
12.
1
2
3
4
5
13.
Þ
Þ
£
£
Extra Practice
Skills Practice p. S30
ä
amplitude:
Ý
Ý
û
û
û
ä
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not periodic
periodic; 2π
û
6.
9.
y
û
x-intercepts: nπ;
π
right
phase shift:
2
_
x
û
x-intercepts: nπ;
3π
phase shift: _ left
2
x
û
û
y
û
995
x
û
amplitude: 1; period: 2
x
y
8.
û
10.
y
4
y
x
14-1 Graphs of Sine and Cosine
7.
_1 ; period: 2π
Using f (x) = sin x or f (x) = cos x as a guide, graph each function. Identify the
amplitude and period.
3 sin x
1x
15. g(x) = _
17. j(x) = 6 sin _
16. g(x) = -cos 4x
14. f (x) = 4 cos x
2
3
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. 40 Hz
Application Practice p. S45
û
û
_π + πn;
4
π
phase shift: _ right
x-intercepts:
4
11, 14–18. For graphs, see
p. A50.
14. amplitude: 4; period: 2π
3
15. amplitude: ; period: 2π
2
π
16. amplitude: 1; period:
2
17. amplitude: 6; period: 6π
_
_
KEYWORD: MB7 Resources
Lesson 14-1
995
Using f (x) = sin x or f (x) = cos x as a guide, graph each function. Identify the
x-intercepts and phase shift.
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( )
3
_
-1 + 23 and _ (1) + 23.
2
2
19. f (x) = sin(x + π)
(
(
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? ≈ 0.8 s
b. The pulse rate is the number of beats
M6
in one minute. What is the pulse rate
indicated by the EKG? 75 beats/min
c. What is the frequency of the EKG? 1.25 Hz
h
(
)
27. h(x) = cos(2πx) - 2
29. sin 160° ≈ 0.3
$EPTHFT
31. sin 15° ≈ 0.25
32. cos 95° ≈ -0.1
f (x) = 6 sin 2x ; f (x) = 6 cos 2x
24d. The pulse rate is measured in
beats per minute and the frequency is measured in cycles,
or beats, per second. They both
measure the same quantity.
Write both a sine and a cosine function that could be used to represent each graph.
35.
36.
Þ
Þ
Ó
£
Ý
25–28. See p. A50.
2π
1
34. f (x) = sin x +
4
3
(
30. cos 50° ≈ 0.6
Write both a sine and a cosine function for each set of conditions.
1 , phase shift of _
2 π left
33. amplitude of 6, period of π
34. amplitude of _
4
3
max.: 24.5 ft; min.: 21.5 ft
_
28. j(x) = -3 sin 3x
Estimation Use a graph of sine or cosine to estimate each value.
ft
S
Determine the amplitude and period for each function. Then describe the
transformation from its parent function.
3 cos _
π -1
πx
25. f (x) = sin x + _
26. h(x) = _
4
4
4
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d. How does the pulse rate relate to the
frequency in hertz?
An EKG measures the
electrical signals that
control the rhythm of a
beating heart. EKGs are
used to diagnose and
monitor heart disease.
19–22. See p. A50.
)
( )
Medicine
Answers
4IMEH
)
3π
π
21. g (x) = sin x + _
22. j(x) = cos x + _
4
4
23. Oceanography The depth d in feet of the water in a bay at any time is given by
5π
d(t) = _32_ sin ___
t + 23, where t is the time in hours. Graph the depth of the water.
31
What are the maximum and minimum depths of the water?
Exercise 37 involves
graphing and interpreting sine functions. This exercise prepares students
for the Multi-Step Test Prep on page
1004.
23.
20. h(x) = cos(x - 3π)
ÚÚ
ûÓÊÊ
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ä°x
û
ÚÚ
ÊÊ
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ä
Ý
Ó
ä
û
û
35. f (x) = -4 sin 2x ;
( _π4 )
g (x) = 4 cos 2 x +
37. This problem will prepare you for the Multi-Step Test Prep on page 1004.
_ _
1
36. f (x) = - 1 sin x + 1;
2
4
1
1
g (x) = cos (x - π) + 1
4
2
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? period: 12.2; amplitude: 1.5; max.: 3; min.: 0
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.
_ _
14-1 PRACTICE A
2π
_
t + 1.5
h(t) = 1.5 cos
14-1 PRACTICE C
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b. h(0) = 3; h(6.1) = 0
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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.
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
40. Which trigonometric function best matches the graph?
1 sin x
1 sin 2x
y=_
y=_
2
2
1x
y = 2 sin x
y = 2 sin _
2
û
43.
û
Þ
£
Ý
ä
û
£
41. What is the amplitude for y = -4 cos 3πx?
-4
û
4
phase shift π right,
3
3π
horizontal compression,
vertical stretch, and 42. Based on the graphs, what is the relationship between f and g?
v
reflection across the
f has twice the amplitude of g.
x-axis; amplitude:
f has twice the period of g.
4; period: π;
f has twice the frequency of g.
π
x-intercepts: 0, ,
f has twice the cycle of g.
2
3π
π,
, and 2π;
2
43. Short Response Using y = sin x as a guide, graph
max.: 4; min.: -4
y = -4 sin 2(x - π) on the interval [0, 2π] and describe the transformations.
}
_
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 vertical shift give the values of the
maxima and minima.
39. The period decreases for b > 1
and increases for b < 1 because
2π
.
the period is given by
b
44, 45. See p. A50.
_
_
Journal
Have students describe how to identify the period and the phase shift of
a sine or cosine function.
CHALLENGE AND EXTEND
44. Graph f (x) = Sin -1 x and g (x) = Cos -1 x. (Hint: Use what you learned about graphs of
inverse functions in Lesson 9-5 and inverse trigonometric functions in Lesson 13-4.)
1
Consider the functions f (θ) = __
sin θ and g (θ) = 2 cos θ for 0° ≤ θ ≤ 360°.
2
45. On the same set of coordinate axes, graph f (θ) and g (θ).
46. (256°, -0.485) 46. What are the approximate coordinates of the points of intersection of f (θ) and g (θ)?
and (76°, 0.485)
47. When is f (θ) > g (θ)? 76° < θ < 256°
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
SPIRAL REVIEW
Use interval notation to represent each set of numbers. (Lesson 1-1)
48. -7 < x ≤ 5
50. 0 ≤ x ≤ 9 [0, 9]
49. x ≤ -2 or 1 ≤ x < 13
(-7, 5]
51. Flowers Adam has $100 to purchase
Roses
6
3
7
4
a combination of roses, lilies, and
Lilies
8
5
carnations. Roses cost $6 each, lilies
10
3
cost $2 each, and carnations cost
Carnations
11
15
13
18
$4 each. (Lesson 3-5)
a. Write a linear equation in three variables to represent this situation.
6r + 2 + 4c = 100
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
a = 30.8; m∠B = 9.7°; m∠C = 5.3°
m∠A = 10°; m∠B = 12.4°; m∠C = 157.6
14-1 Graphs of Sine and Cosine
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Suppose that the height, in
feet, above ground of one of
the cabins of a Ferris wheel at
t minutes is modeled by
( ( ))
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+ 36.
2. Graph the height of the cabin
for two complete revolutions.
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14-1 PROBLEM SOLVING
997
1. Using f(x) = cos x as a guide,
graph g(x) = 1.5cos 2x.
(EIGHTFT
49. (-∞, -2] or
[1, 13)
3. What is the radius of this
Ferris wheel? 30 ft
YY
X
X
Also available on transparency
5SEYOURGRAPHTOPREDICTTHEHEIGHTOFPOINT0ABOVETHESURFACEOFTHE
WATERWHENTHESTOPWATCHSHOWSSECONDS
ABOUTFT
Lesson 14-1
997
14-2 Graphs of Other Trigonometric
14-2 Organizer
Functions
Pacing: Traditional 2 days
Block 1 day
Objectives: Recognize 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.)
Objective
Recognize and graph
trigonometric functions.
Algebra Lab
The tangent and cotangent functions can
be graphed on the coordinate plane. The
tangent function is undefined when
π
θ = __
+ πn, where n is an integer. The
2
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.
GI
In Algebra Lab Activities
<D
@<I
Online Edition
Graphing Calculator,
Tutorial Videos
To graph tangent and cotangent, let the variable x represent the angle θ in
standard position.
Warm Up
3
If sin A = _, evaluate:
5
4
1. cos A
2. tan A
5
4
4. sec A
3. cot A
3
5
5. csc A
3
_
_3
_
_
_
Characteristics of the Graphs of Tangent and Cotangent
4
5
4
y = tan x
FUNCTION
{
y = cot x
Þ
{
Ó
GRAPH
û
DOMAIN
RANGE
PERIOD
A: When time starts getting
asymptotically close to the end of
the period.
Ó
Ý
Also available on transparency
Q: How do you know when trig class
is almost over?
Þ
ä
û
Ý
ä
û
Ó
Ó
{
{
û
⎧
π + πn,
⎨x⎥ x ≠ _
2
⎫
⎩
where n is an integer⎬
⎭
⎧
⎫
⎨y⎥ -∞ < y < ∞⎬
⎩
⎭
π
⎧
⎨x⎥ x ≠ πn,
⎫
⎩
where n is an integer⎬
⎭
⎧
⎫
⎨y⎥ -∞ < y < ∞⎬
⎩
⎭
π
undefined
undefined
AMPLITUDE
Like sine and cosine, you can transform the tangent function.
Transformations of Tangent Graphs
For the graph of y = a tan bx, where a ≠ 0 and x is in radians,
π +_
πn ,
π .
• the asymptotes are located at x = _
• the period is _
⎪b⎥
⎪
⎥
⎪
2
b
b⎥
where n is an integer.
998
Chapter 14 Trigonometric Graphs and Identities
1 Introduce
Motivate
E X P L O R AT I O N
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In 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.
X
KEYWORD: MB7 Resources
/ Ê Ê-
1--Ê
%XPLAIN WHATHAPPENSWHENXAPPROACHES OR 998
Chapter 14
$ESCRIBE WHATYOUTHINKWOULDHAPPENIFYOUCONTINUEDTHE
GRAPH ANOTHER TO THE RIGHT
Explorations and answers are provided in
Alternate Openers: Explorations Transparencies.
1
EXAMPLE
""
Transforming Tangent Functions
Using f (x) = tan x as a guide, graph g (x) = tan 2x. Identify the period,
x-intercepts, and asymptotes.
Students may forget to include negative values of n when determining
the domain of tangent and cotangent functions. Remind students
that n is an integer, not a natural
number.
Step 1 Identify the period.
π =_
π.
π =_
Because b = 2, the period is _
⎪b⎥ ⎪2⎥ 2
Step 2 Identify the x-intercepts.
π , the
An x-intercept occurs at x = 0. Because the period is _
2
π
_
x-intercepts occur at n, where n is an integer.
2
g(x) = tan2x
Step 3 Identify the asymptotes.
1.
Because b = 2, the asymptotes occur at
πn .
πn , or x = _
π +_
π +_
x=_
4
2
2⎪2⎥ ⎪2⎥
y
x
û û
û
û
Step 4 Graph using all of the information
about the function.
Þ
{
Additional Examples
}
v
Ó
Example 1
Ý
@ûÓ@
ä
Using f(x) = tan x as a guide,
1
1
graph g(x) = tan x .
3
2
Identify the period, x-intercepts,
and asymptotes.
@û@
(_ )
_
Ó
Ó
{
period: 2π; x-intercepts:
2πn; asymptotes:
π + 2πn
Ê,,",
,/
1 x. Identify
1. Using f (x) = tan x as a guide, graph g (x) = 3 tan _
2
the period, x-intercepts, and asymptotes.
y
x
û
Transformations of Cotangent Graphs
û
For the graph of y = a cot bx, where a ≠ 0 and x is in radians,
π .
• the period is _
⎪b⎥
EXAMPLE
πn ,
• the asymptotes are located at x = _
⎪b⎥
where n is an integer.
period: 3π; x-intercepts: 3πn;
3π
+ 3πn
asymptotes: x =
2
_
2 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.
Step 1 Identify the period.
INTERVENTION
π =_
π = 2π.
Because b = 0.5, the period is _
⎪b⎥ ⎪0.5⎥
Step 2 Identify the x-intercepts.
Questioning Strategies
EX AM P LE
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.
Step 3 Identify the asymptotes.
{
Because b = 0.5, the asymptotes occur at
πn = 2πn.
x= _
⎪0.5⎥
1
• What determines the location of
the asymptotes on the graph of a
tangent function?
Þ
Ó
Ý
û
Step 4 Graph using all of the information
about the function.
ä
û
Ó
{
14-2 Graphs of Other Trigonometric Functions
999
2 Teach
A211NLS_c14l02_0998_1003.indd 999
8/14/09 9:50:49 AM
Guided Instruction
Introduce the graphs of tangent and cotangent 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.
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.
Lesson 14-2
999
2. Using f (x) = cot x as a guide, graph g (x) = -cot 2x. Identify the
period, x-intercepts, and asymptotes.
Additional Examples
1
. So, secant is undefined where cosine equals zero and the
Recall that sec θ = ____
cos θ
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.
Example 2
Using f(x) = cot x as a guide,
1
graph g(x) = _ cot 3x. Identify
2
the period, x-intercepts, and
asymptotes.
Characteristics of the Graphs of Secant and Cosecant
y = sec x
FUNCTION
y
y = csc x
Þ
Þ
x
?
û
?û
2.
y
?
π
π
π
period: _; x-intercepts: _ + _n;
3
6
3
π
asymptotes: x = _ n
3
Example 3
Using f(x) = cos x as a guide,
1
1
graph g(x) = sec x .
2
2
Identify the period and
asymptotes.
_
(_ )
û
period:
Ý
Ý
GRAPH
x
Ó
Ó
û
?û
ä
û
ä
û
Ó
Ó
_π ;
2
π
π
+ n;
x-intercepts:
4
2
π
asymptotes: n
2
_ _
_
DOMAIN
RANGE
û
⎧
π + πn,
⎨x⎥ x ≠ _
2
⎫
⎩
where n is an integer⎬
⎭
⎧
⎫
⎨y⎥ y ≤ -1, or y ≥ 1⎬
⎩
⎭
⎧
⎨x⎥ x ≠ πn,
⎫
⎩
where n is an integer⎬
⎭
⎧
⎫
⎨y⎥ y ≤ -1, or y ≥ 1⎬
⎩
⎭
2π
2π
undefined
undefined
PERIOD
AMPLITUDE
y
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.
û
û
x
EXAMPLE
Step 1 Identify the period.
Because sec 2x is the reciprocal of cos 2x, the graphs will have the
same period.
2π = _
2π = π.
Because b = 2 for cos 2x, the period is _
⎪b⎥ ⎪2⎥
INTERVENTION
EX A M P L E
3.
y
2
x
• Compare the graph of cotangent to
the graph of tangent.
EX A M P L E
3
• What determines the location of
the asymptotes on the graph of a
secant or cosecant function?
Graphing Secant and Cosecant Functions
Using f (x) = cos x as a guide, graph g (x) = sec 2x. Identify the period
and asymptotes.
period: 4π; asymptotes:
x = π + 2πn
Questioning Strategies
3
û
û
Step 2 Identify the asymptotes.
Because the period is π, the asymptotes
π +_
π n=_
π+_
π n,
occur at x = _
4
2
⎪2⎥
2⎪2⎥
where n is an integer.
Step 3 Graph using all of the information
about the function.
period: 2π;
asymptotes: πn
1000
Þ
Ó
û
ä
û
Ó
3. Using f (x) = sin x as a guide, graph g (x) = 2 csc x. Identify the
period and asymptotes.
Chapter 14 Trigonometric Graphs and Identities
3 Close
and INTERVENTION
Diagnose Before the Lesson
14-2 Warm Up, TE p. 998
Monitor During the Lesson
Check It Out! Exercises, SE pp. 999–1000
Questioning Strategies, TE pp. 999–1000
Assess After the Lesson
14-2 Lesson Quiz, TE p. 1003
Alternative Assessment, TE p. 1003
1000
Chapter 14
Summarize
Remind 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.
Answers to Think and Discuss
THINK AND DISCUSS
Possible answers:
1. Cosecant is the reciprocal of the
sine function, so the graphs are
related.
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 ORGANIZED
Copy and complete the
graphic organizer.
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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.
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3. See p. A14.
14-2
Exercises
KEYWORD: MB7 14-2
KEYWORD: MB7 Parent
GUIDED PRACTICE
SEE EXAMPLE
1
Using f (x) = tan x as a guide, graph each function. Identify the period, x-intercepts,
and asymptotes.
1x
1. k(x) = 2 tan(3x)
2. g (x) = tan _
3. h(x) = tan 2πx
4
2
Using f (x) = cot x as a guide, graph each function. Identify the period, x-intercepts,
and asymptotes.
3 cot x
5. p(x) = cot 2x
6. g (x) = _
4. j(x) = 0.25 cot x
2
p. 999
SEE EXAMPLE
p. 999
SEE EXAMPLE
3
p. 1000
10–13
14–16
17–19
1
2
3
Extra Practice
Skills Practice p. S30
Application Practice p. S45
Assignment Guide
Assign Guided Practice exercises
as necessary.
Basic 10-33, 35-40, 51-59
Average 10-43, 51-59
Advanced 10-59
Homework Quick Check
Quickly check key concepts.
Exercises: 10, 12, 14, 18
Using f (x) = cos x or f (x) = sin x as a guide, graph each function. Identify the period
and asymptotes.
1 sec x
8. q(x) = sec 4x
9. h(x) = 3 csc x
7. g(x) = _
2
Answers
1–19. See p. A50.
PRACTICE AND PROBLEM SOLVING
Independent Practice
For
See
Exercises Example
14-2 Exercises
Using f (x) = tan x as a guide, graph each function. Identify the period, x-intercepts,
and asymptotes.
3x
π
10. p(x) = tan _
11. g(x) = tan x + _
4
2
πx
1 tan 4x
12. h(x) = _
13. j(x) = -2 tan _
2
2
(
)
Using f (x) = cot x as a guide, graph each function. Identify the period, x-intercepts,
and asymptotes.
1x
14. h(x) = 4 cot x
15. g(x) = cot _
16. j(x) = 0.1 cot x
4
Using f (x) = cos x or f (x) = sin x as a guide, graph each function. Identify the period
and asymptotes.
1 csc x
19. h(x) = csc(-x)
17. g(x) = -sec x
18. k(x) = _
2
14-2 Graphs of Other Trigonometric Functions
1001
Teacher to Teacher
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 Blomquist
Kewaskum, WI
KEYWORD: MB7 Resources
Lesson 14-2
1001
Exercise 20 involves
graphing and interpreting cosecant
functions. This exercise prepares
students for the Multi-Step Test Prep
on page 1004.
20. This problem will prepare you for the Multi-Step Test Prep on page 1004.
Between 1:00 P.M. (t = 1) and 6:00 P.M. (t = 6), the height (in meters) of the tide in a
5π
bay is modeled by h(t) = 0.4 csc ___
t.
31
a. Graph the function for the range 1 ≤ t ≤ 6.
b. At what time does low tide occur? about 3:06 P.M. (t = 3.1)
c. What is the height of the tide at low tide? 0.4 m
d. What is the maximum height of the tide during this time span? When does
this occur? about 3.95 m; 6:00 P.M.
Answers
20a.
h
3π
π 5π
_π ; _
; -_ ; _
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21.
y
,ENGTHFT
x
?û
?û
û
?
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0<x<
R
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R
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dec.
inc.
inc.
dec.
29.
cos x
dec.
dec.
inc.
inc.
30.
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inc.
inc.
dec.
dec.
31.
tan x
inc.
inc.
inc.
inc.
32.
cot x
dec.
dec.
dec.
dec.
33. Critical Thinking Based on the table above, what do you observe about the
increasing/decreasing relationship between reciprocal pairs of trigonometric
functions?
