chapter 14 resource masters - ktl math classes

Chapter 14Resource Masters

Consumable WorkbooksMany of the worksheets contained in the Chapter Resource Masters bookletsare available as consumable workbooks.

Study Guide and Intervention Workbook 0-07-828029-XSkills Practice Workbook 0-07-828023-0Practice Workbook 0-07-828024-9

ANSWERS FOR WORKBOOKS The answers for Chapter 14 of these workbookscan be found in the back of this Chapter Resource Masters booklet.

Copyright © by The McGraw-Hill Companies, Inc. All rights reserved.Printed in the United States of America. Permission is granted to reproduce the material contained herein on the condition that such material be reproduced only for classroom use; be provided to students, teacher, and families without charge; and be used solely in conjunction with Glencoe’s Algebra 2. Any other reproduction, for use or sale, is prohibited without prior written permission of the publisher.

Send all inquiries to:The McGraw-Hill Companies8787 Orion PlaceColumbus, OH 43240-4027

ISBN: 0-07-828017-6 Algebra 2Chapter 14 Resource Masters

1 2 3 4 5 6 7 8 9 10 066 11 10 09 08 07 06 05 04 03 02

© Glencoe/McGraw-Hill iii Glencoe Algebra 2

Contents

Vocabulary Builder . . . . . . . . . . . . . . . . vii

Lesson 14-1Study Guide and Intervention . . . . . . . . 837–838Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 839Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 840Reading to Learn Mathematics . . . . . . . . . . 841Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 842







Chapter 14 AssessmentChapter 14 Test, Form 1 . . . . . . . . . . . 879–880Chapter 14 Test, Form 2A . . . . . . . . . . 881–882Chapter 14 Test, Form 2B . . . . . . . . . . 883–884Chapter 14 Test, Form 2C . . . . . . . . . . 885–886Chapter 14 Test, Form 2D . . . . . . . . . . 887–888Chapter 14 Test, Form 3 . . . . . . . . . . . 889–890Chapter 14 Open-Ended Assessment . . . . . 891Chapter 14 Vocabulary Test/Review . . . . . . 892Chapter 14 Quizzes 1 & 2 . . . . . . . . . . . . . . 893Chapter 14 Quizzes 3 & 4 . . . . . . . . . . . . . . 894Chapter 14 Mid-Chapter Test . . . . . . . . . . . . 895Chapter 14 Cumulative Review . . . . . . . . . . 896Chapter 14 Standardized Test Practice . 897–898Unit 5 Test/Review (Ch. 13–14) . . . . . . 899–900Second Semester Test (Ch. 8–14) . . . . 901–902Final Test (Ch. 1–14) . . . . . . . . . . . . . . 903–904

Standardized Test Practice Student Recording Sheet . . . . . . . . . . . . . . A1

ANSWERS . . . . . . . . . . . . . . . . . . . . . . A2–A38

© Glencoe/McGraw-Hill iv Glencoe Algebra 2

Teacher’s Guide to Using theChapter 14 Resource Masters

The Fast File Chapter Resource system allows you to conveniently file the resourcesyou use most often. The Chapter 14 Resource Masters includes the core materialsneeded for Chapter 14. These materials include worksheets, extensions, andassessment options. The answers for these pages appear at the back of this booklet.

All of the materials found in this booklet are included for viewing and printing in theAlgebra 2 TeacherWorks CD-ROM.

Vocabulary Builder Pages vii–viiiinclude a student study tool that presentsup to twenty of the key vocabulary termsfrom the chapter. Students are to recorddefinitions and/or examples for each term.You may suggest that students highlight orstar the terms with which they are notfamiliar.

WHEN TO USE Give these pages tostudents before beginning Lesson 14-1.Encourage them to add these pages to theirAlgebra 2 Study Notebook. Remind them to add definitions and examples as theycomplete each lesson.

Study Guide and InterventionEach lesson in Algebra 2 addresses twoobjectives. There is one Study Guide andIntervention master for each objective.

WHEN TO USE Use these masters asreteaching activities for students who needadditional reinforcement. These pages canalso be used in conjunction with the StudentEdition as an instructional tool for studentswho have been absent.

Skills Practice There is one master foreach lesson. These provide computationalpractice at a basic level.

WHEN TO USE These masters can be used with students who have weakermathematics backgrounds or needadditional reinforcement.

Practice There is one master for eachlesson. These problems more closely followthe structure of the Practice and Applysection of the Student Edition exercises.These exercises are of average difficulty.

WHEN TO USE These provide additionalpractice options or may be used ashomework for second day teaching of thelesson.

Reading to Learn MathematicsOne master is included for each lesson. Thefirst section of each master asks questionsabout the opening paragraph of the lessonin the Student Edition. Additionalquestions ask students to interpret thecontext of and relationships among termsin the lesson. Finally, students are asked tosummarize what they have learned usingvarious representation techniques.

WHEN TO USE This master can be usedas a study tool when presenting the lessonor as an informal reading assessment afterpresenting the lesson. It is also a helpfultool for ELL (English Language Learner)students.

Enrichment There is one extensionmaster for each lesson. These activities mayextend the concepts in the lesson, offer anhistorical or multicultural look at theconcepts, or widen students’ perspectives onthe mathematics they are learning. Theseare not written exclusively for honorsstudents, but are accessible for use with alllevels of students.

WHEN TO USE These may be used asextra credit, short-term projects, or asactivities for days when class periods areshortened.

© Glencoe/McGraw-Hill v Glencoe Algebra 2

Assessment OptionsThe assessment masters in the Chapter 14Resource Masters offer a wide range ofassessment tools for intermediate and finalassessment. The following lists describe eachassessment master and its intended use.

Chapter Assessment CHAPTER TESTS• Form 1 contains multiple-choice questions

and is intended for use with basic levelstudents.

• Forms 2A and 2B contain multiple-choicequestions aimed at the average levelstudent. These tests are similar in formatto offer comparable testing situations.

• Forms 2C and 2D are composed of free-response questions aimed at the averagelevel student. These tests are similar informat to offer comparable testingsituations. Grids with axes are providedfor questions assessing graphing skills.

• Form 3 is an advanced level test withfree-response questions. Grids withoutaxes are provided for questions assessinggraphing skills.

All of the above tests include a free-response Bonus question.

• The Open-Ended Assessment includesperformance assessment tasks that aresuitable for all students. A scoring rubricis included for evaluation guidelines.Sample answers are provided forassessment.

• A Vocabulary Test, suitable for allstudents, includes a list of the vocabularywords in the chapter and ten questionsassessing students’ knowledge of thoseterms. This can also be used in conjunc-tion with one of the chapter tests or as areview worksheet.

Intermediate Assessment• Four free-response quizzes are included

to offer assessment at appropriateintervals in the chapter.

• A Mid-Chapter Test provides an optionto assess the first half of the chapter. It iscomposed of both multiple-choice andfree-response questions.

Continuing Assessment• The Cumulative Review provides

students an opportunity to reinforce andretain skills as they proceed throughtheir study of Algebra 2. It can also beused as a test. This master includes free-response questions.

• The Standardized Test Practice offerscontinuing review of algebra concepts invarious formats, which may appear onthe standardized tests that they mayencounter. This practice includes multiple-choice, grid-in, and quantitative-comparison questions. Bubble-in andgrid-in answer sections are provided onthe master.

Answers• Page A1 is an answer sheet for the

Standardized Test Practice questionsthat appear in the Student Edition onpages 810–811. This improves students’familiarity with the answer formats theymay encounter in test taking.

• The answers for the lesson-by-lessonmasters are provided as reduced pageswith answers appearing in red.

• Full-size answer keys are provided forthe assessment masters in this booklet.

Reading to Learn MathematicsVocabulary Builder

NAME ______________________________________________ DATE ____________ PERIOD _____

1414

© Glencoe/McGraw-Hill vii Glencoe Algebra 2

Voca

bula

ry B

uild

erThis is an alphabetical list of the key vocabulary terms you will learn in Chapter 14.As you study the chapter, complete each term’s definition or description. Rememberto add the page number where you found the term. Add these pages to your AlgebraStudy Notebook to review vocabulary at the end of the chapter.

Vocabulary Term Found on Page Definition/Description/Example

amplitude

AM·pluh·TOOD

double-angle formula

half-angle formula

midline

phase shift

FAYZ

(continued on the next page)

© Glencoe/McGraw-Hill viii Glencoe Algebra 2

Vocabulary Term Found on Page Definition/Description/Example

trigonometric equation

trigonometric identity

vertical shift

Reading to Learn MathematicsVocabulary Builder (continued)

NAME ______________________________________________ DATE ____________ PERIOD _____

1414

Study Guide and InterventionGraphing Trigonometric Functions

NAME ______________________________________________ DATE ____________ PERIOD _____

14-114-1

© Glencoe/McGraw-Hill 837 Glencoe Algebra 2

Less

on

14-

1

Graph Trigonometric Functions To graph a trigonometric function, make a table ofvalues for known degree measures (0�, 30�, 45�, 60�, 90�, and so on). Round function values tothe nearest tenth, and plot the points. Then connect the points with a smooth, continuouscurve. The period of the sine, cosine, secant, and cosecant functions is 360� or 2� radians.

Amplitude of a FunctionThe amplitude of the graph of a periodic function is the absolute value of half thedifference between its maximum and minimum values.

Graph y � sin � for �360� � � � 0�.First make a table of values.

Graph the following functions for the given domain.

1. cos �, �360� � � � 0� 2. tan �, �2� � � � 0

What is the amplitude of each function?

3. 4.

x

y

O 2

2

x

y

O

y

O

�2

�4

4

2

��2� �3��

2 �� 2

y

O

�1

1

��90��180��270��360�

� �

y

O�0.5

�1.0

1.0

0.5

��90��180��270��360�

y � sin �

� �360° �330° �315° �300° �270° �240° �225° �210° �180°

sin � 0 �12

� ��22�

� ��23�

� 1 ��23�

� ��22�

� �12

� 0

� �150° �135° �120° �90° �60° �45° �30° 0°

sin � ��12

� ��22�

� ��23�

� �1 ��23�

� ��22�

� ��12

� 0

ExampleExample

ExercisesExercises


Variations of Trigonometric Functions

For functions of the form y � a sin b� and y � a cos b�, the amplitude is |a|,

Amplitudes and the period is or .

and Periods For functions of the form y � a tan b�, the amplitude is not defined,

and the period is or .

Find the amplitude and period of each function. Then graph thefunction.

��|b |

180°�|b |

2��|b |

360°�|b |

Study Guide and Intervention (continued)

Graphing Trigonometric Functions

NAME ______________________________________________ DATE ____________ PERIOD _____

14-114-1

ExampleExample

a. y � 4 cos �3�

�

First, find the amplitude.|a| � |4 |, so the amplitude is 4.Next find the period.

� 1080�

Use the amplitude and period to helpgraph the function.

y

O

4

2

�2

�4

�720�540� 1080�900�360�180�

y � 4 cos �–3

360°�

��13��

b. y � ��12� tan 2�

The amplitude is not defined, and the period is �

�2�.

y

O �4

2

–2

–4

4

�2

3�4

� �

ExercisesExercises

Find the amplitude, if it exists, and period of each function. Then graph eachfunction.

1. y � �3 sin � 2. y � 2 tan �2�

�

y

O

�2

2

�2�3�

2� 3�5�2

�2

y

O

2

�2

�360�270�180�90�

Skills PracticeGraphing Trigonometric Functions

NAME ______________________________________________ DATE ____________ PERIOD _____

14-114-1


Less

on

14-

1


1. y � 2 cos � 2. y � 4 sin � 3. y � 2 sec �

4. y � �12� tan � 5. y � sin 3� 6. y � csc 3�

7. y � tan 2� 8. y � cos 2� 9. y � 4 sin �12��

y

O

4

2

�2

�4

�720�540�360�180�

y

O

2

1

�1

�2

�180�135�90�45�

y

O

4

2

�2

�4

�180�135�90�45�

y

O

4

2

�2

�4

�30� 90� 150�

y

O

2

1

�1

�2

�360�270�180�90�

y

O

2

1

�1

�2

�360�270�180�90�

y

O

4

2

�2

�4

�360�270�180�90�

y

O

4

2

�2

�4

�360�270�180�90�

y

O

2

1

�1

�2

�360�270�180�90�



1. y � �4 sin � 2. y � cot �12�� 3. y � cos 5�

4. y � csc �34�� 5. y � 2 tan �

12�� 6. 2y � sin �

FORCE For Exercises 7 and 8, use the following information.An anchoring cable exerts a force of 500 Newtons on a pole. The force hasthe horizontal and vertical components Fx and Fy. (A force of one Newton (N),is the force that gives an acceleration of 1 m/sec2 to a mass of 1 kg.)

7. The function Fx � 500 cos � describes the relationship between theangle � and the horizontal force. What are the amplitude and period of this function?

8. The function Fy � 500 sin � describes the relationship between the angle � and thevertical force. What are the amplitude and period of this function?

WEATHER For Exercises 9 and 10, use the following information.The function y � 60 � 25 sin �

�6�t, where t is in months and t � 0 corresponds to April 15,

models the average high temperature in degrees Fahrenheit in Centerville.

9. Determine the period of this function. What does this period represent?

10. What is the maximum high temperature and when does this occur?

�

500 NFy

Fx

�

�

�

�

�

�

�

�

�

y

O

1

�1

�180�135�90�45�

y

O

4

2

�2

�4

�360�270�180�90�

y

O

4

2

�2

�4

�360�270�180�90�

Practice Graphing Trigonometric Functions

NAME ______________________________________________ DATE ____________ PERIOD _____

14-114-1

Reading to Learn MathematicsGraphing Trigonometric Functions

NAME ______________________________________________ DATE ____________ PERIOD _____

14-114-1


Less

on

14-

1

Pre-Activity Why can you predict the behavior of tides?

Read the introduction to Lesson 14-1 at the top of page 762 in your textbook.

Consider the tides of the Atlantic Ocean as a function of time.Approximately what is the period of this function?

Reading the Lesson1. Determine whether each statement is true or false.

a. The period of a function is the distance between the maximum and minimum points.

b. The amplitude of a function is the difference between its maximum and minimumvalues.

c. The amplitude of the function y � sin � is 2�.

d. The function y � cot � has no amplitude.

e. The period of the function y � sec � is �.

f. The amplitude of the function y � 2 cos � is 4.

g. The function y � sin 2� has a period of �.

h. The period of the function y � cot 3� is ��3�.

i. The amplitude of the function y � �5 sin � is �5.

j. The period of the function y � csc �14�� is 4�.

k. The graph of the function y � sin � has no asymptotes.

l. The graph of the function y � tan � has an asymptote at � � 180�.

m. When � � 360�, the values of cos � and sec � are equal.

n. When � � 270�, cot � is undefined.

o. When � � 180�, csc � is undefined.

Helping You Remember2. What is an easy way to remember the periods of y � a sin b� and y � a cos b�?


BlueprintsInterpreting blueprints requires the ability to select and use trigonometricfunctions and geometric properties. The figure below represents a plan for animprovement to a roof. The metal fitting shown makes a 30� angle with thehorizontal. The vertices of the geometric shapes are not labeled in theseplans. Relevant information must be selected and the appropriate functionused to find the unknown measures.

Find the unknown measures in the figure at the right.

The measures x and y are the legs of a right triangle.

The measure of the hypotenuse

is �1156� in. � �1

56� in. or �

2106� in.

� cos 30� � sin 30�

y � 1.08 in. x � 0.63 in.

Find the unknown measures of each of the following.

1. Chimney on roof 2. Air vent 3. Elbow joint

B

A

4'

t

r

1'–47

40°

D

C

1'–43

1'–41

2'

1'–21

x

y

A

1'–24

1'–29

40°

x��2106�

y��2106�

5"––16

15"––16

13"––16

5"––16

x

y0.09"

top view

side view

metal fitting

Roofing Improvement

30°

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

14-114-1

ExampleExample

Study Guide and InterventionTranslations of Trigonometric Graphs

NAME ______________________________________________ DATE ____________ PERIOD _____

14-214-2


Less

on

14-

2

Horizontal Translations When a constant is subtracted from the angle measure in atrigonometric function, a phase shift of the graph results.

The horizontal phase shift of the graphs of the functions y � a sin b(� � h), y � a cos b(� � h),

Phase Shiftand y � a tan b(� � h) is h, where b � 0.If h � 0, the shift is to the right.If h 0, the shift is to the left.

State the amplitude, period, and phase shift for y � �

12� cos 3��

�2��. Then graph

the function.

Amplitude: a � | �12� | or �

12�

Period: � or �23��

Phase Shift: h � ��2�

The phase shift is to the right since ��2� � 0.

State the amplitude, period, and phase shift for each function. Then graph thefunction.

1. y � 2 sin (� � 60�) 2. y � tan �� 2��

3. y � 3 cos (� � 45�) 4. y � �12� sin 3��

�3��

y

O�0.5

�1.0

1.0

0.5

�2��3

��6

��3

��2

5��6

�

y

O

2

�2

�360� 450�270�180�90�

y

O

�2

2

�2�3��

2��

2

y

O

2

�2

�360��90� 270�180�90�

2��|3 |

2��|b|

y

O�0.5

�1.0

1.0

0.5

�2��3

��6

��3

��2

5��6

�

ExampleExample

ExercisesExercises


Vertical Translations When a constant is added to a trigonometric function, the graphis shifted vertically.

The vertical shift of the graphs of the functions y � a sin b(� � h) � k, y � a cos b(� � h) � k,

Vertical Shiftand y � a tan b(� � h) � k is k.If k � 0, the shift is up.If k 0, the shift is down.

The midline of a vertical shift is y � k.

Step 1 Determine the vertical shift, and graph the midline.Graphing Step 2 Determine the amplitude, if it exists. Use dashed lines to indicate the maximum andTrigonometric minimum values of the function.Functions Step 3 Determine the period of the function and graph the appropriate function.

Step 4 Determine the phase shift and translate the graph accordingly.

State the vertical shift, equation of the midline, amplitude, andperiod for y � cos 2� � 3. Then graph the function.Vertical Shift: k � �3, so the vertical shift is 3 units down.

The equation of the midline is y � �3.

Amplitude: |a| � |1 | or 1

Period: � or �

Since the amplitude of the function is 1, draw dashed lines parallel to the midline that are 1 unit above and below the midline.Then draw the cosine curve, adjusted to have a period of �.

State the vertical shift, equation of the midline, amplitude, and period for eachfunction. Then graph the function.

1. y � �12� cos � � 2 2. y � 3 sin � � 2

y

O�1�2�3�4�5�6

1

�3�2

�2

� 2�

y

O�1�2

321

�3�2

�2

� 2�

2��|2 |

2��|b|

y

O�1

21

�3��2

��2

� 2�


Translations of Trigonometric Graphs

NAME ______________________________________________ DATE ____________ PERIOD _____

14-214-2

ExampleExample

ExercisesExercises

Skills PracticeTranslations of Trigonometric Graphs

NAME ______________________________________________ DATE ____________ PERIOD _____

14-214-2


Less

on

14-

2

State the amplitude, period, and phase shift for each function. Then graph thefunction.

1. y � sin (� � 90�) 2. y � cos (� � 45�) 3. y � tan �� 2��

State the vertical shift, equation of the midline, amplitude, and period for eachfunction. Then graph the function.

4. y � csc � � 2 5. y � cos � � 1 6. y � sec � � 3

State the vertical shift, amplitude, period, and phase shift of each function. Thengraph the function.

7. y � 2 cos [3(� � 45�)] � 2 8. y � 3 sin [2(� � 90�)] � 2 9. y � 4 cot ��43��

�4�� 2

�2�2

O �2�3�

2��

2

y

�2

�4

4

2

y

O

6

4

2

�2

�360�270�180�90�

y

O

6

4

2

�2

�360�270�180�90�

y

O

6

4

2

�2

�360�270�180�90�

y

O

2

1

�1

�720�540�360�180�

y

O

2

�2

�4

�6

�720�540�360�180�

�2�2

O �2�3�

2��

2

y

�2

�4

4

2

y

O

2

1

�1

�2

�360�270�180�90�

y

O

2

1

�1

�2

�360�270�180�90�


State the vertical shift, amplitude, period, and phase shift for each function. Thengraph the function.