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4ANGENTANDCOTANGENTGRAPHSHAVEDIFFERENTCHARACTERISTICS
24. j (θ) = csc θ
Chapter 14 Trigonometric Graphs and Identities
Y
R R R R
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.
33.
Possible answer:
For reciprocal pairs
of trigonometric
functions, when one
increases, the other
decreases and vice
versa.
14-2 PRACTICE A
23. h(θ) = sec θ
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. s = h cot θ
b. Graph your answer to part a for a gnomon of
…
ô
height 6 ft.
R R R R 22. g(θ) = cot θ
the time when the light shines parallel to the wall.
2
c. Critical Thinking Identify the location of any asymptotes.
4
4
What do the asymptotes represent? Possible answer: The asymptotes represent
ft
,%33/.
2
2
a. What is the period of a(t)? 3 s
b. Graph the function for 0 ≤ t ≤ 3.
Math History
26b.
2
( )
x
3π
π 5π
_π ; _
; -_ ; _
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 _23_πt .
4IME
25b.
23.
2
21. f (θ) = tan θ
24. -π; 0; π; 2π
t
2
2
Find four values for which each function is undefined.
22. -π; 0; π; 2π
2
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Q??
Q ??
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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
-_
0
2
π
_
2
37. What is the range of f(x) = 3 csc 2θ?
⎧
⎫
⎨y⎥ y ≤ -1 or y ≥ 1⎬
⎩
⎭
⎧
⎫
⎨y⎥ y ≤ -3 or y ≥ 3⎬
⎩
⎭
38. Which could be the equation of the graph?
y = tan 2x
y = 2 tan x
y = cot 2x
y = 2 cot x
3π
_
2
Answers
⎧
⎫
⎨y⎥ y ≤ -2 or y ≥ 2⎬
⎩⎧
⎭⎫
1⎬
1 or y ≥ _
⎨y⎥ y ≤ -_
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2
⎩
34, 35, 41–50. See p. A51.
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2
π
_
2π
2
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Have students explain how the
domain and range of each of the six
trigonometric graphs relates to right
triangles.
Ó
Ó
{
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?
40. The graph of which function has a period of ___
2
3
3
3
_
_
y = sec x
y = csc x
2
2
y = sec 3x
y = csc 3x
CHALLENGE AND EXTEND
Describe the period, local maximum and minimum values, and phase shift.
π
π
1 x-_
41. f (x) = 4 - 3 csc π(x -1) 42. g(x) = 4 cot _
43. h(x) = 0.5 sec 2 x + _
4
2
2
(
)
(
(
)
)
5
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7
2
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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.
Give students the graphs of the
3
functions f(x) = 2 csc _ x ,
4
π
_
)
(
g x = cot x +
, and
4
π
h(x) = 3 sec _ x , and ask them to
2
write the function rule.
(
( )
Graph each trigonometric function and its inverse. Identify the domain and range
of the corresponding inverse function.
π <x<_
π
π
47. f (x) = Sec x for 0 ≤ x ≤ π and x ≠ _
48. f (x) = Tan x for -_
2
2
2
π ≤x≤_
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49. g (x) = Csc x for -_
2
2
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SPIRAL REVIEW
)
14-2
√
5
_
15
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1 1
4
4 9
51. -_
54. _
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; -10 52. 0.2 -0.2; 5
10 10
9 -9; 4
55. Technology Marjorie’s printer prints 30 pages per minute. How many pages does
Marjorie’s printer print in 22 seconds? (Lesson 2-2) 11 pages
_
( )
__
1. Using f(x) = tan x as a guide,
1
graph g(x) = -tan _x .
2
Identify the period,
x-intercepts, and asymptotes.
( )
y
Convert each measure from degrees to radians or from radians to degrees.
(Lesson 13-3)
5π
π
3π radians
π radians
56. 45°
radians 57. _
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radians 59. -_
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Lesson 14-2
1003
SECTION 14A
SECTION
14A
Trigonometric Graphs
The 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.
4π
Organizer
Objective: Assess students’
h (t) = 8.15 cos
GI
ability to apply concepts and skills
in Lessons 14-1 and 14-2 in a
real-world format.
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1. The highest tides in the world have
Resources
www.mathtekstoolkit.org
1–6
Lesson 14-1
4. At time t = 0, the tide is at 16.3 m. What is the tide’s height after 3 hours?
after 9 hours? about 8.66 m; about 6.62 m
5. No; the period 5. Will a high tide occur at the same time each day at the Bay of Fundy?
Why or why not?
of the function
is 12.5 h, which 6. It is possible to write a function that models the height of the tide based
is not a factor
on the sine function. What is the function? What is the phase shift?
of 24 h.
4π
π
π
yes; h (t) = 8.15 sin
+ 8.15;
t+
left
25
2
2
Answers
3.
h
(_
(EIGHTM
Tides at the Bay of Fundy
2. What are the amplitude, period, maximum and minimum values, and
phase shift of the function? amplitude: 8.15; period: 12.5; maximum: 16.3;
minimum: 0; phase shift: none
3. Graph the function.
Algebra II Assessments
Text Reference
25
been measured at the Bay of Fundy, in
Time (h)
Height (m)
Nova Scotia, Canada. As shown in the
table, high tides in the bay can reach
High Tide
t=0
16.3
heights of 16.3 m. Assume that it takes
Low Tide
t = 6.25
0
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.
Online Edition
Problem
_t + 8.15
_)
_
t
4IMEH
1004
Chapter 14 Trigonometric Graphs and Identities
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.
KEYWORD: MB7 Resources
1004
Chapter 14
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 halfway 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
π
left
equivalent to a cosine graph?
2
_
Extension
At what time is the height of the tide exactly
8.15 m? 3.125 hr, or 3 hr 7.5 min
SECTION 14A
SECTION
Quiz for Lessons 14-1 Through 14-2
14A
14-1 Graphs of Sine and Cosine
Identify whether each function is periodic. If the function is periodic, give the period.
1.
2.
Þ
Þ
Organizer
Ó
Ó
Ý
Ý
ä
Objective: Assess students’
û
mastery of concepts and skills in
Lessons 14-1 and 14-2.
Ó
not periodic
3.
ä
û
{
Ó
periodic; 2π
4.
Þ
Þ
GI
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Ó
Ý
ä
Ó
Ý
Ó
Resources
Ó
Assessment Resources
periodic; 4
not periodic
Section 14A Quiz
Using f (x) = sin x or f (x) = cos x as a guide, graph each function. Identify the
amplitude and period.
5. f (x) = sin 4x
6. g(x) = -3 sin x
7. h(x) = 0.25 cos πx
Using f (x) = sin x or f (x) = cos x as a guide, graph each function. Identify the
x-intercepts and phase shift.
(
3π
8. f (x) = cos x - _
2
)
(
)
3π
9. g (x) = sin x - _
4
(
5π
10. h(x) = cos x + _
4
Test & Practice Generator
)
INTERVENTION
Resources
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
π
π
of 500 newtons for 0 ≤ x ≤ __
. What is the torque for an angle of __
?
2
3
Ready to Go On?
Intervention and
Enrichment Worksheets
Ready to Go On? CD-ROM
14-2 Graphs of Other Trigonometric Functions
Ready to Go On? Online
Using f (x) = tan x as a guide, graph each function. Identify the period, x-intercepts,
and asymptotes.
1 tan 4x
12. f (x) = _
2
1x
13. g (x) = -2 tan _
2
1 πx
14. h(x) = tan _
2
Using f (x) = cot x as a guide, graph each function. Identify the period, x-intercepts,
and asymptotes.
15. g (x) = -2 cot x
16. h(x) = cot 0.5x
Answers
5–20. See p. A52.
17. j(x) = cot 4x
Using f (x) = cos x or f (x) = sin x as a guide, graph each function. Identify the
period and asymptotes.
18. f (x) = -2 sec x
1 csc x
19. g (x) = _
4
20. h(x) = sec πx
Ready to Go On?
NO
READY
Ready to Go On?
Intervention
YES
Diagnose and Prescribe
INTERVENE
TO
Worksheets
1005
ENRICH
GO ON? Intervention, Section 14A
CD-ROM
Lesson 14-1
14-1 Intervention
Activity 14-1
Lesson 14-2
14-2 Intervention
Activity 14-2
Online
Diagnose and
Prescribe Online
READY TO GO ON?
Enrichment, Section 14A
Worksheets
CD-ROM
Online
Ready to Go On?
1005
SECTION
14B Trigonometric Identities
One-Minute Section Planner
Lesson
Lab Resources
14-3 Technology Lab Graph Trigonometric Identities
•
Use a graphing calculator to compare graphs and make conjectures
about trigonometric identities.
SAT-10
NAEP ✔ ACT
SAT
SAT Subject Tests
□
□
□
□
Use fundamental trigonometric identities to simplify and rewrite
expressions and to verify other identities.
SAT-10
NAEP ✔ ACT
SAT ✔ SAT Subject Tests
□
□
□
□
Required
Technology Lab Activities
14-3 Technology Lab
Optional
graphing calculator
graphing calculator
□
Lesson 14-4 Sum and Difference Identities
• Evaluate trigonometric expressions by using sum and difference
•
Technology Lab Activities
14-3 Lab Recording Sheet
□
Lesson 14-3 Fundamental Trigonometric Identities
•
Materials
Optional
graphing calculator
identities.
Use matrix multiplication with sum and difference identities to
perform rotations.
SAT-10
NAEP ✔ ACT
SAT ✔ SAT Subject Tests
□
□
□
□
□
Lesson 14-5 Double-Angle and Half-Angle Identities
• Evaluate and simplify expressions by using double-angle and halfangle identities.
SAT-10
NAEP
□
□
Optional
graphing calculator
✔ SAT Subject Tests
□ ACT □ SAT □
Lesson 14-6 Solving Trigonometric Equations
• Solve equations involving trigonometric functions.
✔ ACT □ SAT □
✔ SAT Subject Tests
□ SAT-10 □ NAEP □
Required
graphing calculator
MK = Manipulatives Kit
1006A
Chapter 14
Math Background
IDENTITIES
Lesson 14-3
SUM AND DIFFERENCE IDENTITIES
Lesson 14-4
An 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 difference of two squares, a 2 – b 2 = (a + b)(a – b), is an
identity since the equation is true for all a and b.
The 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.
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 algebraic 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.
The complex number w = cos B + isin B is represented 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.
W (cos B, sin B)
B
Z (cos A, sin A)
A
TRIGONOMETRIC IDENTITIES
Lesson 14-3
The 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 2θ + cos 2θ = 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
π
of __
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.
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 magnitude 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.
1006B
14-3
Organizer
Graph Trigonometric
Identities
Use with Lesson 14-3
Pacing:
1
Traditional __
day
2
1
__
Block 4 day
You can use a graphing calculator to compare graphs and make conjectures
about trigonometric identities.
Objective: Use a graphing
calculator to compare graphs
and make conjectures about
trigonometric identities.
Use with Lesson 14-3
Activity
GI
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@<I
KEYWORD: MB7 Lab14
2
sin x
Determine whether _______
= 1 + cos x is a possible identity.
1 - cos x
Materials: graphing calculator
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.
Online Edition
Graphing Calculator, TechKeys
2
sin x
1 Enter _______
as Y1 and 1 + cos x as Y2. For Y2, select the
1 - cos x
mode represented by the 0 with a line through it. This will
help you see the path of the graph.
Resources
2 Set the graphing window by using
Technology Lab Activities
and 7:ZTrig.
14-3 Lab Recording Sheet
Teach
Discuss
Discuss the limitations of using a
graph to determine whether an
equation is an identity. Include
topics such as domain and viewing
windows.
3 Watch the calculator as the graphs are generated. As Y2 is being
graphed, a circle will move along the path of the graph.
Close
4 The path of the circle, Y2, traced the graph of Y1. The graphs appear
to be the same.
Key Concept
When you graph both sides of a
trigonometric equation together
and their graphs coincide, then the
equation is most likely an identity.
2
sin x
= 1 + cos x is most
Because the graphs appear to be identical, _______
1 - cos x
likely an identity. Use algebra to confirm.
Assessment
Journal Have students explain why
graphing shows that an equation is
only most likely an identity.
Try This
1. Make a Conjecture Determine whether sec x - tan x sin x = cos x is a
possible identity. It is a possible identity.
2. Prove or disprove your answer to Problem 1 by using algebra.
1 + tan x
3. Make a Conjecture Determine whether _______
= tan x is a possible identity.
1 + cot x
It is a possible identity.
4. Prove or disprove your answer to Problem 3 by using algebra.
1006
Chapter 14 Trigonometric Graphs and Identities
Answers to Try This
2. sec x - tan x sin x
sin 2 x
1
= cos x - cos x
1 - sin 2 x
=
cos x
cos 2 x
= cos x = cos x
_ _
_
_
4.
1 + tan x
_
1 + cot x
sin x
1 + ____
cos x
=
cos
x
1 + ____
sin x
_
_________
cos x
_
cos x + sin x
=
sin x + cos x
_________
sin x
cos x + sin x __
sin x
= __
·
cos x
sin x + cos x
sin x
= _ = tan x
cos x
KEYWORD: MB7 Resources
1006
Chapter 14
Angle Relationships
Organizer
Geometry
Angle relationships in circles and polygons can be used to solve problems.
Geometry
Pacing:
1
Traditional __
day
2
1
__
Block 4 day
Objective: Apply trigonometric
,
,
functions to solving problems
involving inscribed and
circumscribed polygons.
,
À
À
ô
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ô
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(
)
n - 2 180°
θ= _
n
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Remember
(
_
r = R cos 180°
n
)
(
(
)
_ = 2R sin 180°
_
s = 2r tan 180°
n
n
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.
(
)
Students 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.
,
Close
Choose a formula relating the radius of
the circumscribed circle to the side
length of the polygon.
Substitute 5 for R and 8 for n.
( )
_
s = 2(5)sin 180°
8
Assess
Ask students to find the value of
θ and the side length for a regular
hexagon circumscribing a circle of
radius 3 cm. θ = 120 ◦; s ≈ 3.46 cm
s = 10 sin 22.5° ≈ 3.83 cm
Try This
Online Edition
Teach
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s = 2R sin 180°
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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.
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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
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? ≈2.59 in.
xäÂ
xxÂ
{xÂ
ÈxÂ
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.
Connecting Algebra to Geometry
1007
Answers
1. 4.62 in.; 6.93 in.
2. 28.28 cm, 28.28 cm, and 40 cm
A211NLS_c14cn1_1007.indd 1007
8/7/09 9:32:02 AM
KEYWORD: MB7 Resources
Connecting Algebra to Geometry
1007
14-3 Fundamental
14-3 Organizer
Trigonometric Identities
Pacing: Traditional 1 day
1
Block __
day
2
Objectives: Use 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.
Technology Lab
GI
In Technology Lab Activities
<D
@<I
Who uses this?
Ski supply manufacturers can use
trigonometric identities to determine
the type of wax to use on skis. (See
Example 3.)
Objective
Use fundamental
trigonometric identities
to simplify and rewrite
expressions and to verify
other identities.
Online Edition
Graphing Calculator, Tutorial
Videos
A derivation for a Pythagorean identity
is shown below.
x2 + y2 = r2
Pythagorean Theorem
y2
x +_
_
=1
r2
r2
cos 2 θ + sin 2 θ = 1
2
Warm Up
Divide both sides by r 2.
y
x and sin θ for _
Substitute cos θ for _
r
r.
Simplify.
(
)(
sin A cos 2 A
1. _ _
cos A sin A
)
Fundamental Trigonometric Identities
cos A
( )( )
sin A
1
2. tan A _ _
tan A sin A
1
Also available on transparency
Reciprocal
Identities
Tangent and Cotangent
Ratio Identities
Pythagorean
Identities
1
csc θ = _
sin θ
1
sec θ = _
cos θ
sin θ
tan θ = _
cos θ
cos 2 θ + sin 2 θ = 1
sin(-θ) = -sin θ
cos θ
cot θ = _
sin θ
1 + tan 2 θ = sec 2 θ
cos(-θ) = cos θ
cot 2 θ + 1 = csc 2 θ
tan(-θ) = -tan θ
1
cot θ = _
tan θ
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.
Student: Superheroes would be in
big trouble if villains knew trig.
EXAMPLE
Parent: Why is that?
Student: Because then the villains
could figure out the superheroes’
secret identities.
Negative-Angle
Identities
1
Proving Trigonometric Identities
Prove each trigonometric identity.
A sec θ = csc θ tan θ
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.
1008
sec θ = csc θ tan θ
( )(
Choose the right-hand side to modify.
sin θ
1
_
= _
sin θ cos θ
)
Reciprocal and ratio identities
1
=_
cos θ
Simplify.
= sec θ
Reciprocal identity
Chapter 14 Trigonometric Graphs and Identities
1 Introduce
E X P L O R AT I O N
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Motivate
Recall 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 expressions; one form may be more useful than another
for a given situation. Trigonometric identities can
be used to rewrite expressions.
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KEYWORD: MB7 Resources
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SEC XTAN X
1008
Chapter 14
Explorations and answers are provided in
Alternate Openers: Explorations Transparencies.
Prove each trigonometric identity.
B csc(-θ) = -csc θ
csc(-θ) = -csc θ
1
_
=
sin(-θ)
1
_
=
-sin θ
1 = -csc θ
- _
sin θ
Additional Examples
Choose the left-hand side to modify.
Reciprocal identity
Example 1
Negative-angle identity
( )
-csc θ = -csc θ
Prove each trigonometric
identity.
sec θ
A. tan θ =
csc θ
1
_
cos θ
=
1
_
sin θ
1
=
· sin θ = tan θ
cos θ
_
( )
_
Reciprocal identity
( )
Prove each trigonometric 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.
EXAMPLE
If you get stuck, try
converting all of
the trigonometric
functions into sine
and cosine functions.
2
B. 1 - cot θ = 1 + cot(-θ)
1
=1+
tan(-θ)
1
=1+
-tan θ
_
Using Trigonometric Identities to Rewrite
Trigonometric Expressions
Rewrite each expression in terms of cos θ, and simplify.
sin 2 θ
A _
B sec θ - tan θ sin θ
1 - cos θ
1 - sin θ · sin θ Substitute.
1 - cos 2 θ Pythagorean identity
_
cos θ
cos θ
1 - cos θ
Factor the
sin 2 θ
1 -_
_
Multiply.
difference
(
)
1
+
cos
θ
1
cos
θ
(
)
cos θ
cos
θ
___
of
two
1 - cos θ
1 - sin 2 θ
_
squares.
Subtract fractions.
cos θ
(1 + cos θ)(1 - cos θ)
__
Simplify.
cos 2 θ
_
1 - cos θ
Pythagorean identity
cos θ
1 + cos θ
cos θ
Simplify.
_
= 1 + (-cot θ)
_ (_)
= 1 - cot θ
Example 2
Rewrite each expression in
terms of cos θ, and simplify.
A. sec θ (1- sin 2 θ)
cos θ
B. sin θ cos θ (tan θ + cot θ) 1
Rewrite each expression in terms of sin θ, and simplify.
cos 2 θ
1 -1
2b. cot 2 θ
2a. _
1 + sin θ
1 - sin θ
2
_
sin θ
INTERVENTION
Questioning Strategies
Graphing to Check for Equivalent Expressions
I like to use a graphing calculator to check for
equivalent expressions.
sin 2 θ
and
For Example 2A, enter y = _
(1 - cosθ)
y = 1 + cosθ. Graph both functions in the
same viewing window.
Julia Zaragoza
Oak Ridge
High School
EX AM P LE
Î
Óû
1
• How can the identity in Example
1B be demonstrated by using the
unit circle?
Óû
EX AM P LE
2
• How do you know when you
should use one of the Pythagorean
identities?
£
The graphs appear to coincide, so the expressions
are most likely equivalent.
Answers to Check It Out!
14-3 Fundamental Trigonometric Identities
1a. sin θ cot θ = sin θ
cos θ
(_
sin θ )
= cos θ
2 Teach
1
_
cos (-θ)
1
=1-_
b. 1 - sec (-θ) = 1 -
Guided Instruction
Guide 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.
1009
Through Cognitive Strategies
cos θ
= 1 - sec θ
Have students work in groups to derive the
Pythagorean identities 1 + cot 2 θ = csc 2 θ
and 1 + tan 2 θ = sec 2 θ so that students
can recall the identities by understanding.
cos 2 θ _
sin 2 θ _
1
+
=
For example, _
2
sin θ
sin 2 θ
sin 2 θ
yields 1 + cot 2 θ = csc 2 θ.
Lesson 14-3
1009
EXAMPLE
Precalculus Many aspects
of trigonometry become
important when students
begin their study of calculus. In calculus, students will encounter situations in which expressions must be
modified in order to solve problems.
3
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.
ô
7AXEDWOOD
mgCOS ô
Additional Examples
mgSIN ô
7ETSNOW
Example 3
ô
At what angle will a wooden
block on a concrete incline start
to move if the coefficient of friction is 0.62?
The symbol μ is
read as “mu.”
θ ≈ 32 ◦
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
sin θ = 0.14 cos θ
sin θ = 0.14
_
cos θ
tan θ = 0.14
INTERVENTION
Questioning Strategies
EX A M P L E
Divide both sides by mg.
3
Substitute 0.14 for μ.
Divide both sides by cos θ.
Ratio identity
θ ≈ 8°
• If you knew the angle at which the
block began to slide, how could
you find the coefficient of friction?
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. θ ≈ 22°
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) = sin 2 θ.
3. GET ORGANIZED Copy and complete the graphic organizer
by writing the three Pythagorean identities.
*Þ̅>}œÀi>˜Ê`i˜ÌˆÌˆiÃ
1010
Chapter 14 Trigonometric Graphs and Identities
Answers to Think and Discuss
3 Close
Summarize
Trigonometric 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.
Possible answers:
and INTERVENTION
Diagnose Before the Lesson
14-3 Warm Up, TE p. 1008
Monitor During the Lesson
Check It Out! Exercises, SE pp. 1009–1010
Questioning Strategies, TE pp. 1009–1010
Assess After the Lesson
14-3 Lesson Quiz, TE p. 1013
Alternative Assessment, TE p. 1013
1010
Chapter 14
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
sin 2 θ + cos 2 θ = 1.
3. See p. A14.
14-3
Exercises
KEYWORD: MB7 14-3
14-3 Exercises
KEYWORD: MB7 Parent
Assignment Guide
GUIDED PRACTICE
SEE EXAMPLE
1
2
6.
Rewrite each expression in terms of cos θ, and simplify.
4. csc θ tan θ
p. 1009
SEE EXAMPLE
3. cos 2 θ (sec 2 θ - 1) = sin 2 θ
2. cot(-θ) = -cot θ
1. sin θ sec θ = tan θ
p. 1008
SEE EXAMPLE
Assign Guided Practice exercises
as necessary.
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
Prove each trigonometric identity.
1
_
5.
cos θ
(1 + sec 2 θ)(1 - sin 2 θ)
1 + cos 2 θ
1
_
cos 2 θ
6. sin 2 θ + cos 2 θ + tan 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. θ ≈ 43°
3
p. 1010
Homework Quick Check
Quickly check key concepts.
Exercises: 8, 10, 12, 14, 16
PRACTICE AND PROBLEM SOLVING
Independent Practice
For
See
Exercises Example
8–11
12–15
16
1
2
3
Prove each trigonometric identity.
sin θ - cos θ = 1 - cot θ
9. __
sin θ
11. sec 2 θ (1 - cos 2 θ) = tan 2 θ
8. sec θ cot θ = csc θ
10. tan θ sin θ = sec θ - cos θ
Extra Practice
Skills Practice p. S31
Application Practice p. S45
Answers
1. sin θ sec θ = sin θ
Rewrite each expression in terms of sin θ, and simplify.
sin 2 θ
cos 2 θ 1 - sin θ
tan θ
13. _
12. _
cot θ 1 - sin 2 θ
1 + sin θ
2
1
θ-1
sec
_
14. cos θ cot θ + sin θ
15.
sin 2 θ
sin θ
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.
_
_ _
cos θ
(_
sin θ )
=-
= -cot θ
Multi-Step Rewrite each expression in terms of a
single trigonometric function.
19. cos θ + sin θ tan θ sec θ
21. cot θ
22. sec θ
23. sin θ
= cos 2 θ
cos θ(tan 2 θ + 1)
20. sin θ csc θ - cos θ sin θ 21. cos θ sec θ csc θ
2
2
23. csc θ(1 - cos 2 θ)
24. csc θ cos θ tan θ 1
sin 2 θ
26. _
1
1 - cos 2 θ
tan θ
27. _
1
sin θ sec θ
22.
sin θ
25. _
csc θ
1 - cos 2 θ
cos
θ
28. _ 1
sin θ cot θ
29. tan θ (tan θ + cot θ)
30. sin 2 θ + cos 2 θ + cot 2 θ
31. sin 2 θ sec θ csc θ tan θ
sec 2 θ
csc 2 θ
= cos 2 θ
Verify each identity.
cos θ -1 = sec θ - sec 2 θ 33. sin 2 θ(csc 2 θ -1) = cos 2 θ 34. tan θ + cot θ = sec θ csc θ
32. _
cos 2 θ
cos θ = sec θ
csc 2 θ
1 - cos 2 θ = sin θ cos θ 37. _
35. _
36. _
= cot 2 θ
tan θ
1 - sin 2 θ
1 + tan 2 θ
14-3 Fundamental Trigonometric Identities
sin θ
_
cos θ
1 - cos θ
=_
2
= sin 2 θ
cos θ
= sec θ - cos θ
=
1
(sin θ)
(_
cos θ )
2
2
sin θ
_
2
cos 2 θ
= tan 2 θ
32.
cos θ
cos θ - 1
1
_
=_ -_
cos θ
cos θ
cos θ
1
1
=_ - _
2
2
cos θ
2
cos θ
(_)
sin 2 θ
cos 2 θ
8. sec θ cot θ =
=
1
cos θ
_
(_
cos θ )( sin θ )
1
_
sin θ
= csc θ
sin θ
cos θ
sin θ - cos θ
__
=_-_
sin θ
sin θ
sin θ
= 1 - cot θ
1011
33. sin 2 θ (csc 2 θ - 1) = sin 2 θ cot 2 θ
2
11. sec 2 θ(1 - cos 2 θ) =
sin θ
(_
)
= sin 2 θ
9.
Prove each fundamental identity without using any of the other fundamental
identities. (Hint: Use the trigonometric ratios with x, y, and r.)
sin θ
cos θ
38. tan θ = _
40. 1 + cot 2 θ = csc 2 θ
39. cot θ = _
cos θ
sin θ
1
1
42. sec θ = _
41. csc θ = _
43. 1 + tan 2 θ = sec 2 θ
cos θ
sin θ
10. tan θ sin θ =
3. cos θ( sec 2 θ - 1) = cos 2 θ (tan 2 θ)
2
18. sin θ cot θ tan θ sin θ
2
cos θ
= tan θ
cos (-θ)
cos θ
2. cot (-θ) =
=
-sin θ
sin (-θ)
θ ≈ 42°
17. tan θ cot θ 1
sin θ
_
=
_
1
(_
cos θ )
2
cos 2 θ
= sec θ - sec 2 θ
(_)
cos 2 θ
sin 2 θ
= cos 2 θ
sin θ
cos θ
_
+_
cos θ
sin θ
sin θ + cos θ
__
=
sin θ cos θ
1
_
=
sin θ cos θ
1
1
= (_)(_
)
34. tan θ + cot θ =
2
2
sin θ cos θ
= sec θ csc θ
35–43. See p. A52.
KEYWORD: MB7 Resources
Lesson 14-3
1011
Exercise 44 involves
solving trigonometric
equations. This exercise prepares students for the
Multi-Step Test Prep on page 1034.
44. This problem will prepare you for the Multi-Step Test Prep on page 1034.
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? ≈ 0.48 s
b. What is the displacement at this time? ≈ 2.31
c. At what other times will the masses have the same displacement?
Answers
≈ 0.48 + πn where n is an integer
sin θ
56. Because tan θ = _,
cos θ
sin(-θ)
tan(-θ) = _ . Use
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.)
cos(-θ)
sin(-θ) = -sinθ and
45. (csc θ - 1)(csc θ + 1) = tan 2 θ no
cos(-θ) = cos θ to get
-sin θ
= -tanθ.
tan(-θ) =
cos θ
cos θ(sec θ + cos θ csc 2 θ) = csc 2 θ yes
47.
_
49. cos θ = 0.99 cos θ no
46. sec θ - cos θ = sin θ no
48. cot θ(cos θ + sin θ tan θ) = csc θ
50. sin θ cos θ = tan θ - tan θ sin 2 θ yes
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
g tan θ
cone is given by the formula r = _____
, where g represents
ω2
the force of gravity and ω represents the angular velocity
of the pendulum.
52. odd: sine,
tangent, cotangent,
cosecant; even:
cosine, secant
ô
Ű
…
/
À
√
g
_____
and fundamental trigonometric
a. Use ω = cos θ
identities to rewrite the formula for the radius. r = sin θ
“
“}
b. Find a formula for in terms of g, ω, and a single
trigonometric function. = g sec θ
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.
_
ω2
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.
{
Þ
b.
odd
Þ
{
Ó
even
Ó
Ý
55. an infinite number
of equivalent forms;
sin θ
tan θ =
,
cos θ
sin θ
,
cos θ =
tan θ
{
ä
Ó
_
_
Ó
Ý
{
{
ä
Ó
Ó
Ó
{
{
Ó
{
sin θ
55. Critical Thinking In how many equivalent forms can tan θ = ____
be expressed?
cos θ
Write at least three of its forms.
sin θ = tan θ cos θ
56. Write About It Use the fact that sin(-θ) = - sin θ and cos(-θ) = cos θ to explain
why tan(-θ) = - tan θ.
14-3 PRACTICE A
14-3 PRACTICE C
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1012
Chapter 14 Trigonometric Graphs and Identities
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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?
tanθ
1
_
_
sec θ csc θ
sinθ cos θ
sin 2 θ
59. Which trigonometric statement is NOT an identity?
1 + cos 2 θ = sin 2 θ
csc 2 θ - 1 = cot 2 θ
Students having
difficulty with
Exercise 58 should
remember to change everything to
sine and cosine. Students having difficulty with Exercise 60 should begin
with the identity 1 + tan 2 θ = sec 2 θ
and try to solve from there.
cos 2 θ
_
cotθ
1 + tan 2 θ = sec 2 θ
1 - sin 2 θ = cos 2 θ
Answers
61. sin θ + cot θ cos θ
cos θ
cos θ Given,
= sin θ +
sin θ
ratio identity
sin 2 θ
cos 2 θ
=
+
Common
sin θ
sin θ
denominators
sin 2 θ + cos 2 θ
Add fractions.
=
sin θ
= 1
Pythagorean identity
sin θ
= csc θ
Ratio identity
(_ )
60. Which is equivalent to 1 - sec 2 θ?
tan 2 θ
-tan 2 θ
cot 2 θ
-cot 2 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 EXTEND
Write each expression as a single fraction.
cos θ + 1
1 +_
1
62. _
cos θ cos 2 θ cos 2 θ
1
sin θ
cos θ + _
63. _
cos θ sin θ cos θ
sin θ
cos θ sin θ - cos θ
64. 1 - _
sin θ
sin θ
1
cos θ
1
-_
65. _
1 - cos θ 1 - cos 2 θ 1 - cos 2 θ
_
_
_
__
Journal
Simplify.
66.
1
____
-1
sin θ
_
1
2
67.
cos 2 θ
_____
1
1
____
+ ____
cos θ
sin θ
_
68.
1
________
sin θ cos θ
sin 2 θ
SPIRAL REVIEW
1
1
____
- ____
cos θ
sin θ
__
_
69.
sin θ
cos θ
____
- ____
cos θ
1
1 - ____
sin θ
sin θ
sin θ
-1
__
sin θ + cos θ
Have 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.
1
1 - ____
2
sin θ
_
sin θ + cos θ
sin θ + 1
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)
B
>

Have students create three of their
own trigonometric identities including 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
1
sin θ = sin θ to _ = tan θ cos θ.
csc θ
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b. High airfares and high temperatures cause tourism to drop off in the summer.
graph A
14-3
Find each probability. (Lesson 11-3)
71. rolling a 4 on a number cube
and a 4 on another number cube
72. getting heads on both tosses
1
when a coin is tossed 2 times
1
_
_
4
36
__
Find four values for which each function is undefined. (Lesson 14-2)
73. y = - tan θ
3π
3π
π
_π , _
, -_, -_
2
2
2
2
74. y = sec(0.5 θ)
75. y = - csc θ
π, 3π, -π, -3π
0,π, -π, 2π
14-3 Fundamental Trigonometric Identities
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(_)
_
2. sec 2 θ = 1 + sin 2 θ sec 2 θ
sin 2 θ
=1+
cos 2 θ
= 1 + tan 2 θ
_
= sec 2 θ
Rewrite each expression in
terms of cos θ, and simplify.
3. sin 2 θ cot 2 θ sec θ
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sin θ cos θ
sin 2 θ
=
sin θ cos θ
sin θ
=
cos θ
= sin θ sec θ
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__
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Lesson 14-3
1013
14-4 Sum and Difference
14-4 Organizer
Identities
Pacing: Traditional 1 day
1
Block __
day
2
trigonometric expressions by using
sum and difference identities.
Objectives
Evaluate trigonometric
expressions by using sum
and difference identities.
Use matrix multiplication with sum
and difference identities to perform
rotations.
Use matrix multiplication
with sum and difference
identities to perform
rotations.
GI
Objectives: Evaluate
<D
@<I
Why learn this?
You can use sum and difference identities and
matrices to form images made from rotations.
(See Example 4.)
Matrix multiplication and sum and difference
identities are tools to find the coordinates of
points rotated about the origin on a plane.
Vocabulary
rotation matrix
Online Edition
Tutorial Videos
Sum and Difference Identities
Warm Up
Sum Identities
Difference Identities
sin(A + B) = sin A cos B + cos A sin B
sin(A - B) = sin A cos B - cos A 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 + tan B
tan(A + B) = __
1 - tan A tan B
tan A - tan B
tan(A - B) = __
1 + tan A tan B
Find each product, if possible.
A=
⎡ 1
√
3⎤
_ -_
⎢
1⎤
1
√
3⎦
2
B=⎢
1
⎣
_
2⎦
2
√3
_
⎣ 2
⎡0
1. AB ⎢
⎣2
EXAMPLE
⎡ √
3
1
Evaluating Expressions with Sum and Difference Identities
Find the exact value of each expression.
A sin 75°
sin 75° = sin(30° + 45° )
-1 ⎤
⎡ √3
-1 ⎤
2. BA ⎢
⎣
2
0⎦
√3
⎦
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).
Also available on transparency
√
√
√2
3 _
2 _
1 ·_
=_
+
·
2 2
2
2
Evaluate.
√
√6
√2
2 + √
6
=_+_=_
4
4
4
Simplify.
( 12 )
π = cos _
cos(-_
( π6 - _π4 )
12 )
π
B cos -_
“A good mathematical joke is better, and better mathematics, than a
dozen mediocre papers.”
In Example 1B, there
is more than one
π
. For
way to get -__
12
π
π
__
example, __
or
4
6
π
π
__
__
.
4
3
(
)
(
)
J. E. Littlewood
= cos
π -_
π.
π as the difference _
Write -_
4
6
12
_π cos _π + sin _π sin _π
6
4
6
4
Apply the identity for cos(A - B).
√
√2
1 _
3 √2
=_·_+_
·
2
2
2 2
Evaluate.
√2
√
√
+ √
6
6
2
=_+_=_
4
4
4
Simplify.
Find the exact value of each expression.
1a. tan 105° -2 -
1014
√
3
(
√
√
2- 6
)_
4
11π
1b. sin -_
12
Chapter 14 Trigonometric Graphs and Identities
1 Introduce
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Have students consider the parabolas y = x 2 and
x = y 2. Have students brainstorm the type of
transformation that would be necessary to transform one of these parabolas into the other. The
transformation is a rotation, and its methods and
techniques will be investigated in this lesson.
/ Ê Ê-
1--Ê
1014
Motivate
Explorations and answers are provided in
Alternate Openers: Explorations Transparencies.
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.
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Additional Examples
Reflection Across x-axis
Phase Shift Right π Radians
ÞÊVœÃ­Ýû®
Example 1
ÞÊVœÃ Ý
Þ
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ä
û
£
Ý
Óû
Óû
û
ä
û
£
ÞÊVœÃ Ý
+ √
√6
2
_
A. cos 15 ◦
Óû
û
Find the exact value of each
expression.
( )
11π
B. tan _
12
ÞÊVœÃ Ý
4
√
3-2
Example 2
Prove the identity.
EXAMPLE
2 Proving Identities with Sum and Difference Identities
(
cos(x - π) = -cos x
Apply the identity for cos(A - B).
-1 · cos x + 0 · sin x =
4
Simplify.
(
)
Example 3
π = -sin x.
2. Prove the identity cos x + _
2
EXAMPLE
1
Find cos(A - B) if sin A = _
3
3
π
_
with 0 < A < and if tan B = _
4
2
+3
π 8 √2
_
with 0 < B < .
2
15
_
3 Using the Pythagorean Theorem with Sum and
Difference Identities
1 - tan θ tan 4
1 + tan θ
=
1 - tan θ
_
Evaluate.
-cos x = -cos x
__π
tan θ + tan 4
π
tan(θ + _) =__
__π
Choose the left-hand side to modify.
cos x cos π + sin x sin π =
)
1 + tan θ
π
tan θ + _ = _
4
1 - tan θ
Prove the identity cos(x - π) = -cos x.
_
_
8
7
with 180° < A < 270° and if cos B =
with
Find tan(A + B) if sin A = 25
17
0° < B < 180°.
Step 1 Find tan A and tan B.
Refer to Lessons 13-2
and 13-3 to review
reference angles.
y
Use reference angles and the ratio definitions sin A = __r and cos B = _xr_.
Draw a triangle in the appropriate quadrant and label x, y, and r for
each angle.
In Quadrant III (QIII),
180° < A < 270°
7.
and sin A = -_
25
In Quadrant I (QI),
0° < B < 180°
8.
and cos B = _
17
Questioning Strategies
EX AM P LE
1
• How can you decide whether to
add or subtract values?
ÀÊÊ£Ç
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EX AM P LE
ÞÊÊÇ
INTERVENTION
Þ
ÝÊÊn
2
x 2 + (-7) = 25 2
8 2 + y 2 = 17 2
x = - √
625 - 49 = -24
y
7.
Thus, tan A = _ = _
x 24
EX AM P LE
y = √
289 - 64 = 15
y
15 .
Thus, tan B = _ = _
8
x
14-4 Sum and Difference Identities
2
• How can you decide what values
to use for A and B in the sum and
difference formulas?
ÀÊÊÓx
1015
3
• How can you check to see if your
answer is reasonable?
Answers to Check It Out!
(_π2 + x)
π
π
= cos( _ ) cos x - sin( _ ) sin x
2
2
2. cos
2 Teach
Guided Instruction
= (0)cos x - (1)sin x
Introduce the sum and difference identities for sine, cosine, and tangent. Practice
with students adding and subtracting angle
measures from the unit circle to help identify what angle measures can be evaluated
with the sum and difference identities.
= -sin x
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
Lesson 14-4
1015
Step 2 Use the angle-sum identity to find tan(A + B).
tan A + tan B
tan(A + B) = __ Apply identity for tan(A + B).
1 - tan A tan B
Additional Examples
( ) ( )
( )( )
7
15
__
+ __
24
8
= __
7 __
__
1 - 24 15
8
Example 4
Find the coordinates to the nearest hundredth of the points (1, 1)
and (2, 0) after a 40 ◦ rotation
about the origin.
(0.12, 1.41), (1.53, 1.29)
15
7
Substitute __
for tan A and __
for tan B.