1. y � �12� tan ��

�2�� 2. y � 2 cos (� � 30�) � 3 3. y � 3 csc (2� � 60�) � 2.5

ECOLOGY For Exercises 4–6, use the following information.The population of an insect species in a stand of trees follows the growth cycle of aparticular tree species. The insect population can be modeled by the function y � 40 � 30 sin 6t, where t is the number of years since the stand was first cut inNovember, 1920.

4. How often does the insect population reach its maximum level?

5. When did the population last reach its maximum?

6. What condition in the stand do you think corresponds with a minimum insect population?

BLOOD PRESSURE For Exercises 7–9, use the following information.Jason’s blood pressure is 110 over 70, meaning that the pressure oscillates between a maximumof 110 and a minimum of 70. Jason’s heart rate is 45 beats per minute. The function thatrepresents Jason’s blood pressure P can be modeled using a sine function with no phase shift.

7. Find the amplitude, midline, and period in seconds of the function.

8. Write a function that represents Jason’s blood pressure P after t seconds.

9. Graph the function.

Time

Jason’s Blood Pressure

Pres

sure

20 4 61 3 5 7 8 9

120

100

80

60

40

20

P

t

y

�

�

y

O

6

4

2

�2

�720�540�360�180��2�2

O �2�3��

2��

2

y

�2

�4

4

2

Practice Translations of Trigonometric Graphs

NAME ______________________________________________ DATE ____________ PERIOD _____

14-214-2

Reading to Learn MathematicsTranslations of Trigonometric Graphs

NAME ______________________________________________ DATE ____________ PERIOD _____

14-214-2


Less

on

14-

2

Pre-Activity How can translations of trigonometric graphs be used to showanimal populations?


According to the model given in your textbook, what would be the estimatedrabbit population for January 1, 2005?

Reading the Lesson

1. Determine whether the graph of each function represents a shift of the parent functionto the left, to the right, upward, or downward. (Do not actually graph the functions.)

a. y � sin (� � 90�) b. y � sin � � 3

c. y � cos �� 3�� d. y � tan � � 4

2. Determine whether the graph of each function has an amplitude change, period change,phase shift, or vertical shift compared to the graph of the parent function. (More thanone of these may apply to each function. Do not actually graph the functions.)

a. y � 3 sin �� 56��

b. y � cos (2� � 70�)

c. y � �4 cos 3�

d. y � sec �12�� 3

e. y � tan �� 4�� 1

f. y � 2 sin ��13��

�6�� 4

Helping You Remember

3. Many students have trouble remembering which of the functions y � sin (� � �) and y � sin (� � �) represents a shift to the left and which represents a shift to the right.Using � � 45�, explain a good way to remember which is which.


Translating Graphs of Trigonometric FunctionsThree graphs are shown at the right:

y � 3 sin 2�

y � 3 sin 2(� � 30�)y � 4 � 3 sin 2�

Replacing � with (� � 30�) translatesthe graph to the right. Replacing ywith y � 4 translates the graph 4 units down.

Graph one cycle of y � 6 cos (5� � 80�) � 2.

Step 1 Transform the equation into the form y � k � a cos b(� � h).

y � 2 � 6 cos 5(� � 16�)

Step 2 Sketch y � 6 cos 5�.

Step 3 Translate y � 6 cos 5� to obtain the desired graph.

Sketch these graphs on the same coordinate system.

1. y � 3 sin 2(� � 45�) 2. y � 1 � 3 sin 2� 3. y � 5 � 3 sin 2(� � 90�)

On another piece of paper, graph one cycle of each curve.

4. y � 2 sin 4(� � 50�) 5. y � 5 sin (3� � 90�)

6. y � 6 cos (4� � 360�) � 3 7. y � 6 cos 4� � 3

8. The graphs for problems 6 and 7 should be the same. Use the sum formula for cosine of a sum to show that the equations are equivalent.

O

y

u

56°

uy 2 2 = 6 cos 5( + 16°)6

–6

y = 6 cos 5( + 16°)

O

y

u72°

uy = 6 cos 56

–6

O

y

u90° 180°

uy = 3 sin 2

uy = 3 sin 2( – 30°)

uy + 4 = 3 sin 2

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

14-214-2

Step 2

Step 3

ExampleExample

Study Guide and InterventionTrigonometric Identities

NAME ______________________________________________ DATE ____________ PERIOD _____

14-314-3


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Find Trigonometric Values A trigonometric identity is an equation involvingtrigonometric functions that is true for all values for which every expression in the equationis defined.

BasicQuotient Identities tan � � �

csoins

��

� cot � � �csoins

��

�

Trigonometric Reciprocal Identities csc � � �sin

1�

� sec � � �co

1s �� cot � � �

ta1n ��

IdentitiesPythagorean Identities cos2 � � sin2 � � 1 tan2 � � 1 � sec2 � cot2 � � 1 � csc2 �

Find the value of cot � if csc � � ��151�; 180� � 270�.

cot2 � � 1 � csc2 � Trigonometric identity

cot2 � � 1 � ��151��2

Substitute ��151� for csc �.

cot2 � � 1 � �12251

� Square ��151�.

cot2 � � �9265� Subtract 1 from each side.

cot � � �4�

56�

� Take the square root of each side.

Since � is in the third quadrant, cot � is positive, Thus cot � � �4�

56�

�.

Find the value of each expression.

1. tan �, if cot � � 4; 180� � 270� 2. csc �, if cos � � ��23�

�; 0� � � 90�

3. cos �, if sin � � �35�; 0� � � 90� 4. sec �, if sin � � �

13�; 0� � � 90�

5. cos �, if tan � � ��43�; 90� � 180� 6. tan �, if sin � � �

37�; 0� � � 90�

7. sec �, if cos � � ��78�; 90� � 180� 8. sin �, if cos � � �

67�; 270� � � 360�

9. cot �, if csc � � �152�; 90� � 180� 10. sin �, if csc � � ��

94�; 270� � 360�

ExampleExample

ExercisesExercises


Simplify Expressions The simplified form of a trigonometric expression is written as anumerical value or in terms of a single trigonometric function, if possible. Any of thetrigonometric identities on page 849 can be used to simplify expressions containingtrigonometric functions.

Simplify (1 � cos2 �) sec � cot � � tan � sec � cos2 �.

(1 � cos2 �) sec � cot � � tan � sec � cos2 � � sin2 � � �co1s ��

csoins �

�� c

soins

��

� � �co1s �� cos2 �

� sin � � sin �� 2 sin �

Simplify � .

�se1c��

sicnot

��

� � �1 �csc

si�n �

� � �

�

�

�

Simplify each expression.

1. 2.

3. 4.

5. � cot � � sin � � tan � � csc � 6.

7. 3 tan � � cot � � 4 sin � � csc � � 2 cos � � sec � 8. 1 � cos2 ��tan � � sin �

csc2 � � cot2 ��tan � � cos �

tan � � cos ��sin �

cos ��sec � � tan �

sin2 � � cot � � tan ��cot � � sin �

sin � � cot ��sec2 � � tan2 �

tan � � csc ��sec �

2�cos2 �

�sin

1�

� � 1 � �sin

1�

� � 1��1 � sin2 �

�sin

1�

�(1 � sin �) � �sin

1�

�(1 � sin �)��(1 � sin �)(1 � sin �)

�sin

1�

�

��1 � sin �

�co

1s ��

csoins

��

�

��1 � sin �

csc ��1 � sin �

sec � cot ��1 � sin �


Trigonometric Identities

NAME ______________________________________________ DATE ____________ PERIOD _____

14-314-3

Example 1Example 1

Example 2Example 2

ExercisesExercises

Skills PracticeTrigonometric Identities

NAME ______________________________________________ DATE ____________ PERIOD _____

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1. sin �, if cos � � ��45� and 90� � 180� 2. cos �, if tan � � 1 and 180� � 270�

3. sec �, if tan � � 1 and 0� � � 90� 4. cos �, if tan � � �12� and 0� � � 90�

5. tan �, if sin � � � and 180� � 270� 6. cos �, if sec � � 2 and 270� � 360�

7. cos �, if csc � � �2 and 180� � 270� 8. tan �, if cos � � � and 180� � 270�

9. cos �, if cot � � ��32� and 90� � 180� 10. csc �, if cos � � �1

87� and 0� � 90�

11. cot �, if csc � � �2 and 180� � 270� 12. tan �, if sin � � ��153� and 180� � 270�


13. sin � sec � 14. csc � sin �

15. cot � sec � 16. �csoesc

��

�

17. tan � � cot � 18. csc � tan � � tan � sin �

19. 20. csc � � cot �

21. 22. 1 � tan2 ��1 � sec �

sin2 � � cos2 ��

1 � cos2 �

1 � sin2 ��sin � � 1

2�5��5

�2��2



1. sin �, if cos � � �153� and 0� � � 90� 2. sec �, if sin � � ��

1157� and 180� � 270�

3. cot �, if cos � � �130� and 270� � 360� 4. sin �, if cot � � �

12� and 0� � � 90�

5. cot �, if csc � � ��32� and 180� � 270� 6. sec �, if csc � � �8 and 270� � 360�

7. sec �, if tan � � 4 and 180� � 270� 8. sin �, if tan � � ��12� and 270� � 360�

9. cot �, if tan � � �25� and 0� � � 90� 10. cot �, if cos � � �

13� and 270� � 360�


11. csc � tan � 12. 13. sin2 � cot2 �

14. cot2 � � 1 15. 16. �csc �co

�s �

sin ��

17. sin � � cos � cot � 18. � 19. sec2 � cos2 � � tan2 �

20. AERIAL PHOTOGRAPHY The illustration shows a plane taking an aerial photograph of point A. Because the point is directly belowthe plane, there is no distortion in the image. For any point B notdirectly below the plane, however, the increase in distance createsdistortion in the photograph. This is because as the distance fromthe camera to the point being photographed increases, theexposure of the film reduces by (sin �)(csc � � sin �). Express (sin �)(csc � � sin �) in terms of cos � only.

21. TSUNAMIS The equation y � a sin �t represents the height of the waves passing abuoy at a time t in seconds. Express a in terms of csc �t.

A B

�

cos ��1 � sin �

cos ��1 � sin �

csc2 � � cot2 ��

1 � cos2 �

sin2 ��tan2 �

Practice Trigonometric Identities

NAME ______________________________________________ DATE ____________ PERIOD _____

14-314-3

Reading to Learn MathematicsTrigonometric Identities

NAME ______________________________________________ DATE ____________ PERIOD _____

14-314-3


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Pre-Activity How can trigonometry be used to model the path of a baseball?


Suppose that a baseball is hit from home plate with an initial velocity of 58 feet per second at an angle of 36� with the horizontal from an initialheight of 5 feet. Show the equation that you would use to find the height ofthe ball 10 seconds after the ball is hit. (Show the formula with theappropriate numbers substituted, but do not do any calculations.)

Reading the Lesson

1. Match each expression from the list on the left with an expression from the list on theright that is equal to it for all values for which each expression is defined. (Some of theexpressions from the list on the right may be used more than once or not at all.)

a. sec2 � � tan2 � i. �sin1

��

b. cot2 � � 1 ii. tan �

c. �csoins

��

� iii. 1

d. sin2 � � cos2 � iv. sec �

e. csc � v. csc2 �

f. �co1s �� vi. cot �

g. �csoins

��

�

2. Write an identity that you could use to find each of the indicated trigonometric valuesand tell whether that value is positive or negative. (Do not actually find the values.)

a. tan �, if sin � � ��45� and 180� � 270�

b. sec �, if tan � � �3 and 90� � 180�


3. A good way to remember something new is to relate it to something you already know.How can you use the unit circle definitions of the sine and cosine that you learned inChapter 13 to help you remember the Pythagorean identity cos2 � � sin2 � � 1?


Planetary OrbitsThe orbit of a planet around the sun is an ellipse with the sun at one focus. Let the pole of a polar coordinatesystem be that focus and the polar axis be toward theother focus. The polar equation of an ellipse is

r � �1 �

2eepcos ��. Since 2p � �

bc

2� and b2 � a2 � c2,

2p � �a2 �

cc2

� � �ac

2��1 � �

ac2

2��. Because e � �ac

�,

2p � a��ac��1 � ��a

c��2� � a��

1e��(1 � e2).

Therefore 2ep � a(1 � e2). Substituting into the polar equation of an ellipse yields an equation that is useful for finding distances from the planet to the sun.

r � �1a�

(1e�

coes

2)�

�

Note that e is the eccentricity of the orbit and a is the length of the semi-major axis of the ellipse. Also, a is the mean distance of the planet from the sun.

The mean distance of Venus from the sun is 67.24 � 106 miles and the eccentricity of its orbit is .006788. Find theminimum and maximum distances of Venus from the sun.

The minimum distance occurs when � � �.

r � � 66.78 � 106 miles

The maximum distance occurs when � � 0.

r � � 67.70 � 106 miles

Complete each of the following.

1. The mean distance of Mars from the sun is 141.64 � 106 miles and theeccentricity of its orbit is 0.093382. Find the minimum and maximumdistances of Mars from the sun.

2. The minimum distance of Earth from the sun is 91.445 � 106 miles andthe eccentricity of its orbit is 0.016734. Find the mean and maximumdistances of Earth from the sun.

67.24 � 106(1 � 0.0067882)��1 � 0.006788 cos 0

67.24 � 106(1 � 0.0067882)��1 � 0.006788 cos �

r

Polar Axis

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

14-314-3

ExampleExample

Study Guide and Intervention

Verifying Trigonometric Identities

NAME ______________________________________________ DATE ____________ PERIOD _____

14-414-4


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4

Transform One Side of an Equation Use the basic trigonometric identities alongwith the definitions of the trigonometric functions to verify trigonometric identities. Often itis easier to begin with the more complicated side of the equation and transform thatexpression into the form of the simpler side.

Verify that each of the following is an identity.ExampleExample

a. � sec � � �cos �

Transform the left side.

� sec � � �cos �

� � �cos �

� � �cos �

� �cos �

� �cos �

�cos � � �cos �

�cos2 ��cos �

sin2 � 1��cos �

1�cos �

sin2 ��cos �

1�cos �

sin ��csoins

��

�

sin ��cot �

sin ��cot � b. � cos � � sec �

Transform the left side.

� cos � � sec �



� sec �

� sec �

sec � � sec �

1�cos �

sin2 � � cos2 ��cos �

sin2 ��cos �

�csoins

��

�

�

�sin

1�

�

tan ��csc �

tan ��csc �

ExercisesExercises

Verify that each of the following is an identity.

1. 1 � csc2 � � cos2 � � csc2 � 2. � �1 � cos3 ��sin3 �

cot ��1 � cos �

sin ��1 � cos �


Transform Both Sides of an Equation The following techniques can be helpful inverifying trigonometric identities.• Substitute one or more basic identities to simplify an expression.• Factor or multiply to simplify an expression.• Multiply both numerator and denominator by the same trigonometric expression.• Write each side of the identity in terms of sine and cosine only. Then simplify each side.

Verify that � sec2 � � tan2 � is an identity.

� sec2 � � tan2 �

� �

�

�

� 1

1 � 1


1��sin2 � � cos2 �

cos2 ��cos2 �

�cos

12 ��

��

�sin2 �

co�s2

c�os2 �

�

1 � sin2 ��cos2 �

�cos

12 ��

��csoins

2

2��

� � 1

sin2 ��cos2 �

1�cos2 �

sec2 ��sin � � �c

soins

��

� � �co1s �� 1

tan2 � � 1��sin � � tan � � sec � � 1

tan2 � � 1��sin � tan � sec � � 1


Verifying Trigonometric Identities

NAME ______________________________________________ DATE ____________ PERIOD _____

14-414-4

ExampleExample

ExercisesExercises

1. csc � � sec � � cot � � tan � 2. �sec ��cos �

tan2 ��1 � cos2 �

3. � 4. � cot2 �(1 � cos2 �)csc2 � � cot2 ��sec2 �

csc ��sin � � sec2 �

cos � � cot ��sin �

Skills PracticeVerifying Trigonometric Identities

NAME ______________________________________________ DATE ____________ PERIOD _____

14-414-4


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1. tan � cos � � sin � 2. cot � tan � � 1

3. csc � cos � � cot � 4. � cos �1 � sin2 ��cos �

5. (tan �)(1 � sin2 �) � sin � cos �2

6. � cot �csc ��sec �

7. � tan2 � 8. � 1 � sin �cos2 ��1 � sin �

sin2 ��1 � sin2 �



Practice Verifying Trigonometric Identities

NAME ______________________________________________ DATE ____________ PERIOD _____

14-414-4

1. � sec2 � 2. � 1cos2 ��1 � sin2 �

sin2 � � cos2 ��

cos2 �

3. (1 � sin �)(1 � sin �) � cos2 � 4. tan4 � � 2 tan2 � � 1 � sec4 �

5. cos2 � cot2 � � cot2 � � cos2 � 6. (sin2 �)(csc2 � � sec2 �) � sec2 �

7. PROJECTILES The square of the initial velocity of an object launched from the ground is

v2 � , where � is the angle between the ground and the initial path, h is the

maximum height reached, and g is the acceleration due to gravity. Verify the identity

� .

8. LIGHT The intensity of a light source measured in candles is given by I � ER2 sec �,where E is the illuminance in foot candles on a surface, R is the distance in feet from thelight source, and � is the angle between the light beam and a line perpendicular to thesurface. Verify the identity ER2(1 � tan2 �) cos � � ER2 sec �.

2gh sec2 ��sec2 � � 1

2gh�sin2 �

2gh�sin2 �

Reading to Learn MathematicsVerifying Trigonometric Identities

NAME ______________________________________________ DATE ____________ PERIOD _____

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Pre-Activity How can you verify trigonometric identities?


For � � ��, 0, or �, sin � � sin 2�. Does this mean that sin � � sin 2� is anidentity? Explain your reasoning.

Reading the Lesson

1. Determine whether each equation is an identity or not an identity.

a. �sin

12 ��

tan1

2 �� 1

b. �sinc�os

ta�n �

�

c. �csoins

��

� � �csoins

��

� � cos � sin �

d. cos2 � (tan2 � � 1) � 1

e. �csoins

2

2�

�� sin � csc � � sec2 �

f. �1 �1sin �� 1 �

1sin �� 2 cos2 �

g. tan2 � cos2 � � �csc

12 ��

h. �sseinc

��

� � �ta1n �� co

1t ��

2. Which of the following is not permitted when verifying an identity?

A. simplifying one side of the identity to match the other side

B. cross multiplying if the identity is a proportion

C. simplifying each side of the identity separately to get the same expression on both sides


3. Many students have trouble knowing where to start in verifying a trigonometric identity.What is a simple rule that you can remember that you can always use if you don’t see aquicker approach?


Heron’s FormulaHeron’s formula can be used to find the area of a triangle if you know thelengths of the three sides. Consider any triangle ABC. Let K represent thearea of �ABC. Then

K � �12�bc sin A

K2 � �b2c2 s

4in2 A� Square both sides.

� �b2c2(1 �

4cos2 A)�

�

� �b2

4c2��1 � �

b2 �2cb

2

c� a2��1 � �

b2 �2cb

2

c� a2�� Use the law of cosines.

� �b �

2c � a� � �

b �2c � a� � �

a �2b � c� � �

a �2b � c� Simplify.

Let s � �a �

2b � c�. Then s � a � �

b �2c � a�, s � b � �

a �2c � b�, s � c � �

a �2b � c�.

K2 � s(s � a)(s � b)(s � c) Substitute.

K � �s(s ��a)(s �� b)(s �� c)�

Use Heron’s formula to find the area of �ABC.