24
8
52
__
416
24
tan(A + B) = _
, or _
35
87
1 - __
Simplify.
64
3. Find sin(A - B) if sin A = _45_ with 90° < A < 180° and if
cos B = __35 with 0° < B < 90°. 24
_
25
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.
INTERVENTION
Questioning Strategies
EX A M P L E
Using a Rotation Matrix
4
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:
⎡cos θ -sin θ⎤ ⎡ x ⎤ ⎡ x ⎤
⎢
⎢ = ⎢ ⎣sin θ cos θ⎦ ⎣ y ⎦ ⎣ y ⎦
• 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
⎡ cos θ sin θ ⎤
multiplying by ⎢
.
⎣ -sin θ cos θ ⎦
Tell students that the clockwise rota◦
tion is equivalent to a (360 - θ)
counterclockwise rotation.
EXAMPLE
4
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.
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Î
Î £®
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Step 1 Write matrices for a 30° rotation
Ó
Ó
and for the points in the figure.
⎡cos 30° -sin 30°⎤
R 30° = ⎢
Rotation matrix
⎣sin 30° cos 30°⎦
⎡0 0 √
3 - √
3⎤
Matrix of point coordinates
S= ⎢
1⎦
⎣2 4 1
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⎡cos 30° -sin 30°⎤ ⎡0 0 √3
- √3
⎤
R 30° × S = ⎢
⎢
1⎦
⎣sin 30° cos 30°⎦ ⎣2 4 1
⎡ -1 -2 1 -2 ⎤
=⎢
2 √
3 √
3 0⎦
⎣ √3
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3 ), B
(-2, 2 √
3 ),
Ó
Ó
C (1, √
3 ), and D
(-2, 0).
Ó
Ý
4. Find the coordinates, to the nearest hundredth, of the points in
the original figure after a 60° rotation about the origin.
(
) (
)
(
A - √
3 , 1 , B -2 √
3 , 2 , C(0, 2), D - √
3 , -1
1016
)
Chapter 14 Trigonometric Graphs and Identities
3 Close
and INTERVENTION
Diagnose Before the Lesson
14-4 Warm Up, TE p. 1014
Monitor During the Lesson
Check It Out! Exercises, SE pp. 1014–1016
Questioning Strategies, TE pp. 1015–1016
Assess After the Lesson
14-4 Lesson Quiz, TE p. 1019
Alternative Assessment, TE p. 1019
1016
Chapter 14
Summarize
By using angle sum and difference formulas, 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.
Answers to Think and Discuss
THINK AND DISCUSS
Possible answers:
1. Evaluate sin(60 ◦ - 45 ◦),
sin(45 ◦ - 30 ◦), or
sin(135 ◦ - 120 ◦).
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?
/>˜}i˜Ì
-ՓÊ>˜`ʈvviÀi˜ViÊ`i˜ÌˆÌˆiÃ
3. GET ORGANIZED Copy and complete the graphic
organizer. For each type of function, give the sum
and difference identity and an example.
14-4
-ˆ˜i
œÃˆ˜i
Exercises
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 multiplied by each other, and the
signs are opposites.
KEYWORD: MB7 14-4
KEYWORD: MB7 Parent
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 R θ assume?
A rotation matrix assumes a counterclockwise rotation about the origin.
SEE EXAMPLE
1
Find the exact value of each expression.
√ √
11π
2. cos 105° 2 - 6 3. sin _
12
4
_
p. 1014
SEE EXAMPLE
2
SEE EXAMPLE
3
p. 1015
p. 1016
√
3
5. cos(-75°)
√
6 - √
2
_
4
)
(
7. tan(π + x) = tan x
_
_
16
_
65
16
-33
_
10. cos(A - B) _ 11. tan(A + B)
63
65
)
56
12. tan(A - B) -_
33
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.
PRACTICE AND PROBLEM SOLVING
14–17
18–20
21–24
25
1
2
3
4
Find the exact value of each expression.
7π √
14. sin _
6 + √2 15. tan 165° √3 - 2 16. sin 195°
12
_
Prove each identity.
3π + x = sin x
18. cos _
2
Extra Practice
Skills Practice p. S31
Application Practice p. S45
)
- √
2 - √6
__
_
4
(
11π
17. cos _
12
√2 - √
6
4
4
3π + x = -cos x
19. sin _
2
20. tan(x - 2π) = tan x
(
)
12
4
Find each value if cos A = -___
with 90° < A < 180° and if sin B = -__
with
5
13
270° < B < 360°.
_
21. sin(A + B) 63
65
_
22. tan(A - B) 33
56
_
(_π2 + x)
6. sin
= sin
_π cos x + cos _π sin x
2
2
= (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
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
Homework Quick Check
Quickly check key concepts.
Exercises: 14, 18, 22, 24
_
23. cos(A + B) - 16 24. cos(A - B) - 56
65
14-4 Sum and Difference Identities
Answers
14-4 Exercises
Assign Guided Practice exercises
as necessary.
A
(-1.73, -1), B
(0.87, 0.5), C
(-1.5, 2.60)
Independent Practice
For
See
Exercises Example
3. See p. A14.
Assignment Guide
4
3π - x = -sin x
8. cos _
2
Find each value if sin A = - 12 with 180° < A < 270° and if sin B = 4 with
5
13
90° < B < 180°.
9. sin(A + B)
SEE EXAMPLE 4
√
6 - √2
_
Prove each identity.
π + x = cos x
6. sin _
2
(
p. 1015
π
4. tan _
212
2. Both the sine and cosine identities are in the form of a sum
or difference of 2 products. The
products each involve the 2 different angles given in the same
order (A then B).
8. cos
65
1017
3π
- x)
(_
2
= cos
3π
3π
_
cos x + sin _ sin x
2
2
= (0)cos x + (-1)sin x
= -sin x
18. cos
3π
+ x)
(_
2
= cos
3π
3π
_
cos x - sin _ sin x
2
2
= (0)cos x - (-1)sin x
= sin x
19, 20. See p. A53.
KEYWORD: MB7 Resources
Lesson 14-4
1017
Language
For Exercise 38, ensure
that students know that,
in this context, to offset is
ENGLISH
to displace or move out of LANGUAGE
LEARNERS
position.
Multiple Representations
Matrices like the one in
Exercise 41 can be used to
represent other transformations. For
example, a 180 ◦ rotation is equivalent to a reflection about the x- and
y-axes.
Exercise 43 involves
applying trigonometric identities. This
exercise prepares students for the
Multi-Step Test Prep on page 1034.
Answers
√
6 - √
2
_
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.
4
A(-1.41,1.41), B(-0.71, 2.12), C(-0.71, 0.71)
6
- √2 - √
28.
4
Find the exact value of each expression.
26. sin 165°
27. tan(-105°) 2 +
√
2 - √
6
29. _
29. sin(-15°)
19π
30. cos _
12
32. sin 255°
33. tan 195° 2 -
__
4
√
3
28. cos 195°
√
3
5π
31. tan _
2+
12
π
34. cos _
12
√
3
√
6 - √
2
_
Find the value for each unknown angle given that 0° ≤ θ ≤ 180°.
√
2
1
1
35. cos (θ - 30°) = _
36. cos(20° + θ) = _
37. sin(180° - θ) = _
2
2
2
θ
=
90°
θ
=
25°
θ
=
30°
or
150°
2
- √6 - √
38. Physics Light enters glass of thickness t at an
32.
4
angle θ i and leaves the glass at the same angle θ i.
ôI
However, the exiting ray of light is offset from the
ˆÀ
√
sin(θ
θ
)
2 + √
6
r
i
initial ray by a distance Δ = _________
t, indicated
34.
ôIôR
sin θ i cos θ r
4
in the figure shown.
T
H
>Ãà ôR
a. Write the formula for Δ in terms of tangent
38a.
Ű
and cotangent by using the difference
Δ = 1 - cot θ i tan θ r t
Ű
ˆÀ
identities and other trigonometric
ôI
$
identities.
ôI
b. Use the figure to write a ratio for sin(θ i - θ r).
30.
4
__
_
(
(
)
)
_
⎡0 -1⎤ ⎡-1
0⎤ ⎡ 0
; ⎢
; ⎢
41a. ⎢
⎣1
0⎦ ⎣ 0 -1⎦ ⎣-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);
sin (θ i - θ r) =
h
Multi-Step Find tan(A + B), cos(A + B), and sin(A - B) for each situation.
253
204
_
; -_;
325
253
36
_
7 with 180° < A < 270° and cos B = _
12 with 0° < B < 90°
39. sin A = -_
25
13
1 with 270° < A < 360° and sin B = _
4 with 0° < B < 90°
40. sin A = -_
5
3
39.
325
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.
2
54 - 25 √
40. _;
P (0, 0), Q (1, -1),
R (0, -4), S (-1, -1)
c.
26.
28
√
4+6 2
;
15
-3 - 8 √
2
15
_
y
Ó +­£]Ê£®
*­ä]Êä®
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 PQRS
to create the logo.
_
x
Þ
{
{
Ý
ä
Ó
,­{]Êä®
Ó -­£]Ê£®
{
( )
11π by using sum or
42. Critical Thinking Is it possible to find the exact value of sin _
24
difference identities? Explain.
11π
_
42. Possible answer: No;
43. This problem will prepare you for the Multi-Step Test Prep on page 1034.
24
cannot be expressed as a sum or
difference of values from the unit
circle.
The displacement y of a mass attached to a spring is modeled by
(
)
π
2π
y(t) = 4.2 sin ___
t - __
, where t is the time in seconds.
2
3
a. What are the amplitude and period of the function? 4.2; 3
b. Use a trigonometric identity to write the displacement, using only the
2π
cosine function.
y (t) = -4.2 cos
t
3
c. What is the displacement of the mass when t = 8 s?
_
14-4 PRACTICE A
2.1
14-4 PRACTICE C
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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°
Þ
{
Ó
45. 60°
47. -30°
46. 120°
Ý
{
ä
Ó
Ó
{
Ó
48. Write About It In general, does
sin(A + B) = sin A + sin B? Give an example
to support your response.
Students having difficulty with
Exercise 50 may wish to refer
to the unit circle as a guide. If
5π
π
π
1
sin _ + x = _, then _ + x = _.
2
2
6
2
{
(
49. Which is the value of cos 15° cos 45° - sin 15° sin 45°?
√2
√2
1
_
_
-_
2
2
2
π +x =_
1?
50. Which gives the value for x if sin _
2
2
π
π
π
_
_
_
4
6
3
(
Students having
difficulty with
Exercise 49 should
condense the expression into
cos(15 ◦ + 45 ◦) rather than
evaluating cos15 ◦ and sin15 ◦.
2 + √2
_
2
)
)
Answers
44–48, 52–55. See p. A53.
π
_
2
51. Given sin A = __12 with 0° < A < 90° and cos B = __35 with 0° < B < 90°, which expression
gives the value of cos(A - B)?
+4
-4
3 √3
3 √3
3 + 4 √3
3 - 4 √3
_
_
_
_
10
10
10
10
52. Short Response Find the exact value for sin(-15°). Show your work.
CHALLENGE AND EXTEND
Journal
53. Verify that the rotation matrix for θ is the inverse of the
rotation matrix for -θ.
*Ī­ÝĪ]ÊÞĪ®
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.
*­Ý]ÊÞ®
À
ô
À
í
"
Find the angle by which a figure ABC with vertices A (1, 0), B (0, 2), and C (-1, 0)
was rotated to get ABC .
√2
√2
√2
√
2
, 2), C -_, -_ 45°
56. A(0, 1), B(-2, 0), C (0, -1) 90°
57. A _, _ , B (- √2
2
2
2
2
58. A(-1, 0), B(0, -2), C (1, 0) 180°
(
(
)
Have students explain how the sum
formulas can be used to show that
adding 360 ◦ has no effect on the
values of sine, cosine, or tangent.
)
(
(
)
)
√3
√3
1
1
59. A _, _
, B (-1, √
3 ), C -_, -_
30°
2
2 2
2
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.
SPIRAL REVIEW
Divide. Assume that all expressions are defined. (Lesson 8-2)
6x 4y 3y 2
9x 3y 2
x2 + x - 2
x 2 + 3x + 2 x - 1
6x 3x
3x 2 ÷ _
60. _
61. __
÷ __
62. _4 ÷ _
3
2
2
2
21y
15xy
7y
x - 2x - 8
x - 3x - 4 x + 2
3x 2y 5 10
2y
_
_
_
Identify the conic section that each equation represents. (Lesson 10-6)
63. x 2 + 2xy + y 2 + 12x - 25 = 0 parabola
65.
1
__
cos θ - cos 3 θ
64. 5x 2 + 5y 2 + 20x - 15y = 0 circle
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1. Find the exact value of cos 75 ◦.
√
6 - √2
4
2. Prove the identity
π
sin _ - θ = cos θ.
2
π
π
= sin _ cos θ - cos _ sin θ
2
2
= cos θ - 0 = cos θ
12
3. Find tan(A - B) for sin A = _
13
π
with 0 < A < _ and
2
π
8
cos B = _ with 0 < B < _.
2
17
21
220
4. Find the coordinates to the
nearest hundredth of the
point (3, 4) after a 60 ◦
rotation about the origin.
_
Rewrite each expression in terms of a single trigonometric function. (Lesson 14-3)
cot θ sec θ
tan θ sin θ
65. _
66. cot θ tan θ csc θ csc θ
67. _
sin 2 θ
sec θ
sin θ cos θ
14-4 Sum and Difference Identities
14-4
(
1019
)
_
⎡ -1.96 ⎤
≈⎢
⎣ 4.60 ⎦
Also available on transparency
Lesson 14-4
1019
14-5 Double-Angle and
14-5 Organizer
Half-Angle Identities
Pacing: Traditional 1 day
1
Block __
day
2
Objectives: Evaluate and
GI
simplify expressions by using
double-angle and half-angle
identities.
<D
@<I
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.)
Objective
Evaluate and simplify
expressions by using
double-angle and
half-angle identities.
You can use sum identities to derive the
double-angle identities.
Online Edition
Tutorial Videos
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
sin 2θ + cos 2θ = 1. It is common to rewrite expressions as functions of θ only.
Warm Up
Find tan θ for 0 ≤ θ ≤ 90 ◦, if
3
3
1. sin θ = _. tan θ =
5
4
√
2
1
_
2. sin θ = . tan θ =
3
4
√
1 - x2
3. cos θ = x. tan θ = _
x
_
Double-Angle Identities
_
cos 2θ = cos 2 θ - sin 2 θ
cos 2θ = 2 cos 2 θ - 1
sin 2θ = 2 sin θ cos θ
cos 2θ = 1 - 2 sin θ
2
Also available on transparency
EXAMPLE
1
2 tan θ
tan 2θ = _
1 - tan 2 θ
Evaluating Expressions with Double-Angle Identities
3
Find sin 2θ and cos 2θ if cos θ = -__
and 90° < θ < 180°.
4
Step 1 Find sin θ to evaluate sin 2θ = 2 sin θ cos θ.
Method 1 Use the reference angle.
θ
Teacher: Why is your answer sin _
2
when the problem asks for sin 2θ?
Student: I guess it’s a case of
mistaken identity.
In QII, 90° < θ < 180°, and cos θ = -__34 .
(-3) + y = 4
2
The signs of x and
y depend on the
quadrant for angle θ.
sin
cos
QI
+
+
QII
+
QIII
QIV
+
2
Use the Pythagorean
Theorem.
2
y = √
16 - 9 = √7
sin θ =
ô
ÝÊÊÎ
Solve for y.
√
7
_
4
Method 2 Solve sin 2 θ = 1 - cos 2 θ.
sin 2 θ = 1- cos 2 θ
3
sin θ = 1- -__
4
( )
2
√7
9
= 1 - __
= ___
4
16
sin θ =
1020
ÀÊÊ{
Þ
Substitute -__34 for cosθ.
Simplify.
√
7
_
4
Chapter 14 Trigonometric Graphs and Identities
1 Introduce
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Motivate
Ask 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.
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1020
Chapter 14
Explorations and answers are provided in
Alternate Openers: Explorations Transparencies.
2/9/10 8:46:34 AM
Step 2 Find sin 2θ.
""
sin 2θ = 2 sin θ cos θ
Apply the identity for sin 2θ.
(_)( _)
=2
√7
4
-
√7
3 for cos θ.
Substitute _ for sin θ and -_
4
2
3
4
7
3 √
= -_
8
Simplify.
Step 3 Find cos 2θ.
cos 2θ = 2 cos 2 θ - 1
2
3 for cos θ.
Substitute -_
4
Additional Examples
Simplify.
Example 1
1
=_
8
Find sin 2θ and tan 2θ if
2
sin θ = _ and 0 ◦ < θ < 90 ◦.
5
1. Find tan 2θ and cos 2θ if cos θ = __13 and 270° < θ < 360°.
4 √
2
7
_
; -_
7
You can use double-angle identities to prove trigonometric identities.
EXAMPLE
2
9
Prove each identity.
A sin 2 θ = 1 (1 - cos 2θ)
2
1
sin 2 θ = _(1 - cos 2θ)
2
1
_
=
1- (1 - 2 sin 2 θ)
2
1 (2 sin 2 θ)
=_
2
A. sin 2θ = 2 tan θ - 2 tan θ sin 2 θ
= 2 tan θ (1 - sin 2 θ)
= sin 2θ
Simplify.
= 2 cos 2 θ - 1
(cos
2
θ + sin θ) + (2 cos θ sin θ) =
sec θ
2
Prove each identity.
( )
2
2a. cos 4 θ - sin 4 θ = cos 2θ
1
= 2 sin θ cos θ = sin 2θ
Regroup.
Rewrite using 1 = cos 2 θ + sin 2 θ and
sin 2θ = 2 sin θ cos θ.
1 + sin 2θ = 1 + sin 2θ
cos θ
sin θ
_____
____
(
(
cos θ ) _
1 )
_
=
·
1
_____ _____
cos θ
14-5 Double-Angle and Half-Angle Identities
2 Teach
INTERVENTION
Questioning Strategies
EX AM P LE
2 tan θ
2b. sin 2θ = _
1 + tan 2 θ
You can use double-angle identities for cosine to derive the half-angle identities
by substituting __2θ for θ. For example, cos 2θ = 2 cos 2 θ - 1 can be rewritten as
cos θ = 2 cos 2 __2θ - 1. Then solve for cos __2θ .
1021
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?
EX AM P LE
2
• How do you decide what to do first
when proving an identity?
Answers to Check It Out!
2a. Possible answer:
Guided Instruction
Introduce 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
θ
as _ = 120 ◦, or 2θ = 120 ◦.
2
= cos 2θ
Choose the left-hand side
to modify.
Expand the square.
2
1 + sin 2θ =
2
B. cos 2θ = (2 - sec 2 θ)(1 - sin 2 θ)
= (2 - sec 2 θ)(cos 2 θ)
cos 2 θ + 2 cos θ sin θ + sin 2 θ =
sin θ
cos θ
cos θ
sin θ _
(_
cos θ )( 1 )
= 2 sin θ cos θ
Apply the identity for cos 2θ.
(cos θ + sin θ)2 = 1 + sin 2θ
2
=2
= 2 (tan θ cos θ) cos θ
B (cos θ + sin θ)2 = 1 + sin 2θ
____)
(
_
=
cos θ
)
= 2 tan θ cos 2 θ
Choose the right-hand side to modify.
sin 2 θ = sin 2 θ
_
2
17
Prove each identity.
2b. Possible answer:
2 tan θ
1 + tan 2 θ
2
25
Proving Identities with Double-Angle Identities
(
2
21 _
21
4 √
4 √
_
;
Example 2
_
Choose to modify
either the left side
or the right side of
an identity. Do not
work on both sides
at once.
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.
Select a double-angle identity.