1. a � 3, b � 4.4, c � 7 2. a � 8.2, b � 10.3, c � 9.5

3. a � 31.3, b � 92.0, c � 67.9 4. a � 0.54, b � 1.32, c � 0.78

5. a � 321, b � 178, c � 298 6. a � 0.05, b � 0.08, c � 0.04

7. a � 21.5, b � 33.0, c � 41.7 8. a � 2.08, b � 9.13, c � 8.99

b2c2(1 � cos A)(1 � cos A)��4

A C

B

c a

b

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

14-414-4

Heron’s FormulaThe area of �ABC is

� , where s � .a � b � c

2s(s � a)(s � b)(s � c)

Study Guide and InterventionSum and Difference of Angles Formulas

NAME ______________________________________________ DATE ____________ PERIOD _____

14-514-5


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5

Sum and Difference Formulas The following formulas are useful for evaluating anexpression like sin 15� from the known values of sine and cosine of 60� and 45�.

Sum and The following identities hold true for all values of � and �.Difference cos (� �) � cos � � cos � � sin � � sin �of Angles sin (� �) � sin � � cos � cos � � sin �

Find the exact value of each expression.

a. cos 345�

cos 345� � cos (300� � 45�)� cos 300�� cos 45� � sin 300� � sin 45�

� �12� � � ��

�

b. sin (�105�)

sin (�105�) � sin (45� � 150�)� sin 45� � cos 150� � cos 45� � sin 150�

� � �� 12�

� �


1. sin 105� 2. cos 285� 3. cos (�75�)

4. cos (�165�) 5. sin 195� 6. cos 420�

7. sin (�75�) 8. cos 135� 9. cos (�15�)

10. sin 345� 11. cos (�105�) 12. sin 495�

�2� � �6��4

�2��2

�3��2

�2��2

�2� � �6��4

�2��2

�3��2

�2��2

ExampleExample

ExercisesExercises


Verify Identities You can also use the sum and difference of angles formulas to verifyidentities.

Verify that cos �� 32�� sin � is an identity.

cos �� 32�� sin � Original equation

cos � � cos �32�� sin � � sin �

32�� sin � Sum of Angles Formula

cos � � 0 � sin � � (�1) � sin � Evaluate each expression.

sin � � sin � Simplify.

Verify that sin �� 2�� cos (� � �) � �2 cos � is an identity.

sin �� 2�� cos (� � �) � �2 cos � Original equation

sin � � cos ��2� � cos � � sin �

�2� � cos � � cos � � sin � � sin � � �2 cos � Sum and Difference of

Angles Formulas

sin � � 0 � cos � � 1 � cos � � (�1) � sin � � 0 � �2 cos � Evaluate each expression.

�2 cos � � �2 cos � Simplify.


1. sin (90� � �) � cos �

2. cos (270� � �) � sin �

3. sin ��23�� cos ��

56�� sin �

4. cos ��34�� sin ��

�4�� 2� sin �


Sum and Difference of Angles Formulas

NAME ______________________________________________ DATE ____________ PERIOD _____

14-514-5

Example 1Example 1

Example 2Example 2

ExercisesExercises

Skills PracticeSum and Difference of Angles Formulas

NAME ______________________________________________ DATE ____________ PERIOD _____

14-514-5


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1. sin 330� 2. cos (�165�) 3. sin (�225�)

4. cos 135� 5. sin (�45)� 6. cos 210�

7. cos (�135�) 8. sin 75� 9. sin (�195�)


10. sin (90� � �) � cos �

11. sin (180� � �) � �sin �

12. cos (270� � � ) � �sin �

13. cos (� � 90�) � sin �

14. sin �� 2�� cos �

15. cos (� � �) � �cos �



1. cos 75� 2. cos 375� 3. sin (�165�)

4. sin (�105�) 5. sin 150� 6. cos 240�

7. sin 225� 8. sin (�75�) 9. sin 195�


10. cos (180� � �) � �cos �

11. sin (360� � �) � sin �

12. sin (45� � �) � sin (45� � �) � �2� sin �

13. cos �x � ��6�� sin �x � �

�3�� sin x

14. SOLAR ENERGY On March 21, the maximum amount of solar energy that falls on asquare foot of ground at a certain location is given by E sin (90� � �), where � is thelatitude of the location and E is a constant. Use the difference of angles formula to findthe amount of solar energy, in terms of cos �, for a location that has a latitude of �.

ELECTRICITY In Exercises 15 and 16, use the following information.In a certain circuit carrying alternating current, the formula i � 2 sin (120t) can be used tofind the current i in amperes after t seconds.

15. Rewrite the formula using the sum of two angles.

16. Use the sum of angles formula to find the exact current at t � 1 second.

Practice Sum and Difference of Angles Formulas

NAME ______________________________________________ DATE ____________ PERIOD _____

14-514-5

Reading to Learn MathematicsSum and Difference of Angles Formulas

NAME ______________________________________________ DATE ____________ PERIOD _____

14-514-5


Less

on

14-

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Pre-Activity How are the sum and difference formulas used to describecommunication interference?


Consider the functions y � sin x and y � 2 sin x. Do the graphs of these twofunctions have constructive interference or destructive interference?

Reading the Lesson

1. Match each expression from the list on the left with an expression from the list on theright that is equal to it for all values of the variables. (Some of the expressions from thelist on the right may be used more than once or not at all.)

a. sin (� � �) i. sin �

b. cos (� � �) ii. sin � cos � � cos � sin �

c. sin (180� � �) iii. �cos �

d. sin (180� � �) iv. cos � cos � � sin � sin �

e. cos (180� � �) v. sin � cos � � cos � sin �

f. sin (� � �) vi. cos � cos � � sin � sin �

g. cos (90� � �) vii. �sin �

h. cos (� � �) viii. cos �

2. Which expressions are equal to sin 15�? (There may be more than one correct choice.)

A. sin 45� cos 30� � cos 45� sin 30� B. sin 45� cos 30� � cos 45� sin 30�

C. sin 60� cos 45� � cos 60� sin 45� D. cos 60� cos 45� � sin 60� sin 45�


3. Some students have trouble remembering which signs to use on the right-hand sides ofthe sum and difference of angle formulas. What is an easy way to remember this?


Identities for the Products of Sines and CosinesBy adding the identities for the sines of the sum and difference of themeasures of two angles, a new identity is obtained.

sin (� � �) � sin � cos � � cos � sin �sin (� � �) � sin � cos � � cos � sin �

(i) sin (� � �) � sin (� � �) � 2 sin � cos �

This new identity is useful for expressing certain products as sums.

Write sin 3� cos � as a sum.In the identity let � � 3� and � � � so that 2 sin 3� cos � � sin (3� � �) � sin (3� � �). Thus,

sin 3� cos � � �12�sin 4� � �

12�sin 2�.

By subtracting the identities for sin (� � �) and sin (� � �),a similar identity for expressing a product as a difference is obtained.

(ii) sin (� � �) � sin (� � �) � 2 cos � sin �

Solve.

1. Use the identities for cos (� � �) and cos (� � �) to find identities for expressing the products 2 cos � cos � and 2 sin � sin � as a sum or difference.

2. Find the value of sin 105� cos 75� without using tables.

3. Express cos � sin �2�

� as a difference.

Enrichment

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ExampleExample

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6Double-Angle Formulas

The following identities hold true for all values of �.Double-Angle sin 2� � 2 sin � � cos � cos 2� � cos2 � � sin2 �

Formulas cos 2� � 1 � 2 sin2 �

cos 2� � 2 cos2 � � 1

Find the exact values of sin 2� and cos 2� if sin � � ��1

90� and 180� � 270�.

First, find the value of cos �.cos2 � � 1 � sin2 � cos2 � � sin2 � � 1

cos2 � � 1 � ��190��2

sin � � ��190�

cos2 � � �11090�

cos � �

Since � is in the third quadrant, cos � is negative. Thus cos � � � .

To find sin 2�, use the identity sin 2� � 2 sin � � cos �.sin 2� � 2 sin � � cos �

� 2��190��

�

The value of sin 2� is .

To find cos 2�, use the identity cos 2� � 1 � 2 sin2 �.cos 2� � 1 � 2 sin2 �

� 1 � 2��190��2

� ��3510�.

The value of cos 2� is ��3510�.

Find the exact values of sin 2� and cos 2� for each of the following.

1. sin � � �14�, 0� � 90� 2. sin � � ��

18�, 270� � 360�

3. cos � � ��35�, 180� � 270� 4. cos � � ��

45�, 90� � 180�

5. sin � � ��35�, 270� � 360� 6. cos � � ��

23�, 90� � 180�

9�19��50

9�19��50

�19��10

�19��10

�19��10

ExampleExample

ExercisesExercises


Half-Angle Formulas

Half-Angle The following identities hold true for all values of �.

Formulas sin ��2

� � ��1 �2cos �� cos �

�2

� � ��1 �2cos ��

Find the exact value of sin ��2� if sin � � �

23� and 90� � 180�.

First find cos �.cos2 � � 1 � sin2 � cos2 � � sin2 � � 1

cos2 � � 1 � ��23��2

sin � � �23

�

cos2 � � �59� Simplify.

cos � � Take the square root of each side.

Since � is in the second quadrant, cos � � � .

sin ��2� � ��1 �

2cos �� Half-Angle formula

� � cos � � �

� � Simplify.

� Rationalize.

Since � is between 90� and 180�, ��2� is between 45� and 90�. Thus sin �

�2� is positive and

equals .

Find the exact value of sin ��2� and cos �

�2� for each of the following.

1. cos � � ��35�, 180� � 270� 2. cos � � ��

45�, 90� � 180�

3. sin � � ��35�, 270� � 360� 4. cos � � ��

23�, 90� � 180�

Find the exact value of each expression by using the half-angle formulas.

5. cos 22�12�� 6. sin 67.5� 7. cos �

78��

�18 ��6�5��6

�18 ��6�5��6

3 � �5��6

�5��

3

1 � ��35�

��2

�5��3

�5��3


Double-Angle and Half-Angle Formulas

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ExampleExample

ExercisesExercises

Skills PracticeDouble-Angle and Half-Angle Formulas

NAME ______________________________________________ DATE ____________ PERIOD _____

14-614-6


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on

14-

6Find the exact values of sin 2�, cos 2�, sin �2�

�, and cos �2�

� for each of the following.

1. cos � � �275�, 0� � 90� 2. sin � � ��

45�, 180� � 270�

3. sin � � �4401�, 90� � 180� 4. cos � � �

37�, 270� � 360�

5. cos � � ��35�, 90� � 180� 6. sin � � �1

53�, 0� � 90�


7. cos 22�12�� 8. sin 165�

9. cos 105� 10. sin ��8�

11. sin �158

�� 12. cos 75�

Verify that each of the following is an

13. sin 2� � �1

2�

ttaann

�2 �

� 14. tan � � cot � � 2 csc 2�

identity.


Find the exact values of sin 2�, cos 2�, sin �2�


� for each of the following.

1. cos � � �153�, 0� � 90� 2. sin � � �1

87�, 90� � 180�

3. cos � � �14�, 270� � 360� 4. sin � � ��

23�, 180� � 270�


5. tan 105� 6. tan 15� 7. cos 67.5� 8. sin ��8��


9. sin2 �2�

� � �tan

2�ta�n

s�in �

�

10. sin 4� � 4 cos 2� sin � cos �

11. AERIAL PHOTOGRAPHY In aerial photography, there is a reduction in film exposure forany point X not directly below the camera. The reduction E� is given by E� � E0 cos4 �,where � is the angle between the perpendicular line from the camera to the ground and theline from the camera to point X, and E0 is the exposure for the point directly below the

camera. Using the identity 2 sin2 � � 1 � cos 2�, verify that E0 cos4 � � E0��12� � �

cos2

2��2.

12. IMAGING A scanner takes thermal images from altitudes of 300 to 12,000 meters. Thewidth W of the swath covered by the image is given by W � 2H� tan �, where H� is the

height and � is half the scanner’s field of view. Verify that �21H�

�csoins 2

2��

� � 2H� tan �.

Practice Double-Angle and Half-Angle Formulas

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Reading to Learn MathematicsDouble-Angle and Half-Angle Formulas

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14-

6Pre-Activity How can trigonometric functions be used to describe music?


Suppose that the equation for the second harmonic is y � sin a�. Then whatwould be the equations for the fundamental tone (first harmonic), thirdharmonic, fourth harmonic, and fifth harmonic?

Reading the Lesson

1. Match each expression from the list on the left with all expressions from the list on theright that are equal to it for all values of �.

a. sin ��

2� i. 2 sin � cos �

b. cos 2� ii. 1 � 2 sin2 �

c. cos ��

2� iii. cos2 � � sin2 �

d. sin 2� iv. ��1 �

2cos ��

v. ��1 �

2cos ��

2. Determine whether you would use the positive or negative square root in the half-angle

identities for sin ��2� and cos �

�2� in each of the following situations. (Do not actually

calculate sin ��2� and cos �

�2�.)

a. sin ��2�, if cos � � �

25� and � is in Quadrant I

b. cos ��2�, if cos � � �0.9 and � is in Quadrant II

c. cos ��2�, if sin � � �0.75 and � is in Quadrant III

d. sin ��2�, if sin � � �0.8 and � is in Quadrant IV


3. Many students find it difficult to remember a large number of identities. How can youobtain all three of the identities for cos 2� by remembering only one of them and using aPythagorean identity?


Alternating CurrentThe figure at the right represents an alternating current generator. A rectangular coil of wire is suspended between the poles of a magnet. As the coil of wire is rotated, it passes through the magnetic fieldand generates current.

As point X on the coil passes through the points A and C, its motion is along the direction of the magnetic field between the poles. Therefore, no current is generated. However, through points Band D, the motion of X is perpendicular to the magnetic field. The maximum current may have a positive

This induces maximum current in the coil. Between A or negative value.

and B, B and C, C and D, and D and A, the current in the coil will have an intermediate value. Thus, the graph of the current of an alternating current generator is closely related to the sine curve.

The actual current, i, in a household current is given by i � IM sin(120�t � �) where IM is the maximum value of the current, t is the elapsed time in seconds,and � is the angle determined by the position of the coil at time tn.

If � � ��2�, find a value of t for which i � 0.

If i � 0, then IM sin (120�t � �) � 0. i � IM sin(120�t � �)

Since IM � 0, sin(120�t � �) � 0. If ab � 0 and a � 0, then b � 0.

Let 120�t � � � s. Thus, sin s � 0.s � � because sin � � 0.120�t � � � � Substitute 120�t � � for s.

120�t � ��2� � � Substitute �

�2

� for �.

� �2140� Solve for t.

This solution is the first positive value of t that satisfies the problem.

Using the equation for the actual current in a household circuit,i � IM sin(120�t � �), solve each problem. For each problem, find thefirst positive value of t.

1. If � � 0, find a value of t for 2. If � � 0, find a value of t for whichwhich i � 0. i � �IM.

3. If � � ��2�, find a value of t for which 4. If � � �

�4�, find a value of t for which

i � �IM. i � 0.

OA

B

C

D

i(amperes)

t(seconds)

XA

B D

C

Enrichment

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Study Guide and InterventionSolving Trigonometric Equations

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7

Solve Trigonometric Equations You can use trigonometric identities to solvetrigonometric equations, which are true for only certain values of the variable.

Find all solutions of 4 sin2 � � 1 � 0 for the interval 0� � 360�.4 sin2 � � 1 � 0

4 sin2 � � 1

sin2 � � �14�

sin � � �12�

� � 30�, 150�, 210�, 330�

Solve sin 2� � cos � � 0for all values of �. Give your answer inboth radians and degrees.

sin 2� � cos � � 02 sin � cos � � cos � � 0

cos � (2 sin � � 1) � 0cos � � 0 or 2 sin � � 1 � 0

sin � � ��12�

� � 90� � k � 180�; � � 210� � k � 360�,� � �

�2� � k � � 330� � k � 360�;

� � �76�� k � 2�,

�11

6�� k � 2�

Example 1Example 1 Example 2Example 2

ExercisesExercises

Find all solutions of each equation for the given interval.

1. 2 cos2 � � cos � � 1, 0 � � 2� 2. sin2 � cos2 � � 0, 0 � � 2�

3. cos 2� � , 0� � � 360� 4. 2 sin � � �3� � 0, 0 � � 2�

Solve each equation for all values of � if � is measured in radians.

5. 4 sin2 � � 3 � 0 6. 2 cos � sin � � cos � � 0

Solve each equation for all values of � if � is measured in degrees.

7. cos 2� � sin2 � � �1

� 8. tan 2� � �1

�3��2


Use Trigonometric Equations

LIGHT Snell’s law says that sin � � 1.33 sin �, where � is the angleat which a beam of light enters water and � is the angle at which the beam travelsthrough the water. If a beam of light enters water at 42�, at what angle does thelight travel through the water?

sin � � 1.33 sin � Original equation

sin 42� � 1.33 sin � � � 42�

sin � � �si

1n.3432�

� Divide each side by 1.33.

sin � � 0.5031 Use a calculator.

� � 30.2� Take the arcsin of each side.

The light travels through the water at an angle of approximately 30.2�.

1. A 6-foot pipe is propped on a 3-foot tall packing crate that sits on level ground. One footof the pipe extends above the top of the crate and the other end rests on the ground.What angle does the pipe form with the ground?

2. At 1:00 P.M. one afternoon a 180-foot statue casts a shadow that is 85 feet long. Write anequation to find the angle of elevation of the Sun at that time. Find the angle ofelevation.

3. A conveyor belt is set up to carry packages from the ground into a window 28 feet abovethe ground. The angle that the conveyor belt forms with the ground is 35�. How long isthe conveyor belt from the ground to the window sill?

SPORTS The distance a golf ball travels can be found using the formula d � sin 2�, where v0 is the initial velocity of the ball, g is the acceleration due

to gravity (which is 32 feet per second squared), and � is the angle that the path ofthe ball makes with the ground.

4. How far will a ball travel hit 90 feet per second at an angle of 55�?

5. If a ball that traveled 300 feet had an initial velocity of 110 feet per second, what angledid the path of the ball make with the ground?

6. Some children set up a teepee in the woods. The poles are 7 feet long from theirintersection to their bases, and the children want the distance between the poles to be 4 feet at the base. How wide must the angle be between the poles?

v02

�g


Solving Trigonometric Equations

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ExampleExample

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7


1. sin � � , 0� � � 360� 2. 2 cos � � ��3�, 90� � 180�

3. tan2 � � 1, 180� � 360� 4. 2 sin � � 1, 0 � � �

5. sin2 � � sin � � 0, � � � 2� 6. 2 cos2 � � cos � � 0, 0 � � �


7. 2 cos2 � � cos � � 1 8. sin2 � � 2 sin � � 1 � 0

9. sin � � sin � cos � � 0 10. sin2 � � 1

11. 4 cos � � �1 � 2 cos � 12. tan � cos � � �12�


13. 2 sin � � 1 � 0 14. 2 cos � � �3� � 0

15. �2� sin � � 1 � 0 16. 2 cos2 � � 1

17. 4 sin2 � � 3 18. cos 2� � �1

Solve each equation for all values of �.

19. 3 cos2 � � sin2 � � 0 20. sin � � sin 2� � 0

21. 2 sin2 � � sin � � 1 22. cos � � sec � � 2

�2��2



1. sin 2� � cos �, 90� � � 180� 2. �2� cos � � sin 2� , 0� � � 360�

3. cos 4� � cos 2�, 180� � � 360� 4. cos � � cos (90 � �) � 0, 0 � � 2�

5. 2 � cos � � 2 sin2 �, � � � � �32�� 6. tan2 � � sec � � 1, �

�2� � � �


7. cos2 � � sin2 � 8. cot � � cot3 �

9. �2� sin3 � � sin2 � 10. cos2 � sin � � sin �

11. 2 cos 2� � 1 � 2 sin2 � 12. sec2 � � 2


13. sin2 � cos � � cos � 14. csc2 � � 3 csc � � 2 � 0

15. �1 �3cos �� 4(1 � cos �) 16. �2� cos2 � � cos2 �

Solve each equation for all values of �.

17. 4 sin2 � � 3 18. 4 sin2 � � 1 � 0

19. 2 sin2 � � 3 sin � � �1 20. cos 2� � sin � � 1 � 0

21. WAVES Waves are causing a buoy to float in a regular pattern in the water. The verticalposition of the buoy can be described by the equation h � 2 sin x. Write an expressionthat describes the position of the buoy when its height is at its midline.

22. ELECTRICITY The electric current in a certain circuit with an alternating current canbe described by the formula i � 3 sin 240t, where i is the current in amperes and t is thetime in seconds. Write an expression that describes the times at which there is nocurrent.

Practice Solving Trigonometric Equations

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Reading to Learn MathematicsSolving Trigonometric Equations

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7

Pre-Activity How can trigonometric equations be used to predict temperature?


Describe how you could use a graphing calculator to determine the months inwhich the average daily high temperature is above 80�F. (Assume that x � 1represents January.) Specify the graphing window that you would use.

Reading the Lesson

1. Identify which equations have no solution.

A. sin � � 1 B. tan � � 0.001 C. sec � � �12�

D. csc � � �3 E. cos � � 1.01 F. cot � � �1000

G. cos � � 2 � �1 H. sec � � 1.5 � 0 I. sin � � 0.009 � 0.99

2. Use a trigonometric identity to write the first step in the solution of each trigonometricequation. (Do not complete the solution.)

a. tan � � cos2 � � sin2 �, 0 � � 2�

b. sin2 � � 2 sin � � 1 � 0, 0� � � 360�

c. cos 2� � sin �, 0� � � 360�

d. sin 2� � cos �, 0 � � 2�

e. 2 cos 2� � 3 cos � � �1, 0� � � 360�

f. 3 tan2 � � 5 tan � � 2 � 0


3. A good way to remember something is to explain it to someone else. How would youexplain to a friend the difference between verifying a trigonometric identity and solvinga trigonometric equation.


Families of Curves

Use these graphs for the problems below.

1. Use the graph on the left to describe the relationship among the curves

y � x�12�, y � x1, and y � x2.

2. Graph y � xn for n � �110�, �

14�, 4, and 10 on the grid with y � x

�12�, y � x1, and

y � x2.

3. Which two regions in the first quadrant contain no points of the graphsof the family for y � xn?

4. On the right grid, graph the members of the family y � emx for which m � 1 and m � �1.

5. Describe the relationship among these two curves and the y-axis.

6. Graph y � emx for m � 0, �14�, �

12�, 2, and 4.

O

y

x

2

3

4

–2–3 –1 1 2 3

The Family y � emx

O

y

x

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8

n = 1

n = 1–2

The Family y � xn

n = 2

Enrichment

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Chapter 14 Test, Form 1

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Ass

essm

ent

Write the letter for the correct answer in the blank at the right of each question.

1. Which equation is graphed?A. y � 4 sin � B. y � 4 cos �C. y � sin 4� D. y � cos 4� 1.

2. Find the amplitude of y � 6 sin �.A. 6 B. � C. �6 D. 2� 2.

3. Find the period of y � 5 cos �.A. �5 B. 5 C. � D. 2� 3.

4. Which equation is graphed?A. y � sin (� � 30�)B. y � sin (� � 30�)C. y � cos (� � 30�)D. y � cos (� � 30�) 4.

5. Which equation is graphed?A. y � cos � � 2 B. y � cos � � 2C. y � sin � � 2 D. y � sin � � 2 5.

6. Find sin � if cos � � �12� and 0� � � � 90�.

A. ��23�

� B. ��23�

� C. �34� D. �

12� 6.

7. Find cot � if tan � � �13� and 0� � � � 90�.

A. 4 B. 3 C. �3 D. ��13� 7.

8. Simplify sin � csc �.A. sin2 � B. �1 C. tan � D. 1 8.

9. Simplify tan � cos �.

A. �csoisn

2

��

� B. cot � C. sin � D. 1 � sec2 � 9.

1414

y

O

2

4

90� 180�

270�

360��2

�4

�

y

O

2

90�

180�

270� 360��2

�

y

O

2

�4

�

� 2�y ��1

y ��2

y ��3


Chapter 14 Test, Form 1 (continued)

10. Simplify cot � sec �.

A. �scions2

��

� B. sin � C. csc � D. sec2 � 10.

11. Which expression is equivalent to �sin2

t�an

�2c�os2 �

�?

A. cot2 � B. cos2 � � cot2 � C. cos2 � � cos4 � D. csc2 � 11.

12. Which expression is equivalent to csc �(csc � � sin �)?A. sec2 � � 1 B. cot2 � C. tan2 � D. 1 12.

13. Find the exact value of cos 135�.

A. ��22�

� B. �12� C. ��

12� D. ��

�22�

� 13.

14. Find the exact value of sin 105�.

A. ��22�

� B. 0 C. ��2� �

4�6�

� D. ��2� �

4�6�

� 14.

15. Which expression is equivalent to sin (90� � �)?A. sin � B. �sin � C. �cos � D. cos � 15.

16. Find the exact value of cos 2� if cos � � �153�

and 0� � � � 90�.

A. �12659�

B. �112609�

C. ��11619

9�D. �1

1619

9�16.

17. Find the exact value of sin 2� if sin � � �45� and 0� � � � 90�.

A. �2245�

B. �1225�

C. �254� D. ��2

75�

17.

18. Find the exact value of cos 22�12�� by using a half-angle formula.

A. B. C. � D. � 18.

19. Which is not a solution of sin 2� � 1?A. 90� B. 45� C. 225� D. �135� 19.

20. LIGHT The length of the shadow S given by a tower that is 100 meters

high is S � �t1a0n0�

�, where � is the angle of inclination of the Sun. If the

angle of inclination is 45�, find the length of the shadow.A. 162 m B. 62 m C. 100 m D. 84 m 20.

Bonus Verify that �1 �co

tsa�n �

� � sec � � sin � sec2 � is an identity. B:

�2 � ��2��

�2 � ��2��

�2 � ��2��

�2 � ��2��

NAME DATE PERIOD

1414

Chapter 14 Test, Form 2A

NAME DATE PERIOD

SCORE


Ass

essm

ent


1. Which equation is graphed?

A. y � 4 sin �32�� B. y � 4 cos �

32��

C. y � 4 sin �23�� D. y � 4 cos �

23�� 1.

2. Find the amplitude of y � 8 sin 2�.A. 2 B. � C. 8 D. 4 2.

3. Find the period of y � tan 3�.

A. �23�� B. �

�3� C. 3� D. 6� 3.


A. y � sin (� � ��4�) B. y � sin ��

�4��

C. y � cos �� 4�� D. y � cos ��

�4�� 4.

5. Find the phase shift of y � cos �� 25��.

A. ��5� B. �

25�� C. ��

�5� D. ��

25�� 5.

6. Which equation is graphed?A. y � 4 sin � � 2B. y � 4 sin � � 2C. y � 4 cos � � 2D. y � 4 cos � � 2 6.

7. Find the vertical shift of y � 3 csc � � 5.A. �3 B. �5 C. 5 D. 3 7.

8. Find csc � if cot � � �13� and 90� � � � 180�.

A. � B. C. D. � 8.

9. Find sin � if cos � � ��23� and 90� � � � 180�.

A. � B. C. � D. 9.�13��3

�13��3

�5��3

�5��3

�10��3

�10��3

2�2��3

2�2��3

1414

y

O

2

4

�2

�4

�

� 2�

y

O

2

�2

�

�

2�

y

O

1

43

7

�3

�1

�

2��

y � 6

y � 2

y � �2


Chapter 14 Test, Form 2A (continued)

10. Simplify �1 �tan

co2s�

2 ��.

A. �cos2 � B. sec2 � C. cos2 � D. sin2 � 10.

11. Simplify �5(cot2 � � csc2 �).A. 5 B. �5 C. �5 csc2 � D. 5 sec2 � 11.

12. Which expression is not equivalent to 1?

A. sin2 � � cot2 � sin2 � B. �1s�in

c2

os�

�� cos �

C. sec2 � � tan2 � D. �cot2

co�s2si

�n2 �

� 12.

13. Which expression is equivalent to tan � � �sseinc

��

�?

A. �cot � B. cot � C. tan � � cot � D. tan � � sec2 � 13.

14. Find the exact value of cos 375�.

A. ��6� �4

�2�� B. ��6� �

4�2�

� C. ��2� �

4�6�

� D. ��2� �

4�6�

� 14.

15. Which expression is equivalent to cos �� 2��?

A. cos � B. �cos � C. sin � D. �sin � 15.

16. Find the exact value of sin 2� if cos � � ��35�

� and 180� � � � 270�.

A. ��19� B. ��

4�9

5�� C. �

19� D. �

4�9

5�� 16.

17. Find the exact value of sin �2�

� if cos � � �23� and 270� � � � 360�.

A. �13� B. ��

13� C. �

�66�

� D. ��66�

� 17.

18. Find the exact value of cos 105� by using a half-angle formula.

A. B. � C. � D. 18.

19. Find the solutions of sin 2� � cos � if 0� � � � 180�.A. 30�, 90� B. 30�, 150� C. 30�, 90�, 150� D. 0�, 90�, 150� 19.

20. BIOLOGY An insect population P in a certain area fluctuates with the

seasons. It is estimated that P � 17,000 � 4500 sin �5�2t�, where t is given in

weeks. Determine the number of weeks it would take for the population to initially reach 20,000.A. 12 weeks B. 692 weeks C. 38 weeks D. 42 weeks 20.

Bonus Verify that �1 �csc

co�t �

� � sin � � cos � is an identity. B:

�2 � ��3��

�2 � ��3��

�2 � ��3��

�2 � ��3��

NAME DATE PERIOD

1414

Chapter 14 Test, Form 2B

NAME DATE PERIOD

SCORE


Ass

essm

ent



A. y � 3 sin �23�� B. y � 3 cos �

23��

C. y � 2 sin �32�� D. y � 2 cos �

32�� 1.

2. Find the amplitude of y � 6 cos 4�.

A. �32� B. 6 C. 4 D. �

�2� 2.

3. Find the period of y � tan 5�.

A. 10� B. �25�� C. 5� D. �

�5� 3.


A. y � sin �� 4�� B. y � cos ��

�4��

C. y � sin �� 4�� D. y � cos ��

�4�� 4.

5. Find the phase shift of y � sin �� 34��.

A. �34�� B. ��

34�� C. �

43�� D. ��

43�� 5.

6. Which equation is graphed?A. y � 2 sin � � 3 B. y � 2 sin � � 3C. y � 3 cos � � 2 D. y � 3 cos � � 2 6.

7. Find the vertical shift of y � �4 sec � � 7.A. �4 B. �7C. 7 D. 4 7.

8. Find sec � if tan � � �14� and 180� � � � 270�.

A. ��

415�� B. ��

�415�� C. �

�417�� D. ��

�417�� 8.

9. Find cos � if sin � � �35� and 90� � � � 180�.

A. �45� B. ��

45� C. �

�534�� D. ��

�534�� 9.

10. Simplify �1 �co

ct2sc

�

2 ��.

A. �1 B. 1 C. tan2 � D. �sin14 �� 10.

1414

y

O

2

�2

�

� 2�

y

O

2

�2

�� 2�

y

O

1

�2

�6

�

� 2�y � �1

y � �3

y � �5


Chapter 14 Test, Form 2B (continued)

11. Simplify �4(sec2 � � tan2 �).A. �4 tan2 � B. 4 tan2 � C. 4 D. �4 11.

12. Which expression is equivalent to 1?

A. �1 �

sinsi

�n �

� B. �sec12 �� csc

12 ��

C. tan2 � � sec2 � D. �cots�ec

cs�c �

� 12.

13. Which expression is equivalent to �1 �sin

co�s �

� � �1 �sin

co�s �

�?

A. �12�

scions�2 �

� B. 2 sin � C. 2 csc � D. �2 csc � 13.

14. Find the exact value of sin (�15�).

A. ��6� �4

�2�� B. ��6� �

4�2�

� C. ��6�4� �2�� D. ��6�

4� �2�� 14.

15. Which expression is equivalent to sin �� 2��?

A. cos �� 2�� B. �cos � C. �sin � D. cos � 15.

16. Find the exact value of cos 2� if sin � � ��23� and 180� � � � 270�.

A. �19� B. ��

4�9

5�� C. ��

19� D. �

4�9

5�� 16.

17. Find the exact value of cos �2�

� if sin � � �14� and 0� � � � 90�.

A. ��

415�� B. ��

�415�� C. D. 17.

18. Find the exact value of sin 105� by using a half-angle formula.

A. B. C. � D. � 18.

19. Find the solutions of 3 sin � � 2 cos2 � if 0� � � � 360�.A. 30�, 150� B. 30�, 120� C. 30�, 330� D. 150�, 330� 19.

20. BIOLOGY An insect population P in a certain area fluctuates with

the seasons. It is estimated that P � 15,000 � 2500 sin �5�2t�, where t is given

in weeks. Determine the number of weeks it would take for the population to initially reach 16,000.A. 21 weeks B. 24 weeks C. 109 weeks D. 7 weeks 20.

Bonus Verify that 1 � csc2 � tan2 � � 2 � tan2 � is an identity. B:

�2 � ��3��

�2 � ��3��

�2 � ��3��

�2 � ��3��

�4 � ��15��

�8 � 2��15��

NAME DATE PERIOD

1414

Chapter 14 Test, Form 2C


1. Graph the function y � �32� cos 2�. 1.

For Questions 2 and 3, find the amplitude, if it exists, and period of each function.

2. y � 3 sin 4� 2.

3. y � �12� tan �

15�� 3.

4. State the phase shift of y � cos �� 23��. Then graph the 4.

function.

5. State the vertical shift and the equation of the midline for 5.y � 3 cos � � 2. Then graph the function.

6. Find sec � if sin � � �35� and 0� � � � 90�. 6.

7. Find cot � if csc � � ��52� and 270� � � � 360�. 7.

8. Simplify �cosc�ot

c�sc �

�. 8.

9. Simplify �1 �cos

co2s�

2 ��. 9.

y

O�

2��

y

O

2

�2

�

� 2�

y

O

1

2

�1

�2

�

� 2�

NAME DATE PERIOD

SCORE 1414

Ass

essm

ent


Chapter 14 Test, Form 2C (continued)

10. Verify that (cos � � sin �)2 � 2 cos � sin � � 1 is an identity. 10.

11. Verify that �1 �csc

co�t �

� � sin � � cos � is an identity. 11.

12. Find the exact value of sin (�195�). 12.

13. Find the exact value of cos 255�. 13.

14. Verify that sin �� 2�� cos � is an identity. 14.

15. Find the exact value of sin 2� if cos � � �14� and 15.

270� � � � 360�.


� if sin � � �13� and 90� � � � 180�. 16.

17. Find the exact value of sin 195� by using a half-angle 17.formula.

18. Verify that sin 2� � �2cs

cco2t��

� is an identity. 18.

19. Solve cos 2� � cos � � 0 for all values of � if � is measured 19.in degrees.

20. BUSINESS The profit P for a product whose sales fluctuate 20.

with the seasons is estimated to be P � 14 � 5 sin �5�2t�,

where t is given in weeks and P is in thousands of dollars.Determine the number of weeks it would take for the profit to initially reach $18,000.

Bonus Find cos 2� if sin �2�

� � . B:�2 � ��3��2

NAME DATE PERIOD

1414

Chapter 14 Test, Form 2D


1. Graph y � �52� sin 2�. 1.

For Questions 2 and 3, find the amplitude, if it exists, and period of each function.

2. y � 2 sin 3� 2.

3. y � �13� tan �

14�� 3.

4. State the phase shift of y � sin �� 23��. Then graph the 4.

function.

5. State the vertical shift and the equation of the midline for 5.y � 3 cos � � 1. Then graph the function.

6. Find csc � if cos � � ��13� and 90� � � � 180�. 6.

7. Find tan � if sec � � �52� and 270� � � � 360�. 7.

8. Simplify �cscs�ec

ta�n �

�. 8.

9. Simplify �1 �sin

s2ec

�

2 ��. 9.

y

O�

2��

y

O

2

�2

�� 2�

y

O

2

�2

�

� 2�

NAME DATE PERIOD

SCORE 1414

Ass

essm

ent


Chapter 14 Test, Form 2D (continued)

10. Verify that cos2 � sec2 � � cos2 � � sin2 � � 0 is an identity. 10.

11. Verify that �tacnot

��se

scec

��

� � �sin

co�t

��

1� is an identity. 11.

12. Find the exact value of sin 165�. 12.

13. Find the exact value of cos (�345�). 13.

14. Verify that cos �� 2�� sin � is an identity. 14.

15. Find the exact value of cos 2� if cos � � �14� and 15.

270� � � � 360�.


� if sin � � �13� and 90� � � � 180�. 16.

17. Find the exact value of cos 195� by using a half-angle 17.formula.

18. Verify that cos 2� � sin2 �(2 cot2 � � csc2 �) is an identity. 18.

19. Solve sin 2� � sin � � 0 for all values of � if � is measured 19.in degrees.

20. BUSINESS The profit P for a product whose sales fluctuate 20.

with the seasons is estimated to be P � 16 � 7 sin �5�2t�,

where t is given in weeks and P is in thousands of dollars.Determine the number of weeks it would take for the profit to initially reach $20,000.

Bonus Find cos 2� if cos �2�

� � . B:�2 � ��2��

NAME DATE PERIOD

1414

Chapter 14 Test, Form 3


1. Graph �12�y � �

34� csc �

12��. 1.

Find the amplitude, if it exists, and period of each function.

2. 5y � �23� cos 4� 3. ��

14�y � ��

38� tan �

15�� 2.

3.

For Questions 4 and 5, state the vertical shift, amplitude,period, and phase shift of each function. Then graph the function.

4. y � 2 tan (2� � 90�) � 3 4.

5. y � �32� � 3 cos �2��

�4�� 5.

6. Find sec � if sin � � �14� and 90� � � � 180�. 6.

7. Find tan � if sec � � �43� and 270� � � � 360�. 7.

8. Simplify �cocto2

t2�

��

cocso

2s2

��

�. 8.

9. Verify that �ccostc2

��si�n

1�

� � cot � csc � is an identity. 9.

y

O�

y

O�

y

O�

NAME DATE PERIOD

SCORE 1414

Ass

essm

ent


Chapter 14 Test, Form 3 (continued)

10. Verify that 1 � cot4 � � �2 si

snin2

4�

�� 1

� is an identity. 10.

11. Find the exact value of cos 75� � cos 15�. 11.

12. Find the exact value of sin 105� � sin 225�. 12.

13. Verify that sin �� 4�� cos ��

34�� 2� cos � 13.

is an identity.

14. Verify that

� 2 � tan � cot � � cot � tan � 14.

is an identity.

15. Find the exact value of sin 2� if cos � � �38� and 15.

270� � � � 360�.


� if sin � � ��1136�

and 16.

180� � � � 270�.

17. Find the exact value of cos �1172�� by using half-angle formulas. 17.

18. Verify that sin2 �2�

� � is an identity. 18.

19. Solve sin �2�

� � cos � � 0 for all values of � if � is measured in 19.

radians.

20. WAVES For a short time after a wave is created by a boat, 20.

its height can be modeled by y � �12�h � �

12�h sin �2P

�t�, where

h is the maximum height of the wave in feet, P is the period in seconds, and t is the propagation of the wave in seconds.If a wave has a maximum height of 3.2 feet and a period of 2.5 seconds, how long after its creation will the wave initially reach a height of 3 feet? Round to the nearest hundredth.

Bonus Find the exact value of if sin � � ��35� and B:

180� � � � 270�.

sin 2� � cos 2��

sin �2�

�

sin2 � � cos � � 1��2 cos �

[sin (� � �)]2��sin � cos � sin � cos �

NAME DATE PERIOD

1414

Chapter 14 Open-Ended Assessment


Demonstrate your knowledge by giving a clear, concise solutionto each problem. Be sure to include all relevant drawings andjustify your answers. You may show your solutions in more thanone way or investigate beyond the requirements of the problem.