( _) -1
9 -1
= 2(_
16 )
3
=2 4
Ê,,",
,/
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 doubleangle identity for sine.
cos 4 θ - sin 4 θ
= (cos 2 θ + sin 2 θ)
(cos 2 θ - sin 2 θ)
= (1)(cos 2θ)
= cos 2θ
Lesson 14-5
1021
Half-Angle Identities
1 - cos θ
θ = ± _
sin _
2
2
Additional Examples
Use half-angle identities to
find the exact value of each
trigonometric expression.
√
3+2
A. cos 15 ◦
2
7π
_
- 3 - 2 √2
B. tan
8
Half-angle identities are useful in calculating exact values for trigonometric
expressions.
_
EXAMPLE
3
Evaluating Expressions with Half-Angle Identities
Use half-angle identities to find the exact value of each trigonometric
expression.
π
A cos 165°
B sin 8
π
1 _
330°
sin _
cos _
2 4
2
_
Example 4
θ
θ
Find cos _ and tan _ if
2
2
π
7
tan θ = _ and 0 < θ < _.
24
2
√
7 2 1
;
10 7
( )
__
√
1 + cos 330°
- __
2
In Example 3, the
√
2 + √
3
expressions - ________
2
√
√
2- 2
and _______
are in
2
-
reduced form and
cannot be simplified
further.
-
INTERVENTION
Negative
in QII
( )
√3
1 + ___
2
_
2
(
)( )
2 + √
3 _
1
_
2
2
Simplify.
• How do you decide which sign
to select when evaluating an
expression by using the half-angle
identity?
3a.
2
√2
1 - ___
2
_
2
(
√2
π =_
cos _
4
2
)( )
_
2 - √2
1
_
2
Positive
in QI
2
Simplify.
Check Use your calculator.
√
7 + 4 √
3
4
• What is the connection between
the half-angle formulas and the
corresponding difference formulas
from Lesson 14-4?
π
1 - cos(__
4)
__
√
2 - √
2
_
2
Check Use your calculator.
3
+
√3
cos 330 ° = _
2
√
2 + √3
-_
2
Questioning Strategies
EX A M P L E
1 - cos θ
θ = ± _
tan _
1 + cos θ
2
θ
.
Choose + or - depending on the location of __
2
Example 3
EX A M P L E
1 + cos θ
θ = ± _
cos _
2
2
Use half-angle identities to find the exact value of each
trigonometric expression.
5π
3a. tan 75°
3b. cos _
8
√
2 - √2
b. 2
_
EXAMPLE
4
Using the Pythagorean Theorem with Half-Angle Identities
5
θ
θ
Find sin __
and tan __
if sin θ = -___
and 180° < θ < 270°.
2
2
13
Step 1 Find cos θ to evaluate the half-angle identities.
Use the reference angle.
Ý
5
In QIII, 180° < θ < 270°, and sin θ = -__
.
13
x + (-5) = 13
2
2
2
x = - √
169 - 25 = -12
Thus, cos θ = - 12 .
13
_
1022
Chapter 14
ô
£Î
Pythagorean Theorem
Solve for the missing
side x.
Chapter 14 Trigonometric Graphs and Identities
a211se_c14l05_1020_1026.indd 1022
1022
x
7/21/09 2:32:00 PM
θ.
Step 2 Evaluate sin _
2
""
θ
sin _
2
1 - cos θ
_
√
2
+
√
In problems such as Example 3,
students may drop or combine the
radicals. The nature of the calculations is such that the answers will
often contain radicals inside of other
radicals. Remind students that these
expressions are in simplest form.
θ where 90° < _
θ < 135°.
Choose + for sin _
2
2
12
1 - -___
13
_
2
( )
Evaluate.
√(_2513 )(_12 )
Simplify.
Math Background To
avoid confusion, you may
wish to encourage students to work on only one side of
an equation at a time when proving
identities. However, for more organized students, it is legitimate to
manipulate both sides of the identity
until they are equal.
√
25
_
26
Be careful to choose
the correct sign for
sin __2θ and cos __2θ . If
180° < θ < 270°, then
90° < __2θ < 135°.
Ê,,",
,/
26
5 √
_
26
θ.
Step 3 Evaluate tan _
2
θ
tan _
2
√
1 - cos θ
- _
1 + cos θ
θ < 135°.
θ where 90° < _
Choose - for tan _
2
2
12
___
-
1 - (- )
_
___
-
√(_2513 )(_131 )
13
Evaluate.
( )
1 + - 12
13
Simplify.
- √
25
-5
4. Find sin __2θ and cos __2θ if tan θ = __43 and 0° < θ < 90°.
√
5 _
5
2 √
_
;
5
5
THINK AND DISCUSS
1. EXPLAIN which double-angle identity you would use to
cos 2θ
simplify _________
.
sin θ + cos θ
2. DESCRIBE how to determine the sign of the value for sin __2θ and
for cos __2θ .
3. GET ORGANIZED Copy and complete the graphic organizer. In each
box, write one of the identities.
œÕLi‡˜}iÊ`i˜ÌˆÌÞÊvœÀÊ
œÃˆ˜i
14-5 Double-Angle and Half-Angle Identities
Answers to Think and Discuss
3 Close
Summarize
Expressions in terms of twice an angle or
half an angle can be rewritten as expressions in terms of the angle by using double-angle and half-angle identities.
1023
Possible answers:
and INTERVENTION
Diagnose Before the Lesson
14-5 Warm Up, TE p. 1020
Monitor During the Lesson
Check It Out! Exercises, SE pp. 1021–1023
Questioning Strategies, TE pp. 1021–1022
Assess After the Lesson
14-5 Lesson Quiz, TE p. 1026
Alternative Assessment, TE p. 1026
1. cos 2θ = cos 2 θ - sin 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 lies in
2
based on the quadrant that θ lies in,
and identify the sign of sine (or cosine)
in that quadrant.
3. See p. A14.
Lesson 14-5
1023
14-5 Exercises
14-5
Exercises
KEYWORD: MB7 14-5
KEYWORD: MB7 Parent
Assignment Guide
Assign Guided Practice exercises
as necessary.
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
Homework Quick Check
GUIDED PRACTICE
SEE EXAMPLE
1
Find sin 2θ, cos 2θ, and tan 2θ for each set of conditions.
5 and _
π <θ<π
4 and 0° < θ < 90°
1. cos θ = -_
2. sin θ = _
5
13
2
2
Prove each identity.
p. 1020
SEE EXAMPLE
119 120
120
_
; -_; _
169
3. 2 cos 2θ = 4 cos 2 θ - 2
1 + cos 2θ
5. _ = cot θ
sin 2θ
7.
p. 1021
SEE EXAMPLE
169 119
√
2 - √
2
_
2
24
7
24 ; -_
_
; -_
p. 1022
√
_
_
_
_
√2 + √2
_
_
θ
θ
θ
Find sin , cos , and tan for each set of conditions.
2
2
2
2
24
1
_
_
11. sin θ = and 180° < θ < 270°
12. cos θ = and 270° < θ < 360°
4
25
SEE EXAMPLE 4
p. 1022
_4 ; -_3 ; -_4
5
5
√
√
√
6
10
15
_
; - _; - _
3
4
PRACTICE AND PROBLEM SOLVING
Answers
3. 2 cos 2 θ = 2(2 cos 2 θ - 1)
= 4 cos 2 θ - 2
4. sin θ = 1 - cos θ
2 cos 2 θ
=12
(2 cos 2 θ - 1) + 1
=12
cos 2θ + 1
=12
1 + (2 cos 2 θ - 1)
1 + cos 2θ
5.
=
sin 2θ
(2 sin θ cos θ)
2 cos 2 θ
=
2 sin θ cos θ
cos θ
=
sin θ
2
_
__
_
_ __
__
_
= cot θ
6, 15–18. See p. A53.
27. cos θ (1 - 4sin 2θ)
28. sin 4θ + cos 4θ - 6 sin 2θ cos 2θ
1
2
3
4
-
Extra Practice
840 _
840
_
; 41 ; _
625 527
841 841
1 (1 + cos 2θ)
16. cos 2 θ = _
2
sin 2θ
18. tan θ = _
1 + cos 2θ
Skills Practice p. S31
Application Practice p. S45
19.
√
2 + √
3
_
20.
√
2 - √
3
_
21.
Use half-angle identities to find the exact value of each trigonometric expression.
5π
7π
20. cos _
21. sin 22.5°
22. tan 15°
19. sin _
12
12
√
3 √
10
10
θ
θ
θ
Find sin , cos , and tan for each set of conditions.
;; -3
24.
2
2
2
10
10
3 and 180° < θ < 270°
3π < θ < 2π
12 and _
23. tan θ = -_
24. sin θ = -_
5
35
2
_
2
2
√
2 - √
2
_
2
2 - √3
22. _
2 + √
3
37
1024
_
_ _
_
25. 3 sin θ cos 2 θ - sin 3 θ
26. 4 sin θ cos 3 θ - 4 cos θ sin 3 θ
25. sin 3θ
26. sin 4θ
27. cos 3θ
28. cos 4θ
29. cos 2θ + 2 sin 2 θ 1
30. cos 2θ + 1 2 cos 2 θ
cos 2θ
32. __
cos θ - sin θ
cos θ + sin θ
Multi-Step Rewrite each expression in terms of trigonometric functions of θ
rather than multiples of θ. Then simplify.
31. tan 2θ(2 - sec 2 θ) 2 tan θ
cos θ sin 2θ
_
sin θ
1 + cos 2θ
33.
6 √
37
1
_; -_
; -_
cos 2θ - 1 -2
34. _
sin 2 θ
6
37
Chapter 14 Trigonometric Graphs and Identities
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336
_
; -_; _
Prove each identity. 625
sin 2θ = 2 cos θ
15. _
sin θ
1 - cos 2θ
17. tan θ = _
sin 2θ
23.
√37
KEYWORD: MB7 Resources
4
Find sin 2θ, cos 2θ, and tan 2θ for each set of conditions.
7 and 90° < θ < 180°
20 and 0 ≤ θ ≤ _
π
13. cos θ = -_
14. tan θ = _
25
21
2
Independent Practice
For
See
Exercises Example
13–14
15–18
19–22
23–24
7
25
25
cos 2θ + 1
4. sin 2 θ = 1 - _
2
2 tan θ
6. sin 2θ = _
2
1 + tan θ
Use half-angle identities to find the exact value of each trigonometric expression.
π
7. cos 67.5°
8. cos _
2 + √3
12
√
2
+
2
3π
2
9. tan _
10. sin 112.5°
8
2 - √
2
3
Quickly check key concepts.
Exercises: 14, 16, 18, 20, 24
2
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41
Exercise 35 involves
applying doubleangle identities. This
exercise prepares students for the
Multi-Step Test Prep on page 1034.
35. This problem will prepare you for the Multi-Step Test Prep on page 1034.
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. y (t) = 6.2 sin t cos t
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? about 0.66 s
c. What is the displacement at this time? about 3.00 m
√
2 - √
3
42. - _
2
43. -
√
2 - √
3
_
2
Answers
36. -
23
32
32
√
5 √
11 √
55
;
;
4
4
5
√
18 + 6 √
5
1
4 √5
37.
5;
; ; 4 √
;
6
9
9
√
√
3 + √5
18 - 6 √
5
;6
√3 - √5
___
_
_
_
θ
θ
θ
Multi-Step Find sin 2θ, cos 2θ, tan 2θ, sin , cos , and tan for each set of
2
2
2
conditions.
√
5
3 and _
π <θ<π
36. cos θ = _
37. cos θ = -_ and 180° < θ < 270°
3
8
2
3π < θ < 2π
1 and _
2 and 0° < θ < 90°
39. tan θ = -_
38. sin θ = _
5
2
2
__
√
_
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.
38.
√
_
Prove each identity.
sin 2 θ
θ = __
45. cos 2 _
2
2(1 - cos θ)
ΔP = 2P f
2
25
25
17
10
√
√
50 + 10 √
21 __
5 - √
21
__
;
10
√
5 + √
21
5
5 - 2 √
_
_ _ _ √
;
3
39. - 4 ; ; - 4 ;
5 5
3
√_
1 - cos θ
2
-
tan θ + sin θ
θ
47. __ = cos 2 _
2
2 tan θ
1 - tan θ
46. cos 2θ = _
1 + tan 2 θ
21
4 √
21 _
4 √
17 _
_
;
;
;
√
50 - 10 √
21
__
;
44. Physics The change in momentum of a scattered nuclear
«Êv
particle is given by ΔP = P f - P i , where P f is the final
ã«
ô
@@
Ó
momentum, and P i is the initial momentum.
Ê
a. Use the diagram and the Pythagorean Theorem to write
«Êˆ
ô
@@
a formula for ΔP in terms of P i . Then write a formula for
Ó
ΔP in terms of P f .
b. Compare your two answers to part a. What does this tell you about the It does not
magnitude, or size, of the momentum before and after the “collision”? change.
c. Write the formula for ΔP in terms of cos θ.
__
__ _
Use half-angle identities to find the exact value of each trigonometric expression.
7π
11π
41. sin _
40. cos _
2 + √
2
2 - √
3
8 12
2
2
42. cos 105°
43. sin(-15°)
Physics
55
3 √
55
23 3 √
_
; - _; _;
10
√_ √ _
(_ )
(_ )
5
5 + 2 √
5 - 2 √5
;10
5 + 2 √5
θ
;
2
44a. ΔP = 2P i sin
(cos x)(1 - cos 2x)
48. Graphing Calculator Graph y = _____________
to discover an identity.
sin 2x
Then prove the identity.
θ
2
ΔP = 2P f sin
49. Multi-Step A golf ball is hit with an initial velocity of v 0 in feet per second at
45–48. See p. A53.
v 2 sin θ cos θ
0
an angle of elevation θ. The function d (θ) = __________
gives the horizontal distance
16
d in feet that the ball travels.
2
v 0 sin 2θ
d (θ) =
a. Rewrite the function in terms of the double angle 2θ.
32
b. Calculate the horizontal distance for an initial velocity of 80 ft/s for angles of
15°, 30°, 45°, 60°, and 75°. 100 ft; ≈ 173 ft; 200 ft; ≈ 173 ft; 100 ft
c. For a given velocity, what angle gives the maximum horizontal distance? 45°
50. First find sin 15° by using
30°
, and then find
sin
2
15°
.
sin 7.5° by using sin
2
_
_
_
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? 30.52° < θ < 59.48°
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?
14-5 PRACTICE A
Possible answer: If θ is multiplied by 2, use a double-angle identity. If θ is divided
by 2, use a half-angle identity.
14-5 Double-Angle and Half-Angle Identities
1025
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Lesson 14-5
1025
In Exercise 55, students who have difficulty 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.
√2
52. What is the value of sin 2θ if cos θ = -_ and 90° < θ < 180°?
2
√2
1
_
_
1
2
2
53. What is the value for cos 2θ if sin θ = cos θ?
0
1
Answers
56. Possible answer:
(cos θ - sin θ)
cos 2θ
__
= __
2
sin θ + cos θ
2
sin θ + cos θ
(___
cos θ - sin θ)(cos θ + sin θ)
=
_(_)
2 cos 2 θ
θ if cos θ = -_
12 and 90° < θ < 180°?
54. What is the value for sin _
2
13
√26
√26
5 √26
_
_
-_
26
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
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.
θ by using the ratio identity for tangent.
58. Derive the half-angle formula for tan _
2
2 + √
3
2 - √
__
2+
2 sin 2 θ
cos 2 θ
π
56. Short Response Verify that _________
= cos θ - sinθ for 0 ≤ θ ≤ __
. Show each
2
sinθ + cos θ
step in your justification process.
sin θ + cos θ
= cos θ - sin θ
⎤
⎡
1 30°
59. tan ⎢
2 2 ⎦
⎣
=
-1
√
2 + √
3
Use half-angle identities to find the exact value of each expression.
π
π
59. tan 7.5°
60. tan _
61. sin _
62. cos 11.25°
16
24
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.
Journal
Have students show that, given
the first double-angle identity for
cosines, cos 2θ = cos 2 θ - sin 2 θ,
they can derive the other two
double-angle identities by using the
Pythagorean identities.
64. Derive the product-to-sum formulas sin A sin B = __12 ⎡⎣cos(A - B) - cos(A + B)⎤⎦ and
cos A cos B = _12_⎡⎣cos(A + B) + cos(A - B)⎤⎦ by using the angle sum and difference
formulas.
SPIRAL REVIEW
Use the vertical-line test to determine whether each relation is a function. (Lesson 1-6)
65.
Have students present, preferably to
the class, a derivation of one doubleangle identity. They should also
include a numerical example of the
identity in use.
66.
n
yes
Þ
{
67.
5x + 12
_
; x ≠ -7
ä
n
Ý
x+7
n { ä
{
n
{
θ
1. Find cos _ and cos 2θ if
2
π
5
sin θ = _ and 0 < θ < _.
2
8
√
39 + 8 7
;
4
32
__ _
2. Prove the following identity:
sin 2θ
__
2 - 2 sin 2 θ
2 sin θ cos θ
= __
n
n
n
7x - 2
_
; x≠0
Add or subtract. Identify any x-values for which the expression is undefined. (Lesson 8-3)
2x + 14
3x - 2 + _
4x - 1 + _
6x - 2
67. _
68. _
x
2
x
+
7
x
+
7
2x
2x - 30x - 20
69.
;
-x 3 + x 2 + 11x + 18
5x
+
8
7x
+
4
x
+
9
x
_
_
_
_
(x + 1)(x - 3) 69. x + 1 - x - 3
70.
;
2
x
+
2
x
x 2(x + 2)
x
__
14-5
x ≠ -1, 3
√
2 - √
6
71. _
4
1026
Chapter 14
__
(
)
√
2 + √6
_
2 2
4 ⎦
2+
√
2 - √6
_
4
4
y
⎤
2 + √
2
2 - √
1 1 π
60. tan ⎢ _(_ · _) = __
⎣
74. cos 255°
√
2 - √6
_
4
Trigonometric Graphs and Identities
⎡
a207se_c14l05_1020_1026.indd 1026
x ≠ -2, 0
Find the exact value of each expression. (Lesson 14-4)
π
7π
72. sin 105°
73. cos _
71. sin -_
12
12
√
2 + √2
x
û
2/16/07 12:48:44 PM
û
2(1-sin 2 θ)
sin θ cos θ
=
cos 2 θ
⎤
1 1 π
1 61. sin ⎢ _(_ · _) = _ 2 - √
2 + √3
= tan θ
⎤
⎡
1 45°
62. cos ⎢
2 + √
2 + √
2
= 12 2 2 ⎦
⎣
2 sin θ(cos θ - 1) = 0
63. π n, where n is an integer; possible
answer:
So cos θ = 1 or sin θ = 0, which are both
true when θ = πn, where n is an integer.
_
3. Find the exact value of
cos 22.5 ◦.
√
2+2
2
_
Also available on transparency
1026
no
Þ
Ý
68.
tan θ =
n
Chapter 14
⎡
⎣
2 2
6 ⎦
_(_)
2
_
Solve the equation:
sin 2 θ = 2 sin θ
2 sin θ cos θ - 2 sin θ = 0
57, 58, 64. See p. A53.
14-6 Solving Trigonometric
14-6 Organizer
Equations
Pacing: Traditional 1 day
1
Block __
day
2
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.)
Objectives: Solve equations
involving trigonometric functions.
GI
Objectives
Solve equations involving
trigonometric functions.
<D
@<I
Online Edition
Graphing Calculator, Tutorial
Videos, Interactivity, TechKeys
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.
EXAMPLE
1
Warm Up
Solving Trigonometric Equations with Infinitely Many Solutions
Solve.
Find all of the solutions of 3 tan θ = tan θ + 2.
Subtract tan θ from both sides.
1. x 2 + 3x - 4 = 0 x = 1 or -4
2
2. 3x 2 + 7x = 6 x =
or -3
3
Evaluate each inverse
trigonometric function.
Combine like terms.
3. Tan -11
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
2 tan θ = 2
tan θ = 1
Compare Example 1
with this solution:
3x = x + 2
3x-x = 2
2x = 2
x=1
45 ◦
√3
4. Sin -1-60 ◦
2
Divide by 2.