1. Ms. Rollins divided her students into four groups, asking each tosolve the equation sin � cot � � cos2 �. The answers given were:Group A: 0� � k 360�, 90� � k 360�, 270� � k 360�Group B: 0� � k 360�, 90� � k 180�Group C: 90� � k 180�Group D: 90� � k 360�, 270� � k 360�

Do any of the groups have the correct solution? Explain yourreasoning.

2. Write a trigonometric function that has no amplitude, a period of

��2�, a phase shift to the left, and a vertical shift upward. Then

graph your function for 0 � � � 2�.

3. Show two different methods of verifying that

�1 � s

1in2 �� tan2 � � 1 is a trigonometric identity.

4. Select a quadrant, other than Quadrant I, and values for p

and q so that sin � � �pq�. Use your values of p and q to find the

exact values of cos �, tan �, csc �, sec �, cot �, sin 2�, cos 2�,

sin �2�


�.

5. Show how to find the exact value of sin 240� by each methodindicated.a. using a sum of angles formulab. using a difference of angles formulac. using a double-angle formulad. using a half-angle formula

y

O

21

345

�3

�1�2

�

2��

NAME DATE PERIOD

SCORE 1414

Ass

essm

ent


Chapter 14 Vocabulary Test/Review

Tell whether each sentence is true or false. If false, replace the underlined word or words to make a true sentence.

1. For the graph of y � 3 sin �x � ��

2��, the vertical shift is 3. 1.

2. For the graph of y � 2 cos (x � 45�) � 5, the phase shift is 5. 2.

3. For the graph of y � 3 sin �x � ��

6�� 2, the line y � �2 is the 3.

amplitude.

4. sin2 � � cos2 � � 1 is a(n) trigonometric identity. 4.

5. The exact value of sin 15� can be found by using a(n) 5.phase shift.

6. cos 2� � cos2 � � sin2 � is a(n) double-angle formula. 6.

7. 2 cos2 � � cos � � 1 � 0 is a(n) trigonometric equation. 7.

In your own words—Define the term.

8. phase shift

amplitudedouble-angle formula

half-angle formulamidline

phase shifttrigonometric equation

trigonometric identity vertical shift

NAME DATE PERIOD

SCORE 1414

Chapter 14 Quiz (Lessons 14–1 and 14–2)

1414


For Questions 1 and 2, find the amplitude, if it exists, and period of each function. Then graph the function.

1. y � �12� cos � 1.

2. y � tan 2� 2.

3. State the phase shift of y � sin �� 4��. 3.

4. State the vertical shift and the equation of the midline for 4.y � 4 cos � � 2.

y

O

2

�2

��

23��4

��4

y

O

1

90� 180� 270� 360��1

�

NAME DATE PERIOD

SCORE


For Questions 1 and 2, find the value of each expression.

1. cos �, if sin � � �12�; 90� � � � 180� 1.

2. cot �, if tan � � 2; 180� � � � 270� 2.

3. Simplify 4(tan2 � � sec2 �). 3.

4. Simplify �1 �

cstca2n�

2 ��. 4.

5. Standardized Test Practice �se

tcan

�2�

�

1� � 5.

A. �cosco

�s

��

1� B. �sinsi

�n

��

1� C. �sisnin�

2

�

�

1� D. 1

NAME DATE PERIOD

SCORE 1414

Ass

essm

ent



1. sin 75� 2. cos (�225�)

3. tan 210�

Verify that each is an identity.

4. sin ��2� � �� cos � 4.

5. cos (180� � �) � �cos � 5.

For Questions 6–8, find the exact value for each.

6. cos 2�, if cos � � ��25�; 90� � � � 180� 6.

7. sin 2�, if sin � � ��49�; 270� � � � 360� 7.

8. cos �2�

�, if sin � � ��25�; 180� � � � 270� 8.

9. Find the exact value of cos 112�12�� by using a half-angle 9.

formula.

10. Verify that cos 2� � 1 � sin 2� tan � is an identity. 10.

Chapter 14 Quiz (Lesson 14–7)

1. Find all solutions for sin � � cos 2� if 0� � � � 360�. 1.

2. Find all solutions for 4 cos2 � � 1 if 0 � � � 2�. 2.

3. Solve cos 2� � cos � for all values of � if � is measured in 3.degrees.

4. Solve cos 2� � 3 sin � � 1 for all values of � if � is measured 4.in radians.

5. LIGHT The length of the shadow s cast by a 40-foot tree 5.depends on the angle of inclination of the sun, �. Express sas a function of �. Then find the angle of inclination that produces a shadow 30 feet long.

NAME DATE PERIOD

SCORE


1414

NAME DATE PERIOD

SCORE

1414

1.

2.

3.

Chapter 14 Mid-Chapter Test (Lessons 14–1 through 14–4)


For Questions 1–5, write the letter for the correct answer in the blank at the right of each question.

Use the graph shown at the right.

1. Find the period of the function.A. 4 B. 2�

C. � D. 2 1.

2. Find the amplitude of the function.A. 4 B. 8

C. � D. ��4� 2.

For Questions 3 and 4, use the graph shown at the right.

3. Find the phase shift of the function.

A. ��4� B. ��

�4�

C. 1 D. 2 3.

4. Find the vertical shift of the function.

A. 1 B. 2 C. ��4� D. ��

�4� 4.

5. Which expression is equivalent to �1 � sisne2

c2�

�sec2 �

�� cos2 �?

A. 1 B. csc2 � C. sin2 � D. 2 cos2 � 5.

6. Graph the function y � �12� cos 4�. 6.

7. Find the amplitude, if it exists, and period of the function 7.y � 2 tan 4�.

8. Find sin � if cos � � �34� and 0� � � � 90�. 8.

9. Simplify �cos2 �se

�c �

sin2 ��. 9.

10. Simplify �cotc�sc

s�ec ��. 10.

11. Verify that �csc2 �co

�t �

cot2 �� tan � is an identity. 11.

y

O

1

�1

�

� 2�

Part I

NAME DATE PERIOD

SCORE 1414

Ass

essm

ent

y

O

2

4

�2

�4

�

2��

y

O

2

4

�2

�

y � 3

y � 1

y � �1

2��

Part II


Chapter 14 Cumulative Review (Chapters 1–14)

1. Solve 5 � � 2x � 1 � � 10 and graph its solution set. (Lesson 1-6) 1.

2. Use long division to find (x3 � 4x2 � 12x � 25) (x � 1). 2.(Lesson 5-3)

3. Write 13n4 � 52n2 in quadratic form, if possible. 3.Then solve. (Lesson 7-3)

4. Express log820 in terms of common logarithms. Then 4.approximate its value to four decimal places. (Lesson 10-4)

5. Find a1 in a geometric series for which Sn � 315, r � 2, and 5.an � 168. (Lesson 11-4)

6. From a group of 5 students and 3 faculty members, a 6.committee of 3 is selected. Find the probability that all 3 are students or all 3 are faculty. (Lesson 12-5)

7. Six coins are tossed. Find P(at least 4 tails). (Lesson 12-8) 7.

8. Find one angle with positive measure and one angle with 8.

negative measure coterminal with ��71�1�

. (Lesson 13-2)

9. Find the exact value of sin 120�. (Lesson 13-3) 9.

10. P��23�

�, ��12�� is located on the unit circle. Find sin � and 10.

cos �. (Lesson 13-6)

11. Find the amplitude, if it exists, and period of the function 11.

y � 2 cos �13��. (Lesson 14-1)

12. Find tan � if cos � � �1123�

and 270� � � � 360�. (Lesson 14-3) 12.


� if sin � � ��37� and 13.

180� � � � 270�. (Lesson 14-6)

14. Solve cos2 � sin � � sin � for all values of � if � is measured 14.in radians. (Lesson 14-7)

�1�2�3�4 0 1 2 43

NAME DATE PERIOD

1414

Standardized Test Practice (Chapters 1–14)


For Questions 1 and 2, use the bar graph that shows the height, to the nearest hundred feet, of five mountains in Vermont’s Green Mountain National Forest.

1. What is the difference in height between the highest and lowest of the given mountains?A. 16 ft B. 160 ftC. 1600 ft D. 16,000 ft 1.

2. What is the mean height of the given mountains? E. 3200 ft F. 32.6 ft G. 3260 ft H. 320 ft 2.

3. If �xy� � 10 and yz � 12, then xz � _____.

A. �56� B. �

65� C. 22 D. 120 3.

4. A tank that holds 500 gallons of water is filled at a rate of 4.5 gallons per minute. How long, to the nearest minute, will it take the tank to fill if it already contains 325 gallons of water?E. 788 min F. 39 min G. 111 min H. 4 min 4.

5. In the figure, the ratio of AC to CB is 12:5. If the area of triangle ABC is 120 cm2, then AB � ________.A. 26 cm B. 10 cmC. 104 cm D. 24 cm 5.

6. Line � passes through the points (3, �5) and (�2, 10). Which point does not lie on line �?

E. (0, 4) F. (�3, 13) G. ��13�, 1� H. (1, 1) 6.

7. The number 5610 is divisible by which of the following?I. 3 II. 6 III. 15

A. I only B. I and II onlyC. I and III only D. I, II, and III 7.

8. In the figure, the length of arc AB is 8�.What is the length of a radius of circle O?E. 24 F. 48G. 2�6� H. 24� 8. HGFE

DCBA

HGFE

DCBA

HGFE

DCBA

HGFE

DCBA

NAME DATE PERIOD

1414

Ass

essm

ent

A

B C

A

BO

60˚

Robert Frost Mtn.

Gillespie Mtn.

Mt. Abraham

Romance Mtn.

Bread Loaf Mtn.

24 26 28 30 32 34 36 38 40 42Height (100 feet)

Part 1: Multiple Choice

Instructions: Fill in the appropriate oval for the best answer.


Standardized Test Practice (continued)

9. The probability of randomly selecting a white 9. 10.

marble from a bag is �110�

. The probability of

randomly selecting a red marble is �35�. If the bag

also contains 9 blue marbles, what is the totalnumber of marbles in the bag?

10. If the mean of x, x � 2, 3x � 2, x � 7, 2x � 1,2x � 1, and x � 3 is 14, what is the mode?

11. Find the value of n in 11. 12.the figure if � � m.

12. Catherine purchased a hammer for $12, a rake for $17, and a shovel for $26 at a localhardware store. If the state sales tax rate is 6%, how much change did Catherine receive from the $60 she gave to the cashier?

Column A Column B 13. 13.

14. 14.

15. Regular hexagon ABCDEF 15.

yx

DCBA

E D

A B

CF

x˚ y˚3y˚

DCBA1.25a where 32a�1 � 81

DCBAThe 8th term of the sequence�1, 2, �4, 8, …

The 8th term of the sequence16, 32, 48, 64, …

Part 3: Quantitative Comparison

Instructions: Compare the quantities in columns A and B. Shade in if the quantity in column A is greater; if the quantity in column B is greater; if the quantities are equal; or if the relationship cannot be determined from the information given.

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

NAME DATE PERIOD

1414

NAME DATE PERIOD

m

��110˚

n˚3n˚

Part 2: Grid In

Instructions: Enter your answer by writing each digit of the answer in a column boxand then shading in the appropriate oval that corresponds to that entry.

A

D

C

B

Unit 5 Test (Chapters 13–14)


1. Solve �ABC if C � 90�, B � 20�, and b � 10. Round measures of sides to the nearest tenth and measures of 1.angles to the nearest degree.

2. Rewrite �25� in radian measure. 2.

3. Rewrite �95�� radians in degree measure. 3.

4. Find one angle with positive measure and one angle with 4.negative measure coterminal with �310�.

5. Find the exact values of the six trigonometric functions of 5.� if the terminal side of � in standard position contains the point (�5, �4).

6. Sketch the angle with measure ��23�� radians. Then label its 6.

reference angle.

For Questions 7–10, find the exact value of each trigonometric function.

7. cot ��6�� 8. sin 405�

9. tan (�3�) 10. sin 60� � cos 60�

11. Find the area of �ABC if A � 56�, b � 20 feet, and c � 12 feet. Round to the nearest tenth.

12. In �ABC, A � 35�, a � 43, and c � 20. Determine whether �ABC has no solution, one solution or two solutions. Then solve the triangle. Round to the nearest tenth.

For Questions 13 and 14, determine whether each triangle should be solved by beginning with the Law of Sines or Law of Cosines. Then solve each triangle. Round to the nearest tenth.

13. a � 16, b � 13, c � 10 13.

14. A � 56�, B � 38�, a � 13 14.

15. P��1157�, ��

187�� is located on the unit circle. 15.

Find sin � and cos �.

16. Solve x � Arctan (��3�). 16.

17. Verify that �sseinc

��

� �ccsoct �

�� csc � is an identity. 17.

O

y

x

NAME DATE PERIOD

SCORE

7.

8.

9.

10.

11.

12.

Ass

essm

ent


Unit 5 Test (continued)(Chapters 13–14)

18. Graph the function y � 4 sin 2�.

For Questions 19 and 20, find the amplitude, if it exists,and period of each function.

19. y � cos 3� 20. y � tan �14

� �

21. State the phase shift of y � cos�� 3��. Then graph the

function.

22. State the vertical shift and the equation of the midline for y � 4 cos � � 1.

23. Find sec � if sin � � ��45

� and 270� � � � 360.

24. Simplify ��se1c ��

scions2

��

�� cos �.

For Questions 25 and 26, find the exact value of each expression.

25. cos 315� 26. sin 195�

27. Verify that cos ��2� � �� sin � is an identity.

For Questions 28 and 29, use the fact that cos � � �16� and

0� � � � 90� to find the exact value of each expression.

28. sin 2� 29. cos �2�

�

30. The profit P for a product whose sales fluctuate with the

seasons is estimated to be P � 21 � 6 sin �5�2t�, where t is

given in weeks and P is in thousands of dollars. Determine the number of weeks it would take for the profit to initially reach $25,000.

NAME DATE PERIOD

18.

19.

20.

21.

22.

23.

24.

25.

26.

27.

28.

29.

30.

y

O

2

�2

�

� 2�

y

O

2

4

�2

�4

�� 2�

Second Semester Test (Chapters 8–14)

NAME DATE PERIOD

SCORE


Ass

essm

ent


1. Find the distance between (2, 5) and (�4, 1).A. �34� B. 2�13� C. 4�2� D. 6�2� 1.

2. Write the equation 9x2 � 4y2 � 16y � 52 in standard form.

A. �x42� � �

(y �9

2)2� � 1 B. �

x42� � �

(y �9

2)2� � 1

C. �x42� � �

(y �9

2)2� � 1 D. �

x42� � �

(y �9

2)2� � 1 2.

3. Which system of inequalities is graphed?A. x2 � y2 � 16 B. x2 � y2 � 16

x2 � 16y2 � 16 16x2 � y2 � 16C. x2 � y2 � 16 D. x2 � y2 � 16 3.

16x2 � y2 � 16 �1x62� � y2 � 1

4. Simplify �45(tt2

��

34)52� �5

2tt��

165�.

A. �((tt

��

33))2

2� B. �(t �t2

3)�(t

9� 3)� C. �

12� D. �

25� 4.

5. Determine the values of x for any holes in the graph of the rational

function f(x) � �x2 �x

2�x

3� 15�.

A. x � �5, x � 3 B. x � �5 C. x � �3, x � 5 D. x � 3 5.

6. Solve �21m�

� �52m�

� �110�

.

A. m � 0 or m � 1 B. m � 1C. m � �1 or m � 0 D. m � �1 or m � 1 6.

7. Solve log16 n � �54�.

A. 32 B. 20 C. 8 D. 64 7.

8. Use log5 2 0.4307 and log5 3 0.6826 to approximate the value of log5 24.A. 0.7625 B. 0.2760 C. 0.6812 D. 1.9747 8.

9. Write an equivalent logarithmic equation for e3 � 6x.A. 3 � 6 ln x B. 3 � ln 6x C. 6x � ln 3 D. x � ln 2 9.

10. Evaluate 12

k�7(3k � 6).

A. 105 B. 165 C. 135 D. 162 10.

y

x

O


Second Semester Test (continued)(Chapters 8–14)

11. Find the next two terms of the geometric sequence 81, 54, 36, … .A. 54, 81 B. 9, �18 C. 18, 0 D. 24, 16 11.

12. Find the fifth term of the sequence in which a1 � 12 and an�1 � an � 2n.A. 24 B. 32 C. 42 D. 30 12.

13. A password has three letters followed by three digits. How many different passwords are possible?A. 12,812,904 B. 13,824,000 C. 11,232,000 D. 17,576,000 13.

14. The odds that an event will occur are 5:3. What is the probability that the event will not occur?

A. �38� B. �

58� C. �

35� D. �

52� 14.

15. On a geometry test, �15� of the students earned an A. Find the probability

that 4 of 5 randomly-selected students earned an A.

A. �31425� B. �6

425�

C. �6125�

D. �1125�

15.

16. In a survey of 550 residents, 42% favored the expansion of the town library.Find the margin of sampling error.A. 8% B. 2% C. 4% D. 6% 16.

17. In �ABC, a � 15, b � 25, and c � 30. Find C.A. 56� B. 30� C. 94� D. 98� 17.

18. Find the exact value of 4(cos 150�)(tan 120�).

A. ��33�

� B. �3� C. 2�3� D. 6 18.

19. Which equation is graphed?A. y � 4 cos 3� B. y � 3 cos 4�

C. y � 3 sin 4� D. y � 4 sin 3� 19.

20. Find csc � if cos � � ��27� and 90� � � � 180�.

A. �71�55�

� B. �3�

75�

� C. ��71�55�

� D. ��3�

75�

� 20.

NAME DATE PERIOD

y

O

2

4

�2

�4

�

� 2��2

3��2



21. Write an equation for the parabola with focus (2, 5) and 21.directrix y � 1.

22. Write an equation for a circle with center at (10, �3) and 22.

radius �15� unit.

23. Find the coordinates of the vertices and foci and the 23.equations of the asymptotes for the hyperbola with equation 9y2 � x2 � 9. Then graph the hyperbola.

24. Write the equation x2 � y2 � �2x � 2y � 23 in standard 24.form. Then state whether the graph of the equation is a parabola, circle, ellipse, or hyperbola.

25. Find the LCM of 4t � 20 and 6t � 30. 25.

26. State whether the equation �3p.1� � r represents a direct, joint, 26.

or inverse variation. Then name the constant of variation.

27. Solve �t �2

3� � �t2 �2t

2�t �

115� � �t �

65�. 27.

For Questions 28–30, solve each equation. 28.

28. ��215��m

� 625m�2 29. ln (2x � 1) � 2 29.

30. 4 log8 3 � �12� log8 9 � log8 x 30.

31. Express log7 32 in terms of common logarithms. Then 31.approximate its value to four decimal places.

32. The half-life of carbon-14 is 5760 years. A scientist 32.unearthed a fossil whose bones contained only 2% as much carbon-14 as they would have contained when the animal was alive. Find the constant k for carbon-14 for t in years,and write the equation for modeling this exponential decay.Then determine how long ago the animal died.

33. Find the three arithmetic means between �2 and 10. 33.

y

xO

NAME DATE PERIOD

Ass

essm

ent



34. Find a1 in a geometric series for which Sn � 153, an � �3, 34.

and r � ��14�.

35. Write 0.7�2� as a fraction. 35.

36. Use Pascal’s triangle to expand (3x � y)5. 36.

37. Find a counterexample to the statement 37.

12 � 22 � 32 � … � n2 � �n(5n

4� 1)�.

38. How many ways can you choose three books from a locker 38.containing seven books?

39. Elias, Alisa, and Drew each roll a die. What is the 39.probability that Elias rolls a 5, Alisa rolls an even number,and Drew does not roll a 1 or 2?

40. At a local gym with 800 members, 450 members take an 40.aerobics class, 200 members do weight training, and 125 members do both weight training and take an aerobics class.What is the probability that a randomly-selected member takes an aerobics class or does weight training?