θ = tan -1 1
Apply the inverse tangent.
θ = 45°
Find θ when tan θ = 1.
_
Also available on transparency
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°.
{
™ä
Use the intersect feature of your
graphing calculator to find the points
of intersection.
™ä
“Do not worry about your difficulties
in mathematics. I can assure you
that mine are still greater.”
Albert Einstein
{
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.
150° + 360°n, 210° + 360°n
Some trigonometric equations can be solved by applying the same methods
used for quadratic equations.
14-6 Solving Trigonometric Equations
1027
1 Introduce
E X P L O R AT I O N
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SINVCOSVSINV
Motivate
Remind students of properties of the real numbers such as the Distributive Property. Point out
that these properties are in fact algebraic identities and were the basic tools used for solving
algebraic equations. In this lesson, the trigonometric 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.
9OUCANALSOSOLVESINVCOSVSINVALGEBRAICALLYBY
ADDINGSINVTOBOTHSIDESOFTHEEQUATIONANDTHENFACTORING
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Explorations and answers are provided in
Alternate Openers: Explorations Transparencies.
Lesson 14-6
1027
EXAMPLE
2
Solving Trigonometric Equations in Quadratic Form
Solve each equation for the given domain.
Additional Examples
A sin 2 θ - 2 sin θ = 3 for 0 ≤ θ < 2π
sin 2 θ - 2 sin θ - 3 = 0
Example 1
Find all solutions for
1
1
sin θ = _ sin θ + _.
4
2
θ = {30° + 360°n, 150° + 360°n}
Example 2
Solve each equation for the
given domain.
(sin θ + 1)(sin θ - 3) = 0
sin θ = -1 or sin θ = 3
sin θ = 3 has no solution because -1 ≤ sin θ ≤ 1.
The only solution will come from
3π
θ=_
sin θ = -1.
2
A trigonometric
equation may have
zero, one, two, or an
infinite number of
solutions, depending
on the equation and
domain of θ.
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.
Substitute 1 for a, 2 for b,
-(2) ± √(
2)2 - 4(1)(-1)
and -1 for c.
cos θ = ___
2(1)
Simplify.
cos θ = -1 ± √2
A. 4 tan 2 θ - 7 tan θ + 3 = 0 for
0 ≤ θ ≤ 360 ◦.
(tan θ - 1)(4 tan θ - 3) = 0
θ = 45 ◦ or 225 ◦, or θ ≈ 36.9
or 216.9
-1 - √
2 < -1 so cos θ = -1 - √
2 has no solution.
)
θ = cos -1(-1 + √2
Apply the inverse cosine.
Use a calculator. Find both
angles for 0° ≤ θ < 360°.
≈ 65.5° or 294.5°
B. 2 cos θ - cos θ = 1 for
0 ≤ θ ≤ π.
2
(2 cos θ + 1) (cos θ - 1) = 0
θ=
Subtract 3 from both sides.
Factor the quadratic expression by
comparing it with x 2 - 2x - 3 = 0.
Apply the Zero Product Property.
Solve each equation for 0 ≤ θ < 2π.
2a. cos 2 θ + 2 cos θ = 3 0
2b. sin 2 θ + 5 sin θ - 2 = 0
2π
_
or θ = 0
3
≈ 21.9°, ≈ 158.1°
You can often write trigonometric equations involving more than one function
as equations of only one function by using trigonometric identities.
EXAMPLE
INTERVENTION
3
Use trigonometric identities to solve each equation for 0 ≤ θ < 2π.
Questioning Strategies
EX A M P L E
A 2 cos 2 θ = sin θ + 1
1
Substitute 1 - sin 2 θ for cos 2 θ by
the Pythagorean identity.
Simplify.
2(1 - sin 2 θ ) - sin θ - 1 = 0
• If a trigonometric equation with no
domain restriction has at least one
solution, will it always have
infinitely many solutions?
EX A M P L E
Solving Trigonometric Equations with Trigonometric Identities
-2 sin 2 θ - sin θ + 2 - 1 = 0
2 sin 2 θ + sin θ - 1 = 0
Multiply by -1.
(2 sin θ - 1)(sin θ + 1) = 0
Factor.
1 or sin θ = -1
sin θ = _
2
5π or θ = _
3π
π
_
θ = or _
6
6
2
2
• What are possible numbers of solutions for a quadratic trigonometric
equation with one variable
0 < θ < 360 ◦?
Apply the Zero Product Property.
Check Use the intersect feature of your graphing calculator. A graph supports
your answer.
Ó
ä
1028
Ú
ä û
È
xû
Ú
È
Îû
Ú
Óû
Ó
Chapter 14 Trigonometric Graphs and Identities
2 Teach
Guided Instruction
Before solving quadratic trigonometric
equations, some students may need to
review solving basic quadratic equations.
Make the connection between a trigonometric equation and an algebraic equation by replacing sine or cosine with x.
Demonstrate that the process of factoring
and isolating the variable will remain
the same.
1028
Chapter 14
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.
Use trigonometric identities to solve each equation for 0° ≤ θ < 360°.
B cos 2θ + 3 cos θ + 2 = 0
Substitute 2 cos 2 θ - 1 for cos 2θ by
the double-angle identity.
Combine like terms.
2 cos 2 θ - 1 + 3 cos θ + 2 = 0
2 cos θ + 3 cos θ + 1 = 0
2
(2 cos θ + 1)(cos θ + 1) = 0
Additional Examples
Example 3
Factor.
1
cos θ = -_
2
or
Use trigonometric identities to
solve each equation.
Apply the Zero Product Property.
A. tan 2 θ + sec 2 θ = 3 for
0 ≤ θ ≤ 2π.
⎧ π 3π 5π 7π ⎫
⎬
θ=⎨ ,
,
,
4
4 ⎭
⎩4 4
2
2
B. cos θ = 1 + sin θ for
0 ≤ θ ≤ 360 ◦.
θ = 0 ◦ or 180 ◦ or 360 ◦
cos θ = -1
____
θ = 120° or 240° or θ = 180°
£
Check Use the intersect feature of your
graphing calculator. A graph supports
your answer.
ä
ÎÈä
ä°Ó
£näc
£Óäc
Ó{äc
INTERVENTION
Use trigonometric identities to solve each equation for the
given domain.
3a. 4 sin 2 θ + 4 cos θ = 5 for 0° ≤ θ < 360° 60°, 300°
Questioning Strategies
3b. sin 2θ = -cos θ for 0 ≤ θ < 2π 90°, 210°, 270°, 330°
EXAMPLE
4
EX AM P LE
• How do you determine which
identity will help solve an equation
or whether one is even necessary?
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
π
t(m) = 1.665 sin __
(m + 3) + 5.485,
6
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
3
N
W
E
• Is it possible to solve an equation
containing more than one trigonometric function without converting
everything to the same function?
S
MAINE
Augusta
The answer will be months of the
year.
List the important information:
• The function model is
π
(m + 3) + 5.485.
t(m) = 1.665 sin __
6
• Sunrise is at 7 A.M., which is represented by t = 7.
• m represents the number of months after January 1.
Mt. Desert
Island
Atlantic
Ocean
2 Make a Plan
Substitute 7 for t in the model. Then solve the equation for m by
using algebra.
14-6 Solving Trigonometric Equations
a207se_c14l06_1027_1033.indd 1029
1029
2/15/07 2:44:54 PM
Lesson 14-6
1029
3 Solve
Additional Examples
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
INTERVENTION
Questioning Strategies
EX A M P L E
Be sure to have
your calculator in
radian mode when
working with angles
expressed in radians.
π (m + 3) + 5.485
Substitute 7 for t.
7 = 1.665 sin _
6
7 - 5.485 = sin _
π (m + 3)
_
Isolate the sine term.
1.665
6
−−−
π
sin -1(0.9099) = _(m + 3)
Apply the inverse sine.
6
Sine is positive in Quadrants I and II. Compute both values.
−−−
−−−
π (m + 3)
π (m + 3)
QII: π - sin -1(0.9099) = _
QI: sin -1(0.9099) = _
6
6
π (m + 3)
π (m + 3)
1.143 ≈ _
π - 1.143 ≈ _
6
6
6
6 (
)
π 1.143 ≈ m + 3
π π - 1.143 ≈ m + 3
(_)
(_)
-0.817 ≈ m
0.817 ≈ m
The value m = 0.817 corresponds to late January and the value
m = -0.817 corresponds to early December.
4 Look Back
4
Check your answer by using a graphing
π
(x + 3) + 5.485
calculator. Enter y = 1.665 sin __
6
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.
• How do you know what month the
value of m corresponds to?
4. The number of hours h of sunlight in a day at
Cadillac Mountain can be modeled by
π (
h(d) = 3.31 sin ____
d - 85.25) + 12.22, where
182.5
d is the number of days after January 1. When are
there 12 hours of sunlight?
late March and late September
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.
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1030
Chapter 14 Trigonometric Graphs and Identities
Answers to Think and Discuss
3 Close
Summarize
Review with students the different methods for solving trigonometric equations,
such as graphing, factoring, and substituting by using trigonometric identities.
Possible answers:
and INTERVENTION
Diagnose Before the Lesson
14-6 Warm Up, TE p. 1027
Monitor During the Lesson
Check It Out! Exercises, SE pp. 1027–1030
Questioning Strategies, TE pp. 1028–1030
Assess After the Lesson
14-6 Lesson Quiz, TE p. 1033
Alternative Assessment, TE p. 1033
1030
Chapter 14
1. First solve the equation for a restricted
domain equal to the period of the
given function. Then use an understanding of the periodicity of the given
function to find all solutions.
2. See p. A14.
14-6
Exercises
14-6 Exercises
KEYWORD: MB7 14-6
KEYWORD: MB7 Parent
GUIDED PRACTICE
SEE EXAMPLE
1
1. 6 cos θ - 1 = 2
p. 1027
SEE EXAMPLE
2
Assignment Guide
Find all of the solutions of each equation.
60° + 360°n, 300° + 360°n
2. 2 sin θ - √
3=0
60° + 360°n, 120° + 360°n
Solve each equation for the given domain.
4. 2 sin 2 θ + 3 sin θ = -1 for 0 ≤ θ < 2π
p. 1028
SEE EXAMPLE
3
p. 1028
30°, 150°, 210°, 330°
p. 1029
5. cos 2 θ - 4 cos θ + 1 = 0 for 0° ≤ θ < 360°
≈ 74.5° or 285.5°
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°
SEE EXAMPLE 4
30° + 360°n, 330° + 360°n
- cos θ
3. cos θ = √3
7. sin 2 θ + cos θ = -1 for 0 ≤ θ < 2π π
Homework Quick Check
Quickly check key concepts.
Exercises: 10, 12, 14, 16, 17
PRACTICE AND PROBLEM SOLVING
9–12
13–14
15–16
17
150° + 360°n, 210° + 360°n
12. 2 sin θ + 1 = 2 + sin θ 90° + 360°n
Solve each equation for the given domain.
13. 2 cos 2 θ + cos θ - 1 = 0 for 0 ≤ θ < 2π
Extra Practice
Skills Practice p. S31
Application Practice p. S45
5π
π
13. _, π, or _
3
π
24. 0, _, π
2
11π
7π _
_
,
27.
6
tan θ - 3 = 0 60° + 180°n
10. √3
60° + 360°n, 300° + 360°n
11. 2 cos θ + √
3=0
1
2
3
4
3
Find all of the solutions of each equation.
9. 1 - 2 cos θ = 0
6
28. no solution
14. sin 2 θ + 2 sin θ - 2 = 0 for 0° ≤ θ < 360°
≈ 47.1°, ≈ 132.9°
Multi-Step Use trigonometric identities to solve each equation for the
given domain.
90°, 120°, 240°, 270°
15. cos 2θ + cos θ + 1 = 0 for 0° ≤ θ < 360°
5π _
3π
_π , _
,
16. cos 2θ = sin θ for 0 ≤ θ < 2π 6
6
2
17. Multi-Step The amount of energy used by a large office building is modeled by
π(
t - 8) + 800, where E is the energy in kilowatt-hours, and t is the
E(t) = 100 sin __
12
time in hours after midnight.
17a. 10:00 A.M. and 6:00 P.M.
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.
4.
18. 2 sin 2 θ = sin θ 0°, 30°, 150°, 180°
19. 2 cos 2 θ = sin θ + 1 30°, 150°, 270°
20. cos 2θ - 2 sin θ + 2 = 0
21. 2 cos 2 θ + 3 sin θ = 3 30°, 90°, 150°
22. cos θ + sin θ - 1 = 0 0°, 90°, 180°
23. 2 sin 2 θ + sin θ = 0 0°, 180°, 210°, 330°
≈ 55.4°,
≈ 124.6°
2
Solve each equation algebraically for 0 ≤ θ < 2π.
25. cos 2 θ - 3 cos θ = 4 π
3π
4π
2π
π
26. cos θ (0.5 + cos θ) = 0 ,
27. 2 sin 2 θ - 3 sin θ = 2
,
,
3
2 3
2
1
2
2
_
____
cos θ = 5
2
30. cos θ + 4 cos θ - 3 = 0
28. cos θ +
2
≈ 0.869, ≈ 5.414
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 temperature of 1 pound of water 1 degree
Fahrenheit.
Answers
Solve each equation algebraically for 0° ≤ θ < 360°.
24. sin 2 θ - sin θ = 0
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
8. Heating The amount of energy from natural gas used for heating a manufacturing
π
(m + 1.5) + 650, where E is the energy used in
plant is modeled by E(m) = 350 sin __
6
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.
mid-April and mid-December
Independent Practice
For
See
Exercises Example
Assign Guided Practice exercises
as necessary.
3π _
7π _
11π
_
,
,
2
6
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.
29. sin θ + 3 sin θ + 3 = 0 no solution
_
_
3
3
tan θ 0, π , π, 4π
31. tan 2 θ = √3
14-6 Solving Trigonometric Equations
1031
KEYWORD: MB7 Resources
Lesson 14-6
1031
Science Link
Exercise 34 points out
that the length of a tidal
day, a period containing two complete 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 actually about 24 hours and 50 minutes.
Exercise 38 involves
solving and interpreting a trigonometric
equation. This exercise prepares students for the Multi-Step Test Prep on
page 1034.
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
v 2 sin 2θ . At what angle(s) with the horizontal will the ball travel 250 feet?
R(θ) = _
32
b. The maximum vertical height H max in feet that the ball travels upward can be
v 2 sin 2 θ . At what angle(s) with the horizontal will the ball
modeled by H max(θ) = _
64
travel 50 feet?
32a. ≈ 30° and ≈ 60°
Performing Arts
33. Performing Arts A theater has a rotating stage
Ãi}“i˜Ì
that can be turned for different scenes. The stage has
À
ô
a radius of 18 feet, and the area in square feet of the
À
segment of the circle formed by connecting two radii
2
r
_
as shown is A = (θ - sin θ), with θ in radians.
2
a. What angle gives a segment area of 92 square feet?
How many such sets can simultaneously fit on the
π
2π
a. ≈ ; 4 sets b. ≈
; 5 sets
full rotating stage?
5
2
b. What angle gives a segment area of 50 square feet? About how many such sets
can simultaneously fit on the full rotating stage?
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.
_
34a. 3:25 A.M.,
7:20 A.M., 3:55 P.M.,
and 7:50 P.M.
34. Oceanography The height of the water on a certain day at a pier in Cape Cod,
π (t + 4) + 7.5, where h is the
Massachusetts, can be modeled by h(t) = 4.5 sin _
6.25
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? 6.25 h
d. No; the period of
the model is 12.5 h.
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 functions 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
sin θ
tan θ = ____
cos θ.
_
c. What is the period for the tide? 12.5 h
d. Does the cycle of tides fit evenly in a 24-hour day? Explain.
35.
35. B is incorrect;
Possible answer:
Division by sin θ
eliminates the
solution when
sin θ = 0.
/////ERROR ANALYSIS///// Below are two solution procedures for solving
sin 2 θ - __12 sin θ = 0 for 0° ≤ θ < 360°. Which is incorrect? Explain the error.
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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. x ≈ -4.165, -1.797, 1.395, 5.464, 6.831
38. This problem will prepare you for the Multi-Step Test Prep on page 1034.
The displacement in centimeters of a mass attached to a spring is modeled by
2π
π
y (t) = 2.9 cos ___
t + __
+ 3, where t is the time in seconds.
4
3
(
)
a. What are the maximum and minimum displacements of the mass? 5.9 cm; 0.1 cm
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? 0.74 s
c. At what other times will the displacement be 1 cm?
14-6 PRACTICE A
0.74 + 3n and 1.51 + 3n where n is an integer
14-6 PRACTICE C
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Chapter 14 Trigonometric Graphs and Identities
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Chapter 14
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Estimation Use a graphing calculator to approximate the solution to each
equation to the nearest tenth of a degree for 0° ≤ θ < 360°.
≈ 197.1°, ≈ 252.9°
39. tan θ - 12 = -1 ≈ 84.8°, ≈ 264.8°
40. sin θ + cos θ + 1.25 = 0
43. sin 2 θ + 5 sin θ = 3.5 38.5°, 141.5°
44. cos 2 θ - cos 2 θ + 1 = 0 no solution
52.5°, 113.5°, 232.5°, 293.5°
41. 4 sin 2(2θ - 30) = 4 60°, 150°, 240°, 330°42. tan 2 θ + tan θ = 3
In Exercises 46 and
50, students can
substitute the answer
choices into the equations to eliminate choices.
45. Write About It How many solutions can a trigonometric equation have? Explain
by using examples. 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.
Answers
51. cos θ =
__
θ = cos -1
= tan θ for -90° ≤ θ ≤ 90°?
47. Which gives an approximate solution to 5 tan θ - √3
-19.1°
19.1°
2
2(2)
-1 ± √
17
=
4
= 2 √3
for 0° ≤ θ < 360°?
46. Which values are solutions of 2 cos θ + √3
30° or 150°
60° or 120°
30° or 330°
60° or 320°
-23.4°
(1) - 4(2)(-2)
-1 ± √
___
23.4°
θ = cos -1
48. Which value for θ is NOT a solution to sin 2 θ = sin θ?
0°
90°
180°
270°
( _)
(__)
-1 + √17
4
or
17
-1 - √
4
θ ≈ 38.7° or 321.3° or no solution. Thus, θ ≈ 38.7° or 321.3°.
49. Which gives all of the solutions of cos θ - 1 = -__12 for 0 ≤ θ < 2π?
5π
4π
2π or _
2π or _
_
_
3
3
3
3
5π
2π
π or _
π or _
_
_
3
3
3
3
50. Which gives the solution to sin 2 θ - sin θ - 2 = 0 for 0° ≤ θ < 360°?
90°
270°
90° or 270°
No solution
51. Short Response Solve 2 cos 2 θ + cos θ - 2 = 0 algebraically. Show the steps in the
solution process.
CHALLENGE AND EXTEND
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°
Solve each equation algebraically for 0° ≤ θ < 360°.
52. 9 cos 3 θ - cos θ = 0
55. sin 2 θ - 4.5 sin θ = 2.5
210°, 330°
53. 4 cos 3 θ - cos θ = 0
1
56. ⎪sin θ⎥ = _
2
54. 16 sin 4 θ - 16 sin 2 θ + 3 = 0
√
3
57. ⎪cos θ⎥ = _
2
30°, 150°, 210°, 330°
__
−−
19 π
Order the given numbers from least to greatest. (Lesson 1-1) √
2 5 , 4.47, √
21 ,
,
√
−
3
−−
√
−
5
19
π
4 0.65
3
_
_
_
_
5
58.
, -1, 0.86, 1, -1, ,
59. 2 √
5,
21 ,
, 4.47, √
6
,
1
,
0.8
4
2
6
0.65
6 2
__
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)
cos 2θ + 1
2
sin 2θ cos θ 63. cos 2θ + sin 2 θ
62. _
64. _ cos θ
61. cos 2θ - 2 cos 2 θ
2
2 sin θ
2
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1033
14-6
14-6 CHALLENGE
^
%QUATION
Have students create three different trigonometric equations with
solutions worked out. At least one
equation should be quadratic,
and at least one should involve a
Pythagorean or double-angle identity. At least two of them should be
solvable without a calculator.
cos θ
14-6 Solving Trigonometric Equations
Have 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 solution is possible.