41. Determine whether the data {2, 1, 5, 9, 2, 3, 1, 7, 3, 2, 4, 8, 41.3, 6, 4, 3} appear to be positively skewed, negatively skewed,or normally distributed.

42. On a multiple-choice quiz with eight questions, each 42.question has four answer choices. If Noreen randomly guesses at all eight questions, find P(more than 6 correct).

43. Find the exact values of the six trigonometric functions of � 43.if the terminal side of � in standard position contains the point (�8, �15).

44. Determine whether �ABC with A � 35�, a � 20, and b � 13 44.has no solution, one solution, or two solutions. Then, if possible, solve the triangle. Round measures of sides to the nearest tenth and measures of angles to the nearest degree.

45. Verify that �scisnc

��

ctaotn

��

� � cos2 � is an identity. 45.

46. Find the exact value of cos 2� if sin � � ��56� and 46.

180� � � � 270�.

NAME DATE PERIOD

Final Test (Chapters 1–14)

NAME DATE PERIOD

SCORE


Ass

essm

ent


1. The five fastest roller coasters in the world are Fujiyama (Japan),Goliath (CA), Millennium Force (OH), Steel Dragon 2000 (Japan), and Superman the Escape (CA). The speeds, in miles per hour, of the first four coasters are 83, 85, 92, and 95, respectively. How fast can Superman the Escape travel if the average speed of all five coasters is no more than 91 miles per hour? Source: World Almanac

A. no more than 100 mph B. at least 93 mphC. at least 100 mph D. no more than 93 mph 1.

2. Write an equation of the line that passes through (9, 6) and is perpendicular

to the line whose equation is y � ��13�x � 7.

A. y � ��13�x � 9 B. y � �3x � 33

C. y � 3x � 21 D. y � �13�x � 3 2.

3. Find x in the solution of the system 3x � y � 2 and 2x � 3y � 16.

A. 2 B. �4 C. �1181�

D. �1101�

3.

4. Find the coordinates of the vertices of the figure formed by y � x � 2,x � y � 6, and y � �2.A. (0, 0), (2, 4), (8, �2) B. (�4, �2), (2, 4), (8, �2)C. (�4, �2), (4, 2), (8, �2) D. (�2, �4), (2, 4), (8, �2) 4.

5. Solve � � � � � for y.

A. 1 B. 3 C. �3 D. �1 5.

6. The vertices of �ABC are A(�3, �4), B(�1, 3), and C(3, �2). The triangle is

rotated 90� counterclockwise. Use the rotation matrix � � to find the

coordinates of C .A. (�3, 2) B. (4, �3) C. (�3, �1) D. (2, 3) 6.

7. Simplify �yy2

2��

y2y

��208�. Assume that the denominator is not equal to 0.

A. �yy

��

52� B. �

yy

��

52� C. �

52� D. �

yy��

140

� 7.

8. Simplify �12��

ii�.

A. �13� � �

23�i B. �

15� � �

25�i C. �

13� � i D. �

15� � �

35�i 8.

0 �11 0

10

2x � 5yx � 3y


Final Test (continued)(Chapters 1–14)

9. Solve 3x2 � 8x � 4 � 0 by graphing. If exact roots cannot be found, state the consecutive integers between which the roots are located.A. 2; between 0 and 1 B. between 0 and 1; between 7 and 8C. 1, 2 D. between 0 and 1; between 3 and 4 9.

10. Find the exact solutions to 6x2 � 1 � �8x by using the Quadratic Formula.

A. �4 � �10� B. ��4 �6

�22�� C. ��2 �

32�10�� D. ��4 �

6�10�� 10.

11. State the degree of 9 � 4x2 � 6x3 � x4 � 7x.A. 9 B. 1 C. 4 D. 10 11.

12. Which describes the number and type of roots of the equation x4 � 625 � 0?A. 1 real root, 1 imaginary root B. 2 real roots, 2 imaginary rootsC. 2 real roots D. 4 real roots 12.

13. If g(x) � 3x � 8, find g[g( � 4)].A. �68 B. 4 C. �20 D. 52 13.

14. Which equation is graphed?A. y � �x2 � 2x � 1B. x � �y2 � 2y � 1C. y � x2 � 2x � 1D. x � y2 � 2y � 1 14.

15. Write an equation for an ellipse if the endpoints of the major axis are at (�8, 1) and (8, 1) and the endpoints of the minor axis are at (0, �1) and (0, 3).

A. �1x62� � �

(y �4

1)2� � 1 B. �

(x �64

1)2� � �

y42� � 1

C. �(x �

161)2� � �

y42� � 1 D. �6

x42� � �

(y �4

1)2� � 1 15.

16. Find the exact solution(s) of the system �x42� � y2 � 1 and x � y2 � 1.

A. (4, �3�), (4, ��3�), (�4, �3�), (�4, ��3�)B. (4, �3�), (�4, �3�)C. (2, 1), (2, �1), (4, �3�), (4, ��3�)D. (4, �3�), (4, ��3�) 16.

17. Simplify �54nm

2� � �2nm�

.

A. �8m10

2

n�

2m5n3

� B. �8m10

2

n�

2m5n3

� C. �54nm2 �

�2nm� D. �5

2n2� 17.

NAME DATE PERIOD

y

xO


NAME DATE PERIOD


Ass

essm

ent

18. If y varies inversely as x and y � 6 when x � 3, find y when x � 36.

A. 72 B. 2 C. �12� D. 18 18.

19. Write the equation 4�3 � �614�

in logarithmic form.

A. log 64 � 43 B. log�3 64 � 4

C. log4 �614�

� �3 D. log4 (�3) � 64 19.

20. Solve 6n�1 � 10. Round to four decimal places.A. n � 0.2851 B. n � 0.6667 C. n � 1.2851 D. n � �0.7782 20.

21. Find Sn for the arithmetic series in which a1 � 29, n � 17, and an � 131.A. 2720 B. 1360 C. 177 D. 160 21.

22. Find the sum of the infinite geometric series 1 � �35� � �2

95�

� … , if it exists.

A. �53� B. �

52� C. �

35� D. does not exist 22.

23. Use the Binomial Theorem to find the sixth term in the expansion of (m � 2p)7.A. 21m2p5 B. 672m2p5 C. 32m2p5 D. 448mp6 23.

24. How many four-digit numerical codes can be created if no digit may be repeated?A. 10,000 B. 24 C. 3024 D. 5040 24.

25. A bookshelf holds 4 mysteries, 3 biographies, 1 book of poetry, and 2 reference books. If a book is selected at random from the shelf, find the probability that the book selected is a biography or reference book.

A. �12� B. �

16� C. �

56� D. �5

30�

25.

26. Find the standard deviation of the data set to the nearest tenth.{21, 13, 18, 16, 13, 35, 12, 8, 15}A. 16.8 B. 7.8 C. 7.3 D. 5.7 26.

27. Rewrite 100� in radian measure.

A. �59� B. �

59�� C. �

190� D. �

109

�� 27.

28. Find the exact value of sin 165�.

A. B. C. D. 28.��6� �� 2��

�2� � �6��4

�6� � �2��4

�6� � �2��4



29. Solve 5 � 2a � 5 � � 4 � 6 and graph the solution set. 29.

For Questions 30 and 31, use the data in the table below that shows the relationship between the distance traveled and the elapsed time for a trip.

30. Draw a scatter plot for the data. 30.

31. Use two ordered pairs to write a prediction equation. Then 31.use your prediction equation to predict the distance traveled in an elapsed time of 6 hours.

32. Classify the system x � 9y � 10 and 2x � y � 1 as consistent 32.and independent, consistent and dependent, or inconsistent.

For Questions 33 and 34, use the following information.A manufacturer produces badminton and tennis rackets. The profit on each badminton racket is $10 and on each tennis racket is $25. The manufacturer can make at most 600 rackets. Of these, at least 100 rackets must be badminton rackets.

33. Let b represent the number of badminton rackets and 33.t represent the number of tennis rackets. Write a system of inequalities to represent the number of rackets that can be produced.

34. How many tennis rackets should the manufacturer produce 34.to maximize profit?

35. Solve the system of equations. 2x � y � 3z � 9 35.x � 2y � z � �8x � 3y � 2z � 11

36. Perform the indicated operations. If the matrix does not 36.exist, write impossible.

� � � � � 4� �

37. Evaluate � � using expansion by minors. 37.3 4 02 5 �10 3 �7

�5 12 �1

�4 20 �3

�5 1

2 �1 33 0 �4

d

tO

75

150

225

1 2 3 4Time (h)

Dis

tan

ce (

mi)

0 1�3 �2 �1�4

� 72 � 5

2 � 32

12

32

NAME DATE PERIOD

Time t (h) 0 1 2 3

Distance d (mi) 0 55 100 150 260

4



38. Find the inverse of M � � �, if it exists. 38.

39. Simplify �(3x2y0)2 � �x1�1��(2x2 � 5). Assume that no variable 39.

equals 0.

40. Simplify �3 �5�6��. 40.

41. Write the radical �327t8u6� using rational exponents. 41.

42. Solve �2x � 7� � 2 � 5. 42.

43. Write a quadratic equation with �23� and �3 as its roots. 43.

Write the equation in the form ax2 � bx � c � 0, where a, b, and c are integers.

44. Write the equation y � 4x2 � 16x � 7 in vertex form. 44.

45. Use synthetic substitution to find f(�4) for 45.f(x) � 2x3 � 5x2 � 3x � 8.

46. List all of the possible rational zeros of 46.f(x) � 3x4 � 5x3 � 2x � 12.

47. Find the inverse of the function g(x) � 2x � 1. 47.

48. Graph y � �2x � 6�. 48.

49. Write an equation for a circle if the endpoints of a diameter 49.are at (�1, �5) and (5, 3).

50. Write an equation for the hyperbola with vertices (0, 4) and 50.(0, �4) if the length of the conjugate axis is 6 units.

51. Write the equation y � 12x � 3x2 � 19 in standard form. 51.Then state whether the graph of the equation is a parabola,circle, ellipse, or hyperbola.

y

xO

1 5�2 0

NAME DATE PERIOD

Ass

essm

ent



52. Simplify . 52.

53. Determine the equations of any vertical asymptotes and the 53.

values of x for any holes in the graph of f(x) � �x2 �x �

x �3

12�.

54. Solve �mm��

43� � �

mm

��

43� � �m

2� 3�. 54.

55. Solve log5 n � �14� log5 81 � �

12� log5 64. 55.

56. In a certain lake, it is estimated that the fish population has 56.been doubling in size every 80 weeks. Write an exponential growth equation of the form y � aekt that models the growth of the fish population, where t is given in weeks, if the initial population was 5000.

57. Find the eighth term of the arithmetic sequence in which 57.a1 � �4 and d � 7.

58. Find the sum of the geometric series for which a1 � 2058, 58.

a4 � 6, and r � �17�.

59. Find the first three iterates x1, x2, x3 of f(x) � 7x � 3 for an 59.initial value x0 � 0.

60. How many different ways can the letters of the word 60.AMERICA be arranged?

61. Three students are selected from a group of four male 61.students and six female students. Find the probability of selecting a male, a female, and another female in that order.

62. The heights of a group of high school students were found 62.to be normally distributed. The mean height was 65 inches and the standard deviation was 2.5 inches. What percent of the students were between 65 inches and 70 inches tall?

63. In �ABC, A � 25�, a � 7, and b � 4. Determine whether 63.the triangle has no solution, one solution, or two solutions.Then solve the triangle. Round measure of sides to the nearest tenth and measures of angles to the nearest degree.

64. Find the value of cot �Cos�1 ��22�

��. 64.

�f �

6g

�

��f2 �

2g2

�

NAME DATE PERIOD

Standardized Test PracticeStudent Record Sheet (Use with pages 810–811 of the Student Edition.)

© Glencoe/McGraw-Hill A1 Glencoe Algebra 2

NAME DATE PERIOD

1414

An

swer

s

Select the best answer from the choices given and fill in the corresponding oval.

1 4 7 9

2 5 8 10

3 6

Solve the problem and write your answer in the blank.

For Questions 13–19, also enter your answer by writing each number or symbol ina box. Then fill in the corresponding oval for that number or symbol.

11 14 16 18

12

13 15 17 19

Select the best answer from the choices given and fill in the corresponding oval.

20 22 24

21 23 DCBADCBA

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DCBADCBA

DCBADCBADCBADCBA

DCBADCBADCBADCBA

Part 2 Short Response/Grid InPart 2 Short Response/Grid In

Part 1 Multiple ChoicePart 1 Multiple Choice

Part 3 Quantitative ComparisonPart 3 Quantitative Comparison


Answers (Lesson 14-1)

Stu

dy G

uid

e a

nd I

nte

rven

tion

Gra

ph

ing

Tri

go

no

met

ric

Fu

nct

ion

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

14-1

14-1

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ra 2

Lesson 14-1

Gra

ph

Tri

go

no

met

ric

Fun

ctio

ns

To

grap

h a

tri

gon

omet

ric

fun

ctio

n,m

ake

a ta

ble

ofva

lues

for

kn

own

deg

ree

mea

sure

s (0

�,30

�,45

�,60

�,90

�,an

d so

on

).R

oun

d fu

nct

ion

val

ues

to

the

nea

rest

ten

th,a

nd

plot

th

e po

ints

.Th

en c

onn

ect

the

poin

ts w

ith

a s

moo

th,c

onti

nu

ous

curv

e.T

he

peri

odof

th

e si

ne,

cosi

ne,

seca

nt,

and

cose

can

t fu

nct

ion

s is

360

�or

2�

radi

ans.

Am

plit

ud

e o

f a

Fu

nct

ion

The

am

plit

ud

eof

the

gra

ph o

f a

perio

dic

func

tion

is t

he a

bsol

ute

valu

e of

hal

f th

edi

ffere

nce

betw

een

its m

axim

um a

nd m

inim

um v

alue

s.

Gra

ph

y�

sin

�fo

r �

360�

��

�0�

.F

irst

mak

e a

tabl

e of

val

ues

.

Gra

ph

th

e fo

llow

ing

fun

ctio

ns

for

the

give

n d

omai

n.

1.co

s �,�

360�

��

�0�

2.ta

n �

,�2�

��

�0

Wh

at i

s th

e am

pli

tud

e of

eac

h f

un

ctio

n?

3.4

4.8

x

y

O2

2

x

y

O

y

O

�2

�44 2

�

y �

tan

�

�2�

�3� 2

��

�� 2

y

O

�11

��

90�

�18

0��

270�

�36

0�

y �

cos

�

y O �0.

5

�1.

0

1.0

0.5

��

90�

�18

0��

270�

�36

0�

y �

sin

�

��

360°

�33

0°�

315°

�30

0°�

270°

�24

0°�

225°

�21

0°�

180°

sin

�0

�1 2�� 22 � �

�� 23 � �1

�� 23 � �� 22 � �

�1 2�0

��

150°

�13

5°�

120°

�90

°�

60°

�45

°�

30°

0°

sin

��

�1 2��

�� 22 � ��

�� 23 � ��

1�

�� 23 � ��

�� 22 � ��

�1 2�0

Exam

ple

Exam

ple

Exer

cises

Exer

cises

©G

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ra 2

Var

iati

on

s o

f Tr

igo

no

met

ric

Fun

ctio

ns

For

fun

ctio

ns o

f th

e fo

rm y

�a

sin

b�

and

y�

aco

s b

�,

the

ampl

itude

is |a

|,

Am

plit

ud

esan

d th

e pe

riod

is

or

.

and

Per

iod

sF

or f

unct

ions

of

the

form

y�

ata

n b

�,

the

ampl

itude

is n

ot d

efin

ed,

and

the

perio

d is

or

.

Fin

d t

he

amp

litu

de

and

per

iod

of

each

fu

nct

ion

.Th

en g

rap

h t

he

fun

ctio

n.

� � | b|

180°

�| b

|

2� � | b|

360°

�| b

|

Stu

dy G

uid

e a

nd I

nte

rven

tion

(c

onti

nued

)

Gra

ph

ing

Tri

go

no

met

ric

Fu

nct

ion

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

14-1

14-1

Exam

ple

Exam

ple

a.y

�4

cos

� 3� �

Fir

st,f

ind

the

ampl

itu

de.

|a|�

|4|,s

o th

e am

plit

ude

is

4.N

ext

fin

d th

e pe

riod

.

�10

80�

Use

th

e am

plit

ude

an

d pe

riod

to

hel

pgr

aph

th

e fu

nct

ion

.y

O4 2

�2

�4

�72

0�54

0�10

80�

900�

360�

180�y

� 4

cos

� – 3

360°

�

��1 3� �

b.

y�

��1 2�

tan

2�

Th

e am

plit

ude

is

not

def

ined

,an

d th

e pe

riod

is

�� 2� .

y O� 4

2 –2 –44

� 23� 4

��

Exer

cises

Exer

cises

Fin

d t

he

amp

litu

de,

if i

t ex

ists

,an

d p

erio

d o

f ea

ch f

un

ctio

n.T

hen

gra

ph

eac

hfu

nct

ion

.

1.y

��

3 si

n �

2.y

�2

tan

� 2� �

amp

litu

de:

3;p

erio

d 2

�o

r 36

0�n

o a

mp

litu

de;

per

iod

2�

or

360�

y

O

�22

�2�

3� 2�

3�5� 2

� 2

y

O2

�2

�36

0�27

0�18

0�90

�


An

swer

s


Skil

ls P

ract

ice

Gra

ph

ing

Tri

go

no

met

ric

Fu

nct

ion

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

14-1

14-1

©G

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Lesson 14-1

Fin

d t

he

amp

litu

de,

if i

t ex

ists

,an

d p

erio

d o

f ea

ch f

un

ctio

n.T

hen

gra

ph

eac

hfu

nct

ion

.

1.y

�2

cos

�2.

y�

4 si

n �

3.y

�2

sec

�

2;36

0�4;

360�

no

am

plit

ud

e;36

0�

4.y

��1 2�

tan

�5.

y�

sin

3�

6.y

�cs

c 3�

no

am

plit

ud

e;18

0�1;

120�

no

am

plit

ud

e;12

0�

7.y

�ta

n 2

�8.

y�

cos

2�9.

y�

4 si

n �1 2� �

no

am

plit

ud

e;90

�1;

180�

4;72

0� y

O4 2

�2

�4

�72

0�54

0�36

0�18

0�

y

O2 1

�1

�2

�18

0�13

5�90

�45

�

y

O4 2

�2

�4

�18

0�13

5�90

�45

�

y

O4 2

�2

�4

�30

�90

�15

0�

y

O2 1

�1

�2

�36

0�27

0�18

0�90

�

y

O2 1

�1

�2

�36

0�27

0�18

0�90

�

y

O4 2

�2

�4

�36

0�27

0�18

0�90

�

y

O4 2

�2

�4

�36

0�27

0�18

0�90

�

y

O2 1

�1

�2

�36

0�27

0�18

0�90

�

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0G

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lgeb

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Fin

d t

he

amp

litu

de,

if i

t ex

ists

,an

d p

erio

d o

f ea

ch f

un

ctio

n.T

hen

gra

ph

eac

hfu

nct

ion

.

1.y

��

4 si

n �

2.y

�co

t �1 2� �

3.y

�co

s 5�

4;36

0�n

o a

mp

litu

de;

360�

1;72

�

4.y

�cs

c �3 4� �

5.y

�2

tan

�1 2� �6.

2y�

sin

��1 2� ;

360�

no

am

plit

ud

e;48

0�n

o a

mp

litu

de;

360�

FOR

CE

For

Exe

rcis

es 7

an

d 8

,use

th

e fo

llow

ing

info

rmat

ion

.A

n a

nch

orin

g ca

ble

exer

ts a

for

ce o

f 50

0 N

ewto

ns

on a

pol

e.T

he

forc

e h

asth

e ho

rizo

ntal

and

ver

tica

l com

pone

nts

F xan

d F

y.(A

for

ce o

f on

e N

ewto

n (N

),is

th

e fo

rce

that

giv

es a

n a

ccel

erat

ion

of

1 m

/sec

2to

a m

ass

of 1

kg.

)

7.T

he

fun

ctio

n F

x�

500

cos

�de

scri

bes

the

rela

tion

ship

bet

wee

n t

he

angl

e �

and

the

hor

izon

tal

forc

e.W

hat

are

th

e am

plit

ude

an

d pe

riod

of

th

is f

un

ctio

n?

500;

360�

8.T

he

fun

ctio

n F

y�

500

sin

�de

scri

bes

the

rela

tion

ship

bet

wee

n t

he

angl

e �

and

the

vert

ical

for

ce.W

hat

are

th

e am

plit

ude

an

d pe

riod

of

this

fu

nct

ion

?50

0;36

0�

WEA

THER

For

Exe

rcis

es 9

an

d 1

0,u

se t

he

foll

owin

g in

form

atio

n.

Th

e fu

nct

ion

y�

60 �

25 s

in �� 6� t

,wh

ere

tis

in

mon

ths

and

t�

0 co

rres

pon

ds t

o A

pril

15,

mod

els

the

aver

age

hig

h t

empe

ratu

re i

n d

egre

es F

ahre

nh

eit

in C

ente

rvil

le.

9.D

eter

min

e th

e pe

riod

of

this

fu

nct

ion

.Wh

at d

oes

this

per

iod

repr

esen

t?12

;a

cale

nd

ar y

ear

10.W

hat

is

the

max

imu

m h

igh

tem

pera

ture

an

d w

hen

doe

s th

is o

ccu

r?85

�F;

July

15

�500

NF y

F x

y

O

1.0

0.5

�0.

5

�1.

0

�36

0�27

0�18

0�90

�

y

O4 2

�2

�4

�72

0�54

0�36

0�18

0�

y

O4 2

�2

�4

�48

0�36

0�24

0�12

0�

y

O1

�1

�18

0�13

5�90

�45

�

y

O4 2

�2

�4

�36

0�27

0�18

0�90

�

y

O4 2

�2

�4

�36

0�27

0�18

0�90

�Pra

ctic

e (

Ave

rag

e)

Gra

ph

ing

Tri

go

no

met

ric

Fu

nct

ion

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

14-1

14-1



Readin

g t

o L

earn

Math

em

ati

csG

rap

hin

g T

rig

on

om

etri

c F

un

ctio

ns

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

14-1

14-1

©G

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Lesson 14-1

Pre-

Act

ivit

yW

hy

can

you

pre

dic

t th

e b

ehav

ior

of t

ides

?

Rea

d th

e in

trod

ucti

on t

o L

esso

n 14

-1 a

t th

e to

p of

pag

e 76

2 in

you

r te

xtbo

ok.

Con

side

r th

e ti

des

of t

he

Atl

anti

c O

cean

as

a fu

nct

ion

of

tim

e.A

ppro

xim

atel

y w

hat

is

the

peri

od o

f th

is f

un

ctio

n?

12 h

ou

rs

Rea

din

g t

he

Less

on

1.D

eter

min

e w

het

her

eac

h s

tate

men

t is

tru

eor

fal

se.

a.T

he

peri

od o

f a

fun

ctio

n i

s th

e di

stan

ce b

etw

een

th

e m

axim

um

an

d m

inim

um

poi

nts

.fa

lse

b.

Th

e am

plit

ude

of

a fu

nct

ion

is

the

diff

eren

ce b

etw

een

its

max

imu

m a

nd

min

imu

mva

lues

.fa

lse

c.T

he

ampl

itu

de o

f th

e fu

nct

ion

y�

sin

�is

2�

.fa

lse

d.

Th

e fu

nct

ion

y�

cot

�h

as n

o am

plit

ude

.tr

ue

e.T

he

peri

od o

f th

e fu

nct

ion

y�

sec

�is

�.

fals

e

f.T

he

ampl

itu

de o

f th

e fu

nct

ion

y�

2 co

s �

is 4

.fa

lse

g.T

he

fun

ctio

n y

�si

n 2

�h

as a

per

iod

of �

.tr

ue

h.

Th

e pe

riod

of

the

fun

ctio

n y

�co

t 3�

is �� 3� .

tru

e

i.T

he

ampl

itu

de o

f th

e fu

nct

ion

y�

�5

sin

�is

�5.

fals

e

j.T

he

peri

od o

f th

e fu

nct

ion

y�

csc

�1 4� �is

4�

.fa

lse

k.

Th

e gr

aph

of

the

fun

ctio

n y

�si

n �

has

no

asym

ptot

es.

tru

e

l.T

he

grap

h o

f th

e fu

nct

ion

y�

tan

�h

as a

n a

sym

ptot

e at

��

180�

.fa

lse

m.W

hen

��

360�

,th

e va

lues

of

cos

�an

d se

c �

are

equ

al.

tru

e

n.

Wh

en �

�27

0�,c

ot �

is u

nde

fin

ed.

fals

e

o.W

hen

��

180�

,csc

�is

un

defi

ned

.tr

ue

Hel

pin

g Y

ou

Rem

emb

er2.

Wh

at i

s an

eas

y w

ay t

o re

mem

ber

the

peri

ods

of y

�a

sin

b�

and

y�

aco

s b�

?S

amp

lean

swer

:Th

e p

erio

d o

f th

e fu

nct

ion

s y

�si

n �

and

y�

cos

�is

360

�o

r 2�

.D

ivid

e 36

0�o

r 2�

by t

he

abso

lute

val

ue

of

the

coef

fici

ent

of

�,d

epen

din

go

n w

het

her

yo

u w

ant

to f

ind

th

e p

erio

d in

deg

rees

or

in r

adia

ns.

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Blu

epri

nts

Inte

rpre

tin

g bl

uep

rin

ts r

equ

ires

th

e ab

ilit

y to

sel

ect

and

use

tri

gon

omet

ric

fun

ctio

ns

and

geom

etri

c pr

oper

ties

.Th

e fi

gure

bel

ow r

epre

sen

ts a

pla

n f

or a

nim

prov

emen

t to

a r

oof.

Th

e m

etal

fit

tin

g sh

own

mak

es a

30�

angl

e w

ith

th

eh

oriz

onta

l.T

he

vert

ices

of

the

geom

etri

c sh

apes

are

not

labe

led

in t

hes

epl

ans.

Rel

evan

t in

form

atio

n m

ust

be

sele

cted

an

d th

e ap

prop

riat

e fu

nct

ion

use

d to

fin

d th

e u

nkn

own

mea

sure

s.

Fin

d t

he

un

kn

own

m

easu

res

in t

he

figu

re a

t th

e ri

ght.

Th

e m

easu

res

xan

d y

are

the

legs

of

a ri

ght

tria

ngl

e.

Th

e m

easu

re o

f th

e h

ypot

enu

se

is �1 15 6�

in.�

� 15 6�in

.or

�2 10 6�in

.

�co

s 30

��

sin

30�

y�

1.08

in.

x�

0.63

in.

Fin

d t

he

un

kn

own

mea

sure

s of

eac

h o

f th

e fo

llow

ing.

1.C

him

ney

on

roo

f2.

Air

ven

t3.

Elb

ow jo

int

y�

3.78

��

C�

63.4

3��

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40�

x�

5.72

��

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26.5

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50�

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63�

r�

4.87

�

B

A 4'

t

r

1' – 47

40°

D

C

1' – 43 1' – 4

1

2'

1' – 21

x

y

A1' – 24

1' – 29 40

°

x � �2 10 6�

y � �2 10 6�

5"

–– 16

15"

––16

13"

–– 16

5"

–– 16

x

y0.

09"

top

view

side

vie

w

met

al fi

tting

Roofi

ng Im

pro

vem

ent

30°

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

14-1

14-1

Exam

ple

Exam

ple


An

swer

s


Stu

dy G

uid

e a

nd I

nte

rven

tion

Tran

slat

ion

s o

f Tri

go

no

met

ric

Gra

ph

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

14-2

14-2

©G

lenc

oe/M

cGra

w-H

ill84

3G

lenc

oe A

lgeb

ra 2

Lesson 14-2

Ho

rizo

nta

l Tra

nsl

atio

ns

Wh

en a

con

stan

t is

su

btra

cted

fro

m t

he

angl

e m

easu

re i

n a

trig

onom

etri

c fu

nct

ion

,a p

has

e sh

ift

of t

he

grap

h r

esu

lts.

The

hor

izon

tal p

hase

shi

ft of

the

gra

phs

of t

he f

unct

ions

y�

asi

n b

(��

h),

y�

aco

s b

(��

h),

Ph

ase

Sh

ift

and

y�

ata

n b

(��

h) is

h,

whe

re b

�0.

If h

�0,

the

shi

ft is

to

the

right

.If

h

0, t

he s

hift

is t

o th

e le

ft.

Sta

te t

he

amp

litu

de,

per

iod

,an

d

ph

ase

shif

t fo

r y

��1 2�

cos

3��

�� 2� �.

Th

en g

rap

h

the

fun

ctio

n.

Am

plit

ude

:a�

|�1 2�|or

�1 2�

Per

iod:

�or

�2 3� �

Ph

ase

Sh

ift:

h�

�� 2�

Th

e ph

ase

shif

t is

to

the

righ

t si

nce

�� 2��

0.

Sta

te t

he

amp

litu

de,

per

iod

,an

d p

has

e sh

ift

for

each

fu

nct

ion

.Th

en g

rap

h t

he

fun

ctio

n.

1.y

�2

sin

(�

�60

�)2.

y�

tan

��

�� 2� �2;

360�

;60

�to

th

e le

ftn

o a

mp

litu

de;

�;

�� 2�to

th

e ri

gh

t

3.y

�3

cos

(��

45�)

4.y

��1 2�

sin

3��

�� 3� �

3;36

0�;

45�

to t

he

rig

ht

�1 2� ;�2 3� �

;�� 3�

to t

he

rig

ht

y

O�

0.5

�1.

0

1.0

0.5

�2� 3

� 6� 3

� 25� 6

�

y

O2

�2

�36

0�45

0�27

0�18

0�90

�

y

O

�22

�2�

3� 2�

� 2

y

O2

�2

�36

0��

90�

270�

180�

90�

2� � | 3|

2� � | b|

y O�

0.5

�1.

0

1.0

0.5

�2� 3

� 6� 3

� 25� 6

�

Exam

ple

Exam

ple

Exer

cises

Exer

cises

©G

lenc

oe/M

cGra

w-H

ill84

4G

lenc

oe A

lgeb

ra 2

Ver

tica

l Tra

nsl

atio

ns

Wh

en a

con

stan

t is

add

ed t

o a

trig

onom

etri

c fu

nct

ion

,th

e gr

aph

is s

hif

ted

vert

ical

ly.

The

ver

tical

shi

ft of

the

gra

phs

of t

he f

unct

ions

y�

asi

n b

(��

h) �

k, y

�a

cos

b(�

�h)

�k,

Ver

tica

l Sh

ift

and

y�

ata

n b

(��

h) �

kis

k.

If k

�0,

the

shi

ft is

up.

If k

0,

the

shi

ft is

dow

n.

Th

e m

idli

ne

of a

ver

tica

l sh

ift

is y

�k.

Ste

p 1

Det

erm

ine

the

vert

ical

shi

ft, a

nd g

raph

the

mid

line.

Gra

ph

ing

Ste

p 2

Det

erm

ine

the

ampl

itude

, if

it ex

ists

. U

se d

ashe

d lin

es t

o in

dica

te t

he m

axim

um a

ndTr

igo

no

met

ric

min

imum

val

ues

of t

he f

unct

ion.

Fu

nct

ion

sS

tep

3D

eter

min

e th

e pe

riod

of t

he f

unct

ion

and

grap

h th

e ap

prop

riate

fun

ctio

n.S

tep

4D

eter

min

e th

e ph

ase

shift

and

tra

nsla

te t

he g

raph

acc

ordi

ngly

.

Sta

te t

he

vert

ical

sh

ift,

equ

atio

n o

f th

e m

idli

ne,

amp

litu

de,

and

per

iod

for

y�

cos

2��

3.T

hen

gra

ph

th

e fu

nct

ion

.V

erti

cal

Sh

ift:

k�

�3,

so t

he

vert

ical

sh

ift

is 3

un

its

dow

n.

Th

e eq

uat

ion

of

the

mid

lin

e is

y�

�3.

Am

plit

ude

:|a

| �| 1

| or

1

Per

iod:

�or

�

Sin

ce t

he

ampl

itu

de o

f th

e fu

nct

ion

is

1,dr

aw d

ash

ed l

ines

para

llel

to

the

mid

lin

e th

at a

re 1

un

it a

bove

an

d be

low

th

e m

idli

ne.

Th

en d

raw

th

e co

sin

e cu

rve,

adju

sted

to

hav

e a

peri

od o

f �

.

Sta

te t

he

vert

ical

sh

ift,

equ

atio

n o

f th

e m

idli

ne,

amp

litu

de,

and

per

iod

for

eac

hfu

nct

ion

.Th

en g

rap

h t

he

fun

ctio

n.

1.y

��1 2�

cos

��

22.

y�

3 si

n �

�2

2 u

p;

y �

2;�1 2� ;

2�2

do

wn

;y

��

2;3;

2�y

O�

1�

2�

3�

4�

5�

61

�3� 2

� 2�

2�

y

O�

1�

23 2 1

�3� 2

� 2�

2�

2� � | 2|

2� � | b|

y

O�

12 1

�3� 2

� 2�

2�

Stu

dy G

uid

e a

nd I

nte

rven

tion

(c

onti

nued

)

Tran

slat

ion

s o

f Tri

go

no

met

ric

Gra

ph

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

14-2

14-2

Exam

ple

Exam

ple

Exer

cises

Exer

cises



Skil

ls P

ract

ice

Tran

slat

ion

s o

f Tri

go

no

met

ric

Gra

ph

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

14-2

14-2

©G

lenc

oe/M

cGra

w-H

ill84

5G

lenc

oe A

lgeb

ra 2

Lesson 14-2

Sta

te t

he

amp

litu

de,

per

iod

,an

d p

has

e sh

ift

for

each

fu

nct

ion

.Th

en g

rap

h t

he

fun

ctio

n.

1.y

�si

n (

��

90�)

2.y

�co

s (�

�45

�)3.

y�

tan

��

�� 2� �1;

360�

;�

90�

1;36

0�;

45�

no

am

plit

ud

e;�

;�� 2�

Sta

te t

he

vert

ical

sh

ift,

equ

atio

n o

f th

e m

idli

ne,

amp

litu

de,

and

per

iod

for

eac

hfu

nct

ion

.Th

en g

rap

h t

he

fun

ctio

n.

4.y

�cs

c �

�2

5.y

�co

s �

�1

6.y

�se

c �

�3

3;y

�3;

�2;

y�

�2;

1;36

0�1;

y�

1;1;

360�

no

am

plit

ud

e;36

0�

Sta

te t

he

vert

ical

sh

ift,

amp

litu

de,

per

iod

,an

d p

has

e sh

ift

of e

ach

fu

nct

ion

.Th

engr

aph

th

e fu

nct

ion

.

7.y

�2

cos

[3(�

�45

�)]

�2

8.y

�3

sin

[2(

��

90�)

] �

29.

y�

4 co

t ��4 3� ��

�� 4� ��

�2

2;2;

120�

;�

45�

2;3;

180�

;90

��

2;n

o a

mp

litu

de;

�3 4� �;�

�� 4�

�2

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3� 2�

� 2

y

�2

�44 2

y

O6 4 2

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�

y

O6 4 2

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0�27

0�18

0�90

�

y

O6 4 2

�2

�36

0�27

0�18

0�90

�

y

O2 1

�1

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0�54

0�36

0�18

0�

y

O2

�2

�4

�6

�72

0�54

0�36

0�18

0�

�2

�2O

�2�

3� 2�

� 2

y

�2

�44 2

y

O2 1

�1

�2

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0�18

0�90

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y

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lenc

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cGra

w-H

ill84

6G

lenc

oe A

lgeb

ra 2

Sta

te t

he

vert

ical

sh

ift,

amp

litu

de,

per

iod

,an

d p

has

e sh

ift

for

each

fu

nct

ion

.Th

engr

aph

th

e fu

nct

ion

.

1.y

��1 2�

tan

��

�� 2� �2.

y�

2 co

s (�

�30

�) �

33.

y�

3 cs

c (2

��

60�)

�2.

5

no

ver

tica

l sh

ift;

no

3;

2;36

0;�

30�

�2.

5;n

o a

mp

litu

de;

amp

litu

de;

�;

180�

;�

60�

ECO

LOG

YF

or E

xerc

ises

4–6

,use

th

e fo

llow

ing

info

rmat

ion

.T

he

popu

lati

on o

f an

in

sect

spe

cies

in

a s

tan

d of

tre

es f

ollo

ws

the

grow

th c

ycle

of

apa

rtic

ula

r tr

ee s

peci

es.T

he

inse

ct p

opu

lati

on c

an b

e m

odel

ed b

y th

e fu

nct

ion

y

�40

�30

sin

6t,

wh

ere

tis

th

e n

um

ber

of y

ears

sin

ce t

he

stan

d w

as f

irst

cu

t in

Nov

embe

r,19

20.

4.H

ow o

ften

doe

s th

e in

sect

pop

ula

tion

rea

ch i

ts m

axim

um

lev

el?

ever

y 60

yr

5.W

hen

did

th

e po

pula

tion

las

t re

ach

its

max

imu

m?

1995

6.W

hat

cond

itio

n in

the

sta

nd d

o yo

u th

ink

corr

espo

nds

wit

h a

min

imum

ins

ect

popu

lati

on?

Sam

ple

an

swer

:Th

e sp

ecie

s o

n w

hic

h t

he

inse

ct f

eed

s h

as b

een

cu

t.

BLO

OD

PR

ESSU

RE

For

Exe

rcis

es 7

–9,u

se t

he

foll

owin

g in

form

atio

n.

Jaso

n’s

bloo

d pr

essu

re is

110

ove

r 70

,mea

ning

tha

t th

e pr

essu

re o

scill

ates

bet

wee

n a

max

imum

of 1

10 a

nd

a m

inim

um

of

70.J

ason

’s h

eart

rat

e is

45

beat

s pe

r m

inu

te.T

he

fun

ctio

n t

hat

repr

esen

ts J

ason

’s b

lood

pre

ssur

e P

can

be m

odel

ed u

sing

a s

ine

func

tion

wit

h no

pha

se s

hift

.

7.F

ind

the

ampl

itu

de,m

idli

ne,

and

peri

od i

n s

econ

ds o

f th

e fu

nct

ion

.20

;P

�90

;1�

1 3�s

8.W

rite

a f

un

ctio

n t

hat

rep

rese

nts

Jas

on’s

blo

od

pres

sure

Paf

ter

tse

con

ds.

P�

20 s

in 2

70t

�90

9.G

raph

th

e fu

nct

ion

.

Tim

e

Jaso

n’s

Blo

od

Pre

ssu

re

Pressure

20

46

13

57

89

120

100 80 60 40 20

P

t

y

O4

�4

�8

�12

�36

0�27

0�18

0�90

�

y

O6 4 2

�2

�72

0�54

0�36

0�18

0��

2�

2O�

2�3� 2

�� 2

y

�2

�44 2

� � 2

Pra

ctic

e (

Ave

rag

e)

Tran

slat

ion

s o

f Tri

go

no

met

ric

Gra

ph

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

14-2

14-2


An

swer

s


Readin

g t

o L

earn

Math

em

ati

csTr

ansl

atio

ns

of T

rig

on

om

etri

c G

rap

hs

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

14-2

14-2

©G

lenc

oe/M

cGra

w-H

ill84

7G

lenc

oe A

lgeb

ra 2

Lesson 14-2

Pre-

Act

ivit

yH

ow c

an t

ran

slat

ion

s of

tri

gon

omet

ric

grap

hs

be

use

d t

o sh

owan

imal

pop

ula

tion

s?