30°, 150°, 210°, 330°
SPIRAL REVIEW
60. f (x) = 0.05x;
60.
$12.95; 5% of $259,
the e-commerce
company’s revenue
from web sales of
$259
Journal
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1. Find all solutions for
- cos θ.
cos θ = √2
θ = 45 ◦ + n · 360 ◦ or 315 ◦ +
n · 360 ◦
2. Solve 3 sin 2 θ - 4 sin θ - 4 = 0
for 0 ≤ θ ≤ 360 ◦.
θ ≈ 221.8 ◦ or 318.2 ◦
3. Solve cos 2θ = 3 sin θ + 2 for
0 ≤ θ ≤ 2π.
___
⎧ 7π 3π 11π ⎫
⎬
θ=⎨
,
,
2
6 ⎭
⎩ 6
Also available on transparency
Lesson 14-6
1033
SECTION 14B
SECTION
14B
Trigonometric Identities
Spring 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.
Organizer
Objective: Assess students’
GI
ability to apply concepts and skills
in Lessons 14-3 through 14-6 in a
real-world format.
<D
@<I
1. The displacement in
inches of a mass attached
to a spring is modeled by
2π
π
y 1(t) = 3 sin ___
t + __
,
5
2
where t is the time in
seconds. What is the
amplitude of the motion?
What is the period? 3 in.; 5 s
Online Edition
\
Resources
Algebra II Assessments
Text Reference
1
Lesson 14-1
2–3
Lesson 14-6
4
Lesson 14-3
5–6
Lesson 14-6
)
Period
Amplitude
0
t
2. What is the initial
displacement when t = 0 s?
How long does it take until the displacement is 1.8 in.? 3 in.; 0.74 s
www.mathtekstoolkit.org
Problem
(
y
y (t) = 3 cos
2π
_
t
5
3. At what other times will the displacement be 1.8 in.?
5n ± 0.74 s where n is an integer
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
2π
y 2(t) = sin ___
t. Both masses are set in motion at t = 0 s. How long does it
5
take until both masses have the same displacement? about 0.99 s
6. The displacement of a third mass attached to a spring is modeled by
π
y 3(t) = cos __
t. The second and third masses are set in motion at
5
t = 0 s. How long does it take until both masses have the same
displacement? about 0.83 s
1034
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
π
t + __
(___
5
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
KEYWORD: MB7 Resources
1034
Chapter 14
5. What equation do you need to solve?
3 cos
2π
2π
___
t = sin ___
t
5
5
6. What equation do you need to solve?
2π
π
t = cos __
t
sin ___
5
5
Extension
At 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 ∈ SECTION 14B
SECTION
Quiz for Lessons 14-3 Through 14-6
14B
14-3 Fundamental Trigonometric Identities
Prove each trigonometric identity.
t 2 θ - 1 = 1 - 2 sin 2 θ
_
3. co
cot 2 θ + 1
2. sin(-θ) sec θ cot θ = -1
1. sin 2θ sec θ csc θ = tan θ
Organizer
Rewrite each expression in terms of a single trigonometric function.
csc 2 θ
1
sec θ
cot θ
4. cot θ sec θ csc θ
5. _
6. __
tan θ + cot θ
cos(-θ)
Objective: Assess students’
mastery of concepts and skills in
Lessons 14-3 through 14-6.
14-4 Sum and Difference Identities
Find the exact value of each expression.
6 - √
2
- √
8. sin(-75°) __
√6 - √
2
_
3
9. tan 75° 2 + √
4
4
1
12
Find each value if sin A = __
with 90° < A < 180° and if cos B = ___
with
4
13
270° < B < 360°.
10. sin(A + B)
15
12 + 5 √
__
11. cos(A + B)
52
15 + 5
-12 √
__
52
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.
{
GI
5π
7. cos _
12
15 - 5
-12 √
__
<D
@<I
Online Edition
Resources
52
Assessment Resources
Þ
Section 14B Quiz
Ý
(0, 0), (-2.87, 2.96), (-1.73, -1), (-0.37, -1.37)
{
ä
Ó
Ó
Test & Practice Generator
{
Ó
{
INTERVENTION
14-5 Double-Angle and Half-Angle Identities
4
Find each expression if cos θ = -__
and 180° < θ < 270°.
5
7
_
15. cos 2θ
25
θ
√
10
18. cos _ -_
2
24
_
14. sin 2θ
25
θ 3 √
10
17. sin _ _
2
10
Resources
24
_
16. tan 2θ
Ready to Go On?
Intervention and
Enrichment Worksheets
7
19.
10
20. Use half-angle identities to find the exact value of cos 22.5°.
14-6 Solving Trigonometric Equations
θ
tan _ -3
2
√
2 + √
2
2
_
Ready to Go On? CD-ROM
Ready to Go On? Online
7π
11π
21. Find all solutions of 1 + 2 sin θ = 0 where θ is in radians. _ + 2πn, _ + 2πn
6
6
Solve each equation for 0° ≤ θ < 360°.
23. 8 sin 2 θ - 2 sin θ = 1
22. cos 2θ + 2 cos θ = 3
Answers
30°, 150°, ≈ 194.5°, ≈ 345.5°
0°
Use trigonometric identities to solve each equation for 0 ≤ θ < 2π.
24. cos 2θ = 3 cos θ + 1
2π _
4π
_
,
3
1–3. See p. A54.
25. sin 2 θ + cos θ + 1 = 0 π
3
26. The average daily minimum temperature for Houston, Texas, can be modeled by
π(
T (x) = -15.85 cos __
x - 1) + 76.85, where T is the temperature in degrees
6
Fahrenheit, x is the time in months, and x = 0 is January 1. When is the temperature
65°F? 85°F? mid-March and mid-December; early June and late September
Ready to Go On?
NO
READY
Ready to Go On?
Intervention
YES
Diagnose and Prescribe
INTERVENE
TO
Worksheets
1035
ENRICH
GO ON? Intervention, Section 14B
CD-ROM
Lesson 14-3
14-3 Intervention
Activity 14-3
Lesson 14-4
14-4 Intervention
Activity 14-4
Lesson 14-5
14-5 Intervention
Activity 14-5
Lesson 14-6
14-6 Intervention
Activity 14-6
Online
Diagnose and
Prescribe Online
READY TO GO ON?
Enrichment, Section 14B
Worksheets
CD-ROM
Online
Ready to Go On?
1035
CHAPTER
Study Guide:
Review
14
Organizer
Vocabulary
periodic function . . . . . . . . . . . . . . . . . . . . . . . . . 990
cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 990
phase shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 993
organize and review key concepts
and skills in Chapter 14.
frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 992
rotation matrix . . . . . . . . . . . . . . . . . . . . . . . . . . 1016
GI
amplitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 991
Objective: Help students
<D
@<I
Online Edition
Multilingual Glossary
period . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 990
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
?
.
−−−−−−
?
gives the length of a complete cycle for a periodic function.
−−−−−−
4. A horizontal translation of a periodic function is known as a(n)
?
.
−−−−−−
3. The
Resources
PuzzleView
Test & Practice Generator
14-1 Graphs of Sine and Cosine (pp. 990–997)
Multilingual Glossary Online
■
KEYWORD: MB7 Glossary
π , the period is _
2π = _
2π = 4.
Because b = _
__π
2
⎪b⎥
⎪2⎥
1. cycle
Step 2 Graph.
2. frequency
3. period
_
Óû û
x
?
û
v
(
(
3π
13. h(x) = sin x - _
2
)
The first x-intercept
π
. Thus,
occurs at __
4
the intercepts
Óû
π
occur at __
+ nπ,
4
where n is an
integer.
x
(
û
_1 ; period: 2π
1036
_1 ; period: π
9. amplitude:
2
x
ä
v
y
Ý
Óû
17. What is the maximum and at what time does
it occur?
8. amplitude: 2; period: 2
û
?û
y
Chapter 14
û
û
y
3π
_
+ πn;
4
π
phase shift: _ left
14. x-intercepts: πn;
3π
left
phase shift:
2
_
4
x
y
y
x
û
1036
x
û
12. x-intercepts:
?
û
_
x
û
π
10. amplitude: ; period: 2
2
_
x
_
π
13. x-intercepts:
+ πn; phase
2
3π
right
shift:
2
2
phase shift: π left
x
û
_π + nπ;
11. x-intercepts:
û
)
16. What is the period of the function?
y
y
(
3π
14. j(x) = cos x + _
2
Chapter 14 Trigonometric Graphs and Identities
3
y
Þ
}
)
)
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 CO 2 per square meter per
π(
second). For a bean plant, C(t) = 1.2 sin __
t - 6) + 7,
12
where t is time in hours starting at midnight.
15. Graph the function for two complete cycles.
5π
5π
-___
indicates a shift ___
units right.
4
4
y
Using f (x) = sin x as a guide, graph
5π
g(x) = sin x - ___
. Identify the x-intercepts
4
and phase shift.
The amplitude is 1. The period is 2π.
6. amplitude: 1; period: 4π
Ý
Óû
û
ä
Ó
■
Using f (x) = sin x or f (x) = cos x as a guide,
graph each function. Identify the x-intercepts
and phase shift.
π
12. g(x) = sin x + _
11. f (x) = cos(x + π)
4
Þ
Ó
y
7. amplitude:
}
The curve is reflected
over the x-axis.
4. phase shift
2π
5. amplitude: 1; period:
3
û
Using f (x) = sin x or f (x) = cos x as a guide, graph
each function. Identify the amplitude and period.
1x
5. f (x) = cos 3x
6. g(x) = cos _
2
1 sin x
7. h(x) = -_
8. j(x) = 2 sin πx
3
π sin πx
1 cos 2x
10. g(x) = _
9. f (x) = _
2
2
Because a = -2, amplitude is ⎪a⎥ = ⎪-2⎥ = 2.
Answers
û
Using f (x) = cos x as a guide, graph
π
g(x) = -2 cos __
x. Identify the amplitude
2
and period.
Step 1 Identify the period and amplitude.
Lesson Tutorial Videos
CD-ROM
?
û
EXERCISES
EXAMPLES
û
?
û
?
û
x
Answers
14-2 Graphs of Other Trigonometric Functions (pp. 998–1003)
EXAMPLE
■
Using f (x) = cot x as a guide, graph
g(x) = cot π2 x. Identify the period,
x-intercepts, and asymptotes.
__
Step 1 Identify the period.
π , the period is _
π =_
π = 2.
Because b = _
π
__
2
⎪b⎥
⎪2⎥
Step 2 Identify the x-intercepts.
Ó
EXERCISES
Using f (x) = tan x or f (x) = cot x as a guide, graph
each function. Identify the period, x-intercepts, and
asymptotes.
1 tan x
18. f (x) = _
19. g(x) = tan πx
4
1 πx
20. h(x) = tan _
21. g(x) = 5 cot x
2
Þ
ä
17. 8.2; noon
18. period: π; x-intercepts: πn;
π
+ πn
asymptotes:
2
_
23. j(x) = cot πx
26. h(x) = 4 csc x
27. j(x) = 0.2 sec x
28. h(x) = sec(-x)
29. j(x) = -2 csc x
16. 24 h
y
x
û
û
19. period: 1; x-intercepts: n;
1
+n
asymptotes:
2
_
Ý
Î
x
Using f (x) = cos x or f (x) = sin x as a guide, graph
each function. Identify the period and asymptotes.
24. f (x) = 2 sec x
25. g(x) = csc 2x
Step 3 Identify the asymptotes.
πn = _
πn = 2n.
The asymptotes occur at x = _
π
⎪b⎥ ⎪ __
2⎥
Step 4 Graph.
y
22. j(x) = -0.5 cot x
The first x-intercept occurs at 1. Thus, the
x-intercepts occur at 1 + 2n, where n is an
integer.
15.
Î
Ó
y
x
14-3 Fundamental Trigonometric Identities (pp. 1008–1013)
tan θ
Prove _______
= sec θ csc θ.
sin θ
)
(____
sin 2(-θ)
31. _ = sin θ cos θ
tan θ
32. (sec θ + 1)(sec θ - 1) = tan 2 θ
Modify the left side. Apply
the ratio and Pythagorean
identities.
( )( )
1
1 =
_
(_
cos θ )( sin θ )
sin θ _
1
_
=
cos θ sin 2 θ
_
π
21. period: π; x-intercepts:
+ πn;
2
asymptotes: πn
36. sin θ tan θ = tan θ - sin θ cos θ
2
tan θ = sec θ csc θ
37. _
1 - cos 2 θ
Rewrite _________ in terms of a single
trigonometric function, and simplify.
cot θ + tan θ
csc θ
(cot θ + tan θ)sin θ
Given.
(
Ratio identities
cos 2 θ + sin 2 θ
__
cos θ
1 = sec θ
_
cos θ
y
x
Rewrite each expression in terms of a single
trigonometric function, and simplify.
sec θ sin θ
38. cot θ sec θ
39. _
cot θ
tan(-θ)
cos
θ cot θ
_
_
41.
40.
cot θ
csc 2 θ - 1
Add fractions and
simplify.
Pythagorean and
reciprocal identities
û
_
π
24. period: 2π; asymptotes:
2
+ πn
_
y
26. period: 2π; asymptotes: π
+ πn
_
y
x
?
û
û
û
_π n
2
y
_
y
x
û
x
û
y
û
1
23. period: 1; x-intercepts: _ + n;
2
asymptotes: n
29. period: 2π; asymptotes: π
+ πn
x
û
π
27. period: 2π; asymptotes:
2
+ πn
?û
y
x
?
û
y
25. period: π; asymptotes:
?
û
x
û
1037
π
28. period: 2π; asymptotes:
2
+ πn
y
û
û
π
22. period: π; x-intercepts:
+ πn;
2
asymptotes: πn
Study Guide: Review
35. tan θ + cot θ = sec θ csc θ
Reciprocal identities
)
x
34. (tan θ + cot θ)2 = sec 2 θ + csc 2 θ
Simplify.
sin θ sin θ
cos θ + _
_
cos θ
sin θ
y
33. cos θ sec θ + cos 2 θ csc 2 θ = csc 2 θ
Multiply by the reciprocal.
sec θ csc θ
20. period: 2; x-intercepts: 2n;
asymptotes: 1 + 2n
Prove each trigonometric identity.
30. sec θ sin θ cot θ = 1
1 - cos 2θ
cos θ
_
=
(sin 2 θ)
■
EXERCISES
EXAMPLES
■
x
x
û
û
Study Guide: Review
1037
Answers
14-4 Sum and Difference Identities (pp. 1014–1019)
30.
cos θ
1
sin θ(_)
(_
cos θ )
sin θ
cos θ sin θ
= (_)(_) = 1
cos θ sin θ
__
1
■ Find sin(A + B) if cos A = - with
3
180° < A < 270° and if sin B = 45 with
(-sin θ)(-sin θ)
sin (-θ) __
_
=
tan θ
Step 1 Find sin A and cos B by using the
Pythagorean Theorem with reference triangles.
180° < A < 270°
90° < B < 180°
1
4
cos A = -_
sin B = _
5
3
sin θ
____
cos θ
(_ )
cos θ
= (sin θ)(sin θ)
sin θ
= sin θ cos θ
ÝÊ£
32. (sec θ + 1)(sec θ - 1)
Þ
33. cos θ sec θ + cos 2 θ csc 2 θ
1
= 1 + cos 2 θ
sin 2 θ
= 1 + cot 2 θ
34. (tan θ + cot θ)2
= tan 2θ + 2 tan θ cot θ + cot 2 θ
= tan θ + 2 + cot θ
2
= (tan 2 θ + 1) + (1 + cot 2 θ)
■
= sec θ + csc θ
2
_ _
__
_
sin θ
cos θ
35. tan θ + cot θ =
+
cos θ
sin θ
sin 2 θ + cos 2 θ
=
sin θ cos θ
1
=
sin θ cos θ
36. sin 2 θ tan θ = (1 - cos 2 θ) tan θ
sin θ
(_
cos θ )
(cos θ )
tan θ
37. _ = _
sin θ
____
(sin θ)
£ ä
£
Ó
2
38. csc θ
39. tan 2 θ
40. -tan 2 θ
- √6
- √2
42.
4
+ √2
√6
44.
4
16
46. 65
56
48.
33
41. sin θ
- √
6
- √2
43.
4
__
3
45. 2 - √
63
_
65
16
49. _
47. -
Chapter 14
63
1038
Chapter 14 Trigonometric Graphs and Identities
50. -
56
_
51. -
33
_
52.
65
65
36 - 5 √
7
_
52
__
7
-15 - 12 √
53.
52
54.
+ 36
5 √7
__
55.
- 36
5 √7
__
15 - 12 √
7
15 + 12 √
7
3
with
Find each value if tan A = __
4 5
0° < A < 90° and if tan B = -___
with
12
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 = ___
with
4 5
0° < A < 90° and if cos B = -___
with
13
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 ABCD
with A(0, 0), B(3, 0), C(4, 2), and D(1, 2)
after each rotation about the origin.
58. 30° rotation
59. 45° rotation
60. 60° rotation
61. 90° rotation
Ý
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).
2
= sec θ csc θ
1038
£ ­ä]Ê£®
⎡ -1.73 -1.23 -0.87 ⎤
≈⎢
1 1.87
0.5 ⎦
⎣
= tan θ - sin θ cos θ
sin θ _
1
_
(cos
θ )(sin θ )
1
1
= (_)(_
cos θ sin θ )
­£]ÊÓ®
Step 2 Find the matrix product.
⎡ cos 60° -sin 60° ⎤ ⎡ 0 1 0 ⎤
R 60° × S = ⎢
⎢
⎣ sin 60° cos 60° ⎦ ⎣ 2 2 1 ⎦
= tan θ - cos 2 tan θ
=
Þ
­ä]ÊÓ®
Step 1 Write matrices for a 60° rotation and for
the points in the figure.
⎡ cos 60° -sin 60° ⎤
R 60° = ⎢
Rotation matrix
⎣ sin 60° cos 60° ⎦
⎡0 1 0⎤
S=⎢
Matrix of points
⎣2 2 1⎦
= sec θ csc θ
= tan θ - cos 2 θ
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 60° rotation
about the origin.
Find the exact value of each expression.
19π
42. sin _
43. cos 165°
12
π
44. cos 15°
45. tan _
12
√7
( )( ) ( )( )
= csc 2 θ
2
Ý
- √8
-3
, sin A = _ x = -3, cos B = _
y = - √8
5
3
Step 2 Use the angle-sum identity.
sin(A + B) = sin A cos B + cos A sin B
-3
- √8
1 _
4
+ -_
= _ _
5
3
3 5
-4
3 √8
=_
15
(_)
__
_
_
_
ÀÊÎ
= tan 2 θ
1 - cos 2 θ
ÀÊÊx
ÞÊÊ{
= sec 2 θ - 1
2
__
90° < B < 180°.
2
31.
EXERCISES
EXAMPLES
sec θ sin θ cot θ =
Find the coordinates, to the nearest
hundredth, of the vertices of figure ABCD
with A(0, 0), B(5, 2), C(0, 4), and D(-5, 2)
after each rotation about the origin.
62. 120° rotation
63. 180° rotation
64. 240° rotation
65. 270° rotation
Answers
14-5 Double-Angle and Half-Angle Identities (pp. 1020–1026)
1
Find each expression if sin θ = __
and
4
270° < θ < 360°.
■
sin 2θ
_
15
1 in QIV, cos θ = - √
.
For sin θ = _
4
4
sin 2θ = 2 sin θ cos θ
Identity for sin 2θ
√
√15
15
1
=2
= -_
8
4
4
( _ )(
■
-36 - 5 √7
__
52
7
-15 + 12 √
57. __
56.
EXERCISES
EXAMPLES
_)
Substitute.
θ
cos _
2
θ =±
cos _
2
1 + cos θ
√_
2
1 + (-____)
= - __
√
15
4
θ
Identity for cos _
2
Negative for
θ in QII
cos _
2
4
Find each expression if tan θ = __
and
3
0° < θ < 90°.
66. sin 2θ
67. cos 2θ
θ
θ
_
68. tan
69. sin _
2
2
⎡0
58. ≈ ⎢
⎣0
⎡ 0 2.12
59. ≈ ⎢
⎣ 0 2.12
3
Find each expression if cos θ = __
and
4
3π
___
<
θ
<
2π.
2
70. tan 2θ
71. cos 2θ
θ
θ
72. cos _
73. sin _
2
2
Use half-angle identities to find the exact
value of each trigonometric expression.
π
74. sin _
75. cos 75°
12
14-6 Solving Trigonometric Equations (pp. 1027–1033)
■
EXERCISES
Find all of the solutions of
3 = cos θ.
3 cos θ - √
3 cos θ - √
3 = cos θ
3
3 cos θ - cos θ = √
Subtract tan θ.
2 cos θ = √
3 Combine like terms.
√
3
cos θ = _
Divide by 2.
2
Apply the inverse
√
3
θ = cos -1 _
cosine.
2
θ = 30° or 330°
Find θ for
( )
θ = 30° + 360°n
0° ≤ θ < 360°.
or 330° + 360°n
■
Solve 6 sin 2 θ + 5 sin θ = -1 for 0° ≤ θ < 360°.
6 sin 2 θ + 5 sin θ + 1 = 0 Set equal to 0.
(2 sin θ + 1)(3 sin θ + 1) = 0
sin θ = -1 or sin θ = 3
θ = 210°, 330°
or ≈ 199.5°, 340.5°
Factor.
Zero Product
Property
sin θ = 3 has no
solution since
-1 ≤ sin θ ≤ 1.
Find all of the solutions of each equation.
cos θ + 1 = 0
77. cos θ = 2 + 3 cos θ
76. √2
1
2
78. tan θ + tan θ = 0
79. sin 2 θ - cos 2 θ = _
2
83. 2 sin 2 θ - sin θ = 3
-1.23 ⎤
1.87 ⎦
-2 ⎤
1⎦
⎡0
63. ≈ ⎢
⎣0
0
-5
-2 -4
⎡0
64. ≈ ⎢
⎣0
-0.77
-5.33
⎡0
65. ≈ ⎢
⎣0
2
-5
66.
24
_
68.
_1
4
0
0.77 ⎤
-5.33 ⎦
5⎤
-2 ⎦
3.46
-2
4.23 ⎤
3.33 ⎦
2⎤
5⎦
_
67. - 7
25
√
5
69.
5
25
_
2
_
√
14
72. 4
2 - √3
74.
2
_
◦
71.
_1
73.
√
2
_
75.
◦
8
4
2- √
3
_
2
76. 135 + 360 n, 225 + 360 ◦n
Use trigonometric identities to solve each equation
for 0 ≤ θ < 2π.
84. cos 2θ = cos θ
85. sin 2θ + cos θ = 0
◦
77. 180 ◦ + 360 ◦n
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?
Study Guide: Review
⎡0
1.5 0.27
60. ≈ ⎢
⎣ 0 2.60 4.46
70. -3 √7
Solve each equation for 0 ≤ θ < 2π.
80. 2 cos 2 θ - 3 cos θ = 2 81. cos 2 θ + 5 cos θ - 6 = 0
82. sin 2 θ - 1 = 0
-0.71 ⎤
2.12 ⎦
⎡ 0 -4.23 -3.46
62. ≈ ⎢
⎣0
3.33
-2
)( )
EXAMPLES
-0.13 ⎤
2.23 ⎦
1.41
4.24
⎡ 0 0 -2
61. ≈ ⎢
⎣0 3
4
2
√
4 - √
15
15 1
4 - √
=- _ _
= -_
4
2
√8
(
52
2.60 2.46
1.50 3.73
1039
78. 0 ◦ + 180 ◦n, 135 ◦ + 180 ◦n
79. 60 ◦ + 180 ◦n, 120 ◦ + 180 ◦n
80.
2π _
4π
_
,
3
3
81. 0
π 3π
82. ,
2 2
3π
83.
2
2π 4π
84. 0,
,
3
3
π 7π 3π 11π
85. ,
,
,
2 6
2
6
86a. 900 min; late June
__
_
__
____
b. 540 min; late December
Study Guide: Review
1039
CHAPTER
14
Organizer
1. Using f (x) = cos x as a guide, graph g (x) = __12 cos 2x. Identify the amplitude and period.
(
Objective: Assess students’
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°. max.: 250 N·m at 90°; min.: 0 N·m at 0°
GI
mastery of concepts and skills
in Chapter 14.
<D
@<I
)
π
2. Using f (x) = sin x as a guide, graph g (x) = sin x + __
. Identify the x-intercepts
3
and phase shift.
Online Edition
4. Using f (x) = tan x as a guide, graph g(x) = 2 tan πx. Identify the period, x-intercepts,
and asymptotes.
Resources
5. Using f (x) = cot x as a guide, graph g(x) = cot 4x. Identify the period, x-intercepts,
and asymptotes.
1 csc x. Identify the period and
6. Using f (x) = sin x as a guide, graph g(x) = _
4
asymptotes.
Assessment Resources
Chapter 14 Tests
• Free Response
(Levels A, B, C)
7. Prove the trigonometric identity cot θ = cos 2 θ sec θ csc θ. cos 2 θ sec θ csc θ = cos 2 θ
• Multiple Choice
(Levels A, B, C)
Rewrite each expression in terms of a single trigonometric function.
8. (sec θ + 1)(sec θ - 1) tan 2 θ
• Performance Assessment
Test & Practice Generator
?
y
?û
û
63
_
11. cos(A - B) -
65
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.
Answers
1.
3
12
Find each value if tan A = __
with 0° < A < 90° and if sin B = -___
with
4
13
180° < B < 270°.
10. sin(A + B) -
x
{
cos θ
_
= cot θ
sin θ
56
_
65
Þ
ä
Ó
Ý
Ó
{
Ó
amplitude:
{
(-0.50, 0.87), (1.23, 1.87),
(1.10, 4.10), (-2.37, 2.10)
=
sin(-θ)
9. _ -tan θ
cos(-θ)
1
1
_
(_
cos θ )( sin θ )
{
_1 ; period: π
2
12
Find each expression if tan θ = -___
and 90° < θ < 180°.
5
120
_
13. sin 2θ -
14. cos 2θ -
169
119
_
θ
15. cos _
2
169
3π .
16. Use half-angle identities to find the exact value of sin _
8
3 = 0.
17. Find all of the solutions of tan θ + √
13
2 √
_
√
2 + √
2
_
2
17. 120° + 180°n,
18. Solve 2 sin 2 θ = sin θ for 0° ≤ θ < 360°. 0°, 30°, 150°, 180°
19. Use trigonometric identities to solve 2 cos 2 θ + 3 sin θ = 0 for 0 ≤ θ < 2π.
2π
_
+ πn
3
11π
7π _
_
,
6
6
period:
_π ;
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?
_
13
_
≈ 0.0021 + n 1 s, ≈ 0.0063 + n 1 s
60
60
1040
Chapter 14 Trigonometric Graphs and Identities
4.
Answers
y
2.
y
6.
2π
_
+ πn;
3
π
phase shift: _ left
period: 1; intercepts: n;
1
asymptotes:
+n
2
5.
x
û
y
3
Chapter 14
y
_
x-intercepts:
1040
4
û
KEYWORD: MB7 Resources
x
û
intercepts:
x
4
_π + _π n;
8
4
π
asymptotes: _ n
?
û
?û
x
period: 2π;
asymptotes: πn
û
CHAPTER
14
Organizer
FOCUS ON SAT MATHEMATICS SUBJECT TESTS
To 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.
È
(C) 0 ≤ f (x) ≤ 3
(D) -3 ≤ f (x) ≤ 3
2. If 2 sin θ + 5 sin θ = 3, what could the value
of θ be?
π
(A) _
6
π
_
(B)
3
2π
_
(C)
3
7π
_
(D)
6
11π
_
(E)
6
3. If sec θ = 4, what is tan 2 θ?
1
(A) _
16
(B) 3
(C) 5
GI
<D
@<I
College Entrance Exam
Practice
Not to scale
£ä
n
Online Edition
Resources
(B) -3 < f (x) < 3
2
college entrance exams such as the
SAT Mathematics Subject Tests.
4. Given the figure, what is the value of cos(A - B)?
(A) -1 ≤ f (x) ≤ 1
(E) -∞ < f (x) < ∞
Objective: Provide practice for
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.
Questions on the SAT Mathematics
Subject Tests Levels 1 and 2 represent the following math content
areas:
(A) 0
7
(B) _
25
24
(C) _
25
Level
1
2
Number and
Operations
10—14%
10—14%
(D) 1
28
(E) _
25
Algebra and
Functions
38—42%
48—52%
Geometry and
Measurement
38—42%
28—32%
5. If sin θ = _79_, what is cos 2θ?
8 √2
(A) -_
9
17
(B) -_
81
17
_
(C)
81
56 √
2
(D) _
81
8 √2
(E) _
9
Plane Euclidean
18—22%
0%
Coordinate
8—12%
10—14%
Threedimensional
4—6%
4—6%
Trigonometry
6—8%
12—16%
6—10%
6—10%
Data Analysis,
Statistics, and
Probability
(D) 15
Items on this page focus on:
(E) 17
• Algebra and Functions
• Trigonometry
Text References:
College Entrance Exam Practice
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 reference angle but forgot that the sine function is negative in the third and fourth
quadrants.
1041
Item
1
2
3
4
5
Lesson
1
6
3
4
5
3. Students may choose answer E because
they have added 1 to sec 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.
College Entrance Exam Practice
1041
CHAPTER
14
Organizer
Multiple Choice: Choose Answer Combinations
Objective: Provide opportunities
You 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.
GI
to learn and practice common testtaking strategies.
<D
@<I
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 statement and answer choices thoroughly. Encourage students to investigate
each possibility.
Which exact solution makes the equation 2 cos 2 θ - 3 cos θ = 2 true?
I. θ = 2°
II. θ = 120°
III. θ = 240°
Look at each statement separately, and
determine if it is true or false.
I only
II only
II and III
I, II, and III
As you consider each statement, mark it true or false.
As students practice this strategy,
remind them to choose the best,
most complete answer, not just the
first correct answer that they discover. Review the elimination strategy, and guide students to use logic
to eliminate any obviously incorrect
answer choice. If students are struggling with organization, show them
how a table can be used to organize
their process.
Consider statement I: Substitute 2° for θ in the equation.
2 cos 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 cos 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 cos 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.
1042
1042
Chapter 14
Statement
True/False
I
II
III
False
True
True
Chapter 14 Trigonometric Graphs and Identities
As you eliminate a statement, cross out the
corresponding answer choice(s).
4. How do you determine the period of a
trigonometric function?
Answers
Possible answers:
5. How do you determine the amplitude of a
trigonometric function?
Read each test item and answer the questions
that follow.
1. Some of the identities that
involve the tangent function are
sin θ
,
tan θ =
cos θ
sec 2 θ = tan 2 θ + 1, and
1
.
cot θ =
tan θ
_
6. Using your response to Problems 4 and 5,
which of the three statements are true?
Explain.
_
Item A
Which expression is equivalent to tan 2 θ?
1
III.
I. sec 2 θ - 1
csc 2θ - 1
1 - cos 2 θ
IV.
II. sec 2 θ + 1
1 - sin 2 θ
_
_
I and II
I and III
II and III
I, III, and IV
Which identities do you need to use to prove
that tan θ csc θ = sec θ?
I. tan θ =
sin θ
_
tan 2 θ = sec 2 θ - 1. Statement III
is true because
1
1
=
= tan 2θ.
csc 2 θ - 1
cot 2 θ
Statement IV is true because
cos θ
II. sec 2 θ = tan 2 θ + 1
III. csc θ =
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?
2. Statement I is true because it is a
fundamental identity. Statement
II is false because
Item C
_ _
1
_
sin θ
I only
I and II
II only
I and III
1 - cos θ
sin θ
_
= _ = tan θ.
2
For the graph of f (x) = 3 sin x + 2, which of the
statements are true?
Þ
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.
Ý
û
ä
10. Which combination of statements is correct?
How do you know?
f(x) = a sin bx is equal to ⎪a⎥.
For the graph of the function f (x) = sec 4x,
which are equations of some of the asymptotes?
Ó
û
2π
I. The function has a period of _.
3
II. The function has an amplitude of 3.
_π
8
π
II. x = _
I. x =
II only
III only
II and III
_
I only
I, II, and III
II and III
I and III
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
sin θ
for tan θ.
can substitute ____
cos θ
Statement II can be eliminated
because there are no squared
terms.
2
3π
III. x = 4
III. The function has a period of 2π.
I only
4. The period of a trigonometric
function in the form
2π
.
f(x) = a sin bx is equal to
⎪b⎥
5. The amplitude of a trigonometric
function in the form
_
Item D
{
2
1 - sin θ
cos 2 θ
3. No; check if Statements I and
IV are true before selecting an
answer choice.
9. Is statement III true or false? Explain.
Item B
2
2
11. Create a table, and determine whether each
statement is true or false.
8. Statement II is false because you
cannot use an expression for
tan 2 θ in this identity. No; It
is necessary to determine if
Statement III is true or false first.
12. Using your table, which choice is the most
accurate?
Test Tackler
Answers to Test Items
A. D
B. J
C. D
D. F
1043
Possible answers:
9. Statement III is true because you can
1
for csc θ.
substitute
sin 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.
12. F
Statement
True/False
I
True
II
False
III
False
KEYWORD: MB7 Resources
Test Tackler
1043
CHAPTER
14
KEYWORD: MB7 TestPrep
Organizer
CUMULATIVE ASSESSMENT, CHAPTERS 1–14
Objective: Provide review and
GI
practice for Chapters 1–14 and
standardized tests.
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@<I
Multiple Choice
1. What is the exact value of tan 15°?
5. What is the value of f(x) = 3x3 + 4x2 + 7x + 10
for x = -2?
-44
√
-√2
6
_
4
Online Edition
-12
0
√
+√2
6
_
4
2 +√3
Resources
2 -√
3
Assessment Resources
36
6. Which graph shows an inverse variation function
for which y = 2 when x = -1?
2. Where do the asymptotes occur in the given
8
equation?
Chapter 14 Cumulative Test
8
y
x
x
1 cot 2x
y=_
3
2πn
KEYWORD: MB7 Testprep
y
4
-8 -4 0
4
-8 -4 0
-4
8
4
3πn
2
πn
_
3
-4
y
0
4
y
x
2
4
-4
3. What is the period of the given equation?
8
-8
-8
πn
_
2
4
x
-4 -2 0
-2
4
-4
1x
y = 5 cos _
3
2π
_
5
7. What is the exact value of cos157.5° using
half-angle identities?
√
2 -√2
-_
2
5
_
3
2π
_
3
√
2 -√2
_
6π
√
2 +√2
-_
2
2
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
_
12
7
_
12
5
_
7
7
_
5
1044
cubic
exponential
quadratic
rational
9. H
10. A
2. G
11. 4
3. D
12. 0.894
4. F
13. 0
5. B
14. 245
6. J
7. C
8. F
Chapter 14
8. What type of function is f(x) = -2x 3 - x + 10?
1. D
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2
Chapter 14 Trigonometric Graphs and Identities
Answers
KEYWORD: MB7 Resources
√
2 +√2
_
2/8/10 11:33:51 AM
9. Which is a solution of 2 cos θ = 2 sin θ for
π ≤ θ ≤ 3π?
π
_
4
Short Response
Short-Response Rubric
15. The chart below shows the names of the students
Items 15–16
on the academic bowl team.
π
Robin
Drew
Jim
5π
_
4
Greg
Sarah
Mindy
Ashley
Tina
Justin
3π
David
Amy
Kevin
10. Which is the equation of a circle with center (3, 2)
2 Points = The student’s answer is
an accurate and complete execution of the task or tasks.
1 Point = The student’s answer contains attributes of an appropriate
response but is flawed.
a. Only 2 students can be chosen for the final
and radius 5?
0 Points = The student’s answer
contains no attributes of an appropriate response.
academic bowl. How many different ways can
the students be selected?
25 = (x - 3) 2 + (y - 2) 2
b. Explain why you solved the problem the way
5 = (x - 3) 2 + (y - 2) 2
that you did.
25 = (x + 3) 2 + (y + 2) 2
5 = (x + 3) 2 + (y + 2) 2
4, 12, 36, 108, 324, . . .
Extended-Response
Rubric
Gridded Response
a. Write the explicit rule for the nth term.
Item 17
11. What is the value of x?
b. Find the 10th term.
4 Points = The student correctly
finds the value of each measure
in parts a–f and rounds to the
nearest tenth correctly.
16. Given the sequence:
2x - 7 + 4 = 9
5 √
12. What is the value of cos θ? Round to the nearest
thousandth.
Extended Response
17. The chart below shows the grades in
3 Points = The student correctly
finds the value of at least 5 of
the values in parts a–f or makes
errors in rounding.
Mr. Bradshaw’s class.
8.05
3.6
θ
7.2
90
85
72
86
94
96
85
95
94
68
71
85
93
98
84
83
80
89
2 Points = The student correctly
finds the value of at least 3 of the
values in parts a–f.
Round each answer to the nearest tenth.
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.
13. What is the y-value of the solution of the
following system of nonlinear equations?
a. Find the mean.
b.
c.
d.
e.
f.
1 Point = The student correctly finds
the value of 1 or 2 of the values in
parts a–f.
Find the median.
Find the mode.
Find the variance.
0 Points = The student does not
answer correctly and does not
attempt all parts of the problem.
Find the standard deviation.
Find the range.
⎧
1 y2
x-4=_
4
⎨
(x + 1) 2 _
y2
_
+
=1
⎩ 25
36
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.
14
14. Find the sum of the arithmetric series ∑(3k - 5).
k=1
16a. 4(3n - 1)
b. 78,732
17a. 86
b. 85.5
Cumulative Assessment Chapters 1–14
1045
c. 85
d. 73.6
e. 8.6
f. 30
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Cumulative Assessment
1045
Organizer
OHIO
Objective: Choose appropriate
Sandusky Bay
GI
problem-solving strategies,
and use them with skills from
Chapters 13 and 14 to solve
real-world problems.
<D
@<I
Cleveland
Online Edition
The Rock and Roll
Hall of Fame
The Rock and Roll
Hall of Fame
Reading
Strategies
ENGLISH
LANGUAGE
LEARNERS
Make 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
The 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.)
£äxÊvÌ
ô
≈ 35°
ÓÈäÊvÌ
2. What is the area of the triangular floor space enclosed by the glass
tetrahedron? about 9964 ft 2
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? ≈ 21.9°
Problem-Solving Focus
Encourage students to use the four-step
probelm-solving process for the problems.
Focus on the first step: (1) Understand the
Problem.
A211NLS_c14psl_1046_1047.indd 1046
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.
KEYWORD: MB7 Resources
1046
Chapter 14
8/10/09 6:53:07 PM
Marblehead
Lighthouse
Reading
Strategies
ENGLISH
LANGUAGE
LEARNERS
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.
Marblehead Lighthouse
Since 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.
L
s
A
Choose one or more strategies to solve each problem.
Problem-Solving Focus
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. about 9.9 mi
Encourage students to use the
four-step problem-solving process
for the problems. Focus on the
second step: (2) Make a Plan.
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
π
x,
the function f (x) = sin __
5
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
6
flash?
MI
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.
E
3. Today the flashing light of
Marblehead Lighthouse can
π
be modeled by g(x) = sin __
x.
3
How many seconds are there
between each flash? Does
the light flash more or less
frequently than in 1897? 6 s; more frequently
Real-World Connections
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Real-World Connections
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