Rea

d th

e in

trod

ucti

on t

o L

esso

n 14

-2 a

t th

e to

p of

pag

e 76

9 in

you

r te

xtbo

ok.

Acc

ordi

ng

to t

he

mod

el g

iven

in

you

r te

xtbo

ok,w

hat

wou

ld b

e th

e es

tim

ated

rabb

it p

opu

lati

on f

or J

anu

ary

1,20

05?

1200

Rea

din

g t

he

Less

on

1.D

eter

min

e w

het

her

th

e gr

aph

of

each

fu

nct

ion

rep

rese

nts

a s

hif

t of

th

e pa

ren

t fu

nct

ion

to t

he

left

,to

the

righ

t,u

pwar

d,o

r d

own

war

d.(

Do

not

act

ual

ly g

raph

th

e fu

nct

ion

s.)

a.y

�si

n (

��

90�)

to t

he

left

b.

y�

sin

��

3 u

pw

ard

c.y

�co

s ��

�� 3� �

to t

he

rig

ht

d.

y�

tan

��

4 d

ow

nw

ard

2.D

eter

min

e w

het

her

th

e gr

aph

of

each

fu

nct

ion

has

an

am

plit

ud

e ch

ange

,per

iod

ch

ange

,ph

ase

shif

t,or

ver

tica

l sh

ift

com

pare

d to

th

e gr

aph

of

the

pare

nt

fun

ctio

n.(

Mor

e th

anon

e of

th

ese

may

app

ly t

o ea

ch f

un

ctio

n.D

o n

ot a

ctu

ally

gra

ph t

he

fun

ctio

ns.

)

a.y

�3

sin

��

�5 6� ��

amp

litu

de

chan

ge

and

ph

ase

shif

t

b.

y�

cos

(2�

� 7

0�)

per

iod

ch

ang

e an

d p

has

e sh

ift

c.y

��

4 co

s 3�

amp

litu

de

chan

ge

and

per

iod

ch

ang

e

d.

y�

sec

�1 2� ��

3p

erio

d c

han

ge

and

ver

tica

l sh

ift

e.y

�ta

n ��

�� 4� �

�1

ph

ase

shif

t an

d v

erti

cal s

hif

t

f.y

�2

sin

��1 3� ��

�� 6� ��

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.

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ph

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e cy

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uat

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.S

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um

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atio

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cos

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os 5

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y

90

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2

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– 3

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2

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rich

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t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

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AT

E__

____

____

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ER

IOD

____

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14-2

14-2

Ste

p 2

Ste

p 3

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ple

Exam

ple



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dy G

uid

e a

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rven

tion

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____

____

____

____

____

____

____

____

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____

____

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____

____

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ER

IOD

____

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14-3

14-3

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at i

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for

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or w

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very

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csc

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Th

e si

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y of

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etr

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age

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plif

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____

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14-3

14-3

Exam

ple1

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____

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14-3

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and

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pli

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in �

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20.A

ERIA

L PH

OTO

GR

APH

YT

he

illu

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sh

ows

a pl

ane

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ng

an a

eria

l ph

otog

raph

of

poin

t A

.Bec

ause

th

e po

int

is d

irec

tly

belo

wth

e pl

ane,

ther

e is

no

dist

orti

on i

n t

he

imag

e.F

or a

ny

poin

t B

not

dire

ctly

bel

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he

plan

e,h

owev

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ease

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dis

tan

ce c

reat

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th

e ph

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s be

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se a

s th

e di

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ce f

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cam

era

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poin

t be

ing

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the

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the

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21.T

SUN

AM

IST

he

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sin

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e t

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____

____

____

____

____

____

____

____

____

____

____

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____

____

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ER

IOD

____

_

14-3

14-3



Readin

g t

o L

earn

Math

em

ati

csTr

igo

no

met

ric

Iden

titi

es

NA

ME

____

____

____

____

____

____

____

____

____

____

____

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AT

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____

____

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ER

IOD

____

_

14-3

14-3

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Lesson 14-3

Pre-

Act

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an t

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nom

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be

use

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odel

th

e p

ath

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a b

aseb

all?

Rea

d th

e in

trod

ucti

on t

o L

esso

n 14

-3 a

t th

e to

p of

pag

e 77

7 in

you

r te

xtbo

ok.

Su

ppos

e th

at a

bas

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l is

hit

fro

m h

ome

plat

e w

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in

itia

l ve

loci

ty o

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fee

t pe

r se

con

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an

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gle

of 3

6�w

ith

th

e h

oriz

onta

l fr

om a

n i

nit

ial

hei

ght

of 5

fee

t.S

how

th

e eq

uat

ion

th

at y

ou w

ould

use

to

fin

d th

e h

eigh

t of

the

ball

10

seco

nds

aft

er t

he

ball

is

hit

.(S

how

th

e fo

rmu

la w

ith

th

eap

prop

riat

e n

um

bers

su

bsti

tute

d,bu

t do

not

do

any

calc

ula

tion

s.)

h�

��1

02�

��1

0 �

5

Rea

din

g t

he

Less

on

1.M

atch

eac

h e

xpre

ssio

n f

rom

th

e li

st o

n t

he

left

wit

h a

n e

xpre

ssio

n f

rom

th

e li

st o

n t

he

righ

t th

at i

s eq

ual

to

it f

or a

ll v

alu

es f

or w

hic

h e

ach

exp

ress

ion

is

defi

ned

.(S

ome

of t

he

expr

essi

ons

from

th

e li

st o

n t

he

righ

t m

ay b

e u

sed

mor

e th

an o

nce

or

not

at

all.)

a.se

c2�

�ta

n2

�iii

i.� si

n1�

�

b.

cot2

��

1v

ii.

tan

�

c.� cs oin s

� ��

iiii

i.1

d.

sin

2�

�co

s2�

iiiiv

.se

c �

e.cs

c �

iv.

csc2

�

f.� co

1 s�

�iv

vi.

cot

�

g.�c so ins

��

vi

2.W

rite

an

ide

nti

ty t

hat

you

cou

ld u

se t

o fi

nd

each

of

the

indi

cate

d tr

igon

omet

ric

valu

esan

d te

ll w

het

her

th

at v

alu

e is

pos

itiv

e or

neg

ativ

e.(D

o n

ot a

ctu

ally

fin

d th

e va

lues

.)

a.ta

n �

,if

sin

��

��4 5�

and

180�

�

27

0�ta

n �

�� cs oin s

� ��

;p

osi

tive

b.

sec

�,i

f ta

n �

��

3 an

d 90

�

�

180�

tan

2�

�1

�se

c2�;

neg

ativ

e

Hel

pin

g Y

ou

Rem

emb

er

3.A

goo

d w

ay t

o re

mem

ber

som

eth

ing

new

is

to r

elat

e it

to

som

eth

ing

you

alr

eady

kn

ow.

How

can

you

use

th

e u

nit

cir

cle

defi

nit

ion

s of

th

e si

ne

and

cosi

ne

that

you

lea

rned

in

Ch

apte

r 13

to

hel

p yo

u r

emem

ber

the

Pyt

hag

orea

n i

den

tity

cos

2�

�si

n2

��

1?S

amp

le a

nsw

er:

On

a u

nit

cir

cle,

x�

cos

�an

d y

�si

n �

.Th

e eq

uat

ion

of

the

un

it c

ircl

e is

x2

�y

2�

1,so

th

is is

eq

uiv

alen

t to

th

e eq

uat

ion

co

s2�

�si

n2

��

1.

sin

36�

� cos

36�

�16

��

582

cos2

36�

©G

lenc

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cGra

w-H

ill85

4G

lenc

oe A

lgeb

ra 2

Pla

net

ary

Orb

its

Th

e or

bit

of a

pla

net

aro

un

d th

e su

n i

s an

ell

ipse

wit

h

the

sun

at

one

focu

s.L

et t

he

pole

of

a po

lar

coor

din

ate

syst

em b

e th

at f

ocu

s an

d th

e po

lar

axis

be

tow

ard

the

oth

er f

ocu

s.T

he

pola

r eq

uat

ion

of

an e

llips

e is

r�� 1

�

2 eep cos

��

.Sin

ce 2

p�

�b c2 �an

db2

�a2

�c2

,

2p�

�a2� c

c2�

��a c2 �

�1 �

� ac2 2��.B

ecau

se e

�� ac � ,

2p�

a ��a c� ��1

�� ac � �2 ��

a ��1 e� �(1 �

e2).

Th

eref

ore

2ep

�a(

1 �

e2).

Su

bsti

tuti

ng

into

th

e po

lar

equ

atio

n o

f an

el

lips

e yi

elds

an

equ

atio

n t

hat

is

use

ful

for

fin

din

g di

stan

ces

from

th

e pl

anet

to

the

sun

.

r�� 1a �(1

e�

coe s2 )�

�

Not

e th

at e

is t

he

ecce

ntr

icit

y of

th

e or

bit

and

ais

th

e le

ngt

h o

f th

e se

mi-

maj

or a

xis

of t

he

elli

pse.

Als

o,a

is t

he

mea

n d

ista

nce

of

the

plan

et

from

th

e su

n.

Th

e m

ean

dis

tan

ce o

f V

enu

s fr

om t

he

sun

is

67.2

4 �

106

mil

es a

nd

th

e ec

cen

tric

ity

of i

ts o

rbit

is

.006

788.

Fin

d t

he

min

imu

m a

nd

max

imu

m d

ista

nce

s of

Ven

us

from

th

e su

n.

Th

e m

inim

um

dis

tan

ce o

ccu

rs w

hen

��

�.

r�

�66

.78

10

6m

iles

Th

e m

axim

um

dis

tan

ce o

ccu

rs w

hen

��

0.

r�

�67

.70

10

6m

iles

Com

ple

te e

ach

of

the

foll

owin

g.

1.T

he

mea

n d

ista

nce

of

Mar

s fr

om t

he

sun

is

141.

64

106

mil

es a

nd

the

ecce

ntr

icit

y of

its

orb

it i

s 0.

0933

82.F

ind

the

min

imu

m a

nd

max

imu

mdi

stan

ces

of M

ars

from

th

e su

n.

max

.dis

tan

ce �

15.4

9 �

107

mi;

min

.dis

tan

ce �

12.8

4 �

107

mi

2.T

he

min

imu

m d

ista

nce

of

Ear

th f

rom

th

e su

n i

s 91

.445

10

6m

iles

an

dth

e ec

cen

tric

ity

of i

ts o

rbit

is

0.01

6734

.Fin

d th

e m

ean

an

d m

axim

um

dist

ance

s of

Ear

th f

rom

th

e su

n.

max

.dis

tan

ce �

93.0

0 �

106

mi;

mea

n d

ista

nce

�91

.47

�10

6m

i

67.2

4

106 (

1 �

0.00

6788

2 )�

��

�1

�0.

0067

88 c

os 0

67.2

4

106 (

1 �

0.00

6788

2 )�

��

�1

�0.

0067

88 c

os �

r

Pol

ar A

xis

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

14-3

14-3

Exam

ple

Exam

ple


An

swer

s


Stu

dy G

uid

e a

nd I

nte

rven

tion

Ver

ifyi

ng

Tri

go

no

met

ric

Iden

titi

es

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

14-4

14-4

©G

lenc

oe/M

cGra

w-H

ill85

5G

lenc

oe A

lgeb

ra 2

Lesson 14-4

Tran

sfo

rm O

ne

Sid

e o

f an

Eq

uat

ion

Use

th

e ba

sic

trig

onom

etri

c id

enti

ties

alo

ng

wit

h t

he

defi

nit

ion

s of

th

e tr

igon

omet

ric

fun

ctio

ns

to v

erif

y tr

igon

omet

ric

iden

titi

es.O

ften

it

is e

asie

r to

beg

in w

ith

th

e m

ore

com

plic

ated

sid

e of

th

e eq

uat

ion

an

d tr

ansf

orm

th

atex

pres

sion

in

to t

he

form

of

the

sim

pler

sid

e.

Ver

ify

that

eac

h o

f th

e fo

llow

ing

is a

n i

den

tity

.Ex

ampl

eEx

ampl

e

a.�

sec

��

�co

s �

Tra

nsf

orm

th

e le

ft s

ide.

�se

c �

��

cos

�

��

�co

s �

��

�co

s �

��

cos

�

��

cos

�

�co

s �

��

cos

�

�co

s2�

�co

s �

sin

2�

1�

�co

s �1

� cos

�si

n2

�� co

s �

1� co

s �

sin

�� c so ins

��

sin

�� co

t �

sin

�� co

t �

b.

�co

s �

�se

c �

Tra

nsf

orm

th

e le

ft s

ide.

�co

s �

�se

c �

�co

s �

�se

c �

�co

s �

�se

c �

�se

c �

�se

c �

sec

��

sec

�

1� co

s �

sin

2�

�co

s2�

��

cos

�

sin

2�

� cos

�

� cs oin s� �

�

� � sin1

��

tan

�� cs

c �

tan

�� cs

c �

Exer

cises

Exer

cises

Ver

ify

that

eac

h o

f th

e fo

llow

ing

is a

n i

den

tity

.

1.1

�cs

c2�

�co

s2�

�cs

c2�

1 �

co

s2�

�cs

c2�

�cs

c2�

�cs

c2�

csc2

��

csc2

�

1� si

n2

�

sin

2�

�co

s2�

��

sin

2�

1� si

n2

�

2.�

�

� � � � � �1

�co

s3�

��

sin

3�

1 �

cos3

��

�si

n3

�

1 �

cos3

��

�si

n3

�1

�co

s �(

�co

s2�)

��

�si

n3

�

1 �

cos3

��

�si

n3

�si

n2

��

cos2

��

cos

�(si

n2

��

1)�

��

��

sin

3�

1 �

cos3

��

�si

n3

�

sin

2�

�si

n2

�co

s �

�co

s �

�co

s2�

��

��

�si

n �

��

��

�si

n2

�

1 �

cos3

��

�si

n3

�

sin

��

sin

�

cos

��

�c so ins��

��

�c so is n2 ��

��

��

�1

�co

s2�

1 �

cos3

��

�si

n3

�

sin

�(1

�co

s �)

��c so ins

��

(1 �

cos

�)�

��

��

(1 �

cos

�)(1

�co

s �)

1 �

cos3

��

�si

n3

�co

t �

��

1 �

cos

�si

n �

��

1 �

cos

�

©G

lenc

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cGra

w-H

ill85

6G

lenc

oe A

lgeb

ra 2

Tran

sfo

rm B

oth

Sid

es o

f an

Eq

uat

ion

Th

e fo

llow

ing

tech

niq

ues

can

be

hel

pfu

l in

veri

fyin

g tr

igon

omet

ric

iden

titi

es.

•S

ubs

titu

te o

ne

or m

ore

basi

c id

enti

ties

to

sim

plif

y an

exp

ress

ion

.•

Fact

or o

r m

ult

iply

to

sim

plif

y an

exp

ress

ion

.•

Mu

ltip

ly b

oth

nu

mer

ator

an

d de

nom

inat

or b

y th

e sa

me

trig

onom

etri

c ex

pres

sion

.•

Wri

te e

ach

sid

e of

th

e id

enti

ty i

n t

erm

s of

sin

e an

d co

sin

e on

ly.T

hen

sim

plif

y ea

ch s

ide.

Ver

ify

that

�

sec2

��

tan

2�

is a

n i

den

tity

.

�se

c2�

�ta

n2

�

��

� � �1

1 �

1

Ver

ify

that

eac

h o

f th

e fo

llow

ing

is a

n i

den

tity

.

1�

�si

n2

��

cos2

�

cos2

�� co

s2�

� cos1 2

��

��

�sin

2� co

� s2c �os

2�

�

1 �

sin

2�

��

cos2

�

� cos1 2

��

��

� cs oin s2 2� �

��

1

sin

2�

� cos2

�1

� cos2

�se

c2�

��

�si

n �

�� cs oin s

� ��

�� co

1 s�

��

1

tan

2�

�1

��

�si

n �

�ta

n �

�se

c �

�1

tan

2�

�1

��

�si

n �

ta

n �

se

c �

�1

Stu

dy G

uid

e a

nd I

nte

rven

tion

(c

onti

nued

)

Ver

ifyi

ng

Tri

go

no

met

ric

Iden

titi

es

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

14-4

14-4

Exam

ple

Exam

ple

Exer

cises

Exer

cises

1.cs

c �

�se

c �

�co

t �

�ta

n �

�

�

� �

2.�

� �1

� cos2

�1

� cos2

�

� co1 s

��

� cos

�

� cs oin s2 2� �

�

� sin

2�

sec

�� co

s �

tan

2�

��

1 �

cos2

�

1�

�si

n �

co

s �

1�

�si

n �

co

s �

cos2

��

sin

2�

��

sin

�

cos

�1

��

sin

�

cos

�

sin

�� co

s �

cos

�� si

n �

1� co

s �

1� si

n �

3.�

� �

4.�

cot2

�(1

�co

s2�)

��c so ins 22

��

(sin

2�)

cos2

��

��co

s2�

cos2

��

��co

s2�

cos2

��

cos2

�

sin

2�

� sin

2�

1 �

cos2

��

�si

n2

�

� sin1 2

��

��c so ins 22

��

��

� cos1 2

��

csc2

��

cot2

��

�se

c2�

cos2

�� si

n2

�co

s2�

� sin

2�

� sin1

��

��

sin

�� co

s1 2�

�

cos

�

�c so ins��

�

��

sin

�

csc

��

�si

n �

�se

c2�

cos

��

cot

��

�si

n �



Skil

ls P

ract

ice

Ver

ifyi

ng

Tri

go

no

met

ric

Iden

titi

es

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

14-4

14-4

©G

lenc

oe/M

cGra

w-H

ill85

7G

lenc

oe A

lgeb

ra 2

Lesson 14-4

Ver

ify

that

eac

h o

f th

e fo

llow

ing

is a

n i

den

tity

.

1.ta

n �

cos

��

sin

�ta

n �

cos

��

sin

�� cs oin s

� ��

co

s �

�si

n �

sin

��

sin

�

2.co

t �

tan

��

1co

t �

tan

��

1�c so ins

��

� cs oin s

� ��

�1

1 �

1

3.cs

c �

cos

��

cot

�cs

c �

cos

��

cot

�

� sin1

��

co

s �

�co

t �

�c so ins��

��

cot

�

cot

��

cot

�

4.�

cos

�

�1� co

s sin �2�

��

cos

�

�c co os s2

��

�co

s �

cos

��

cos

�

1 �

sin

2�

��

cos

�

5.(t

an �

)(1

�si

n2

�)

�si

n �

cos

�(t

an �

)(1

�si

n2

�) �

sin

�co

s �

tan

�co

s2�

�si

n �

cos

�

� cs oin s� �

�

cos2

��

sin

�co

s �

sin

�co

s �

�si

n �

cos

�

6.�

cot

�

�c ss ec c� �

��

cot

�

�co

t �

�c so ins��

��

cot

�

cot

��

cot

�

� sin1

��

� � co1 s

��

csc

�� se

c �

7.�

tan

2�

� 1�si

n s2 in� 2

��

�ta

n2

�

� cs oin s2 2� �

��

tan

2�

�� cs oin s� �

��2

�ta

n2

�

tan

2�

�ta

n2

�

8.�

1 �

sin

�

� 1c �o

s s2 in��

��

1 �

sin

�

�1 1� �

s si in n2

��

�1

�si

n �

�1

�si

n �

1 �

sin

��

1 �

sin

�

(1 �

sin

�)(

1 �

sin

�)

��

�1

�si

n �

cos2

��

�1

�si

n �

sin

2�

��

1 �

sin

2�

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lenc

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w-H

ill85

8G

lenc

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lgeb

ra 2

Ver

ify

that

eac

h o

f th

e fo

llow

ing

is a

n i

den

tity

.

Pra

ctic

e (

Ave

rag

e)

Ver

ifyi

ng

Tri

go

no

met

ric

Iden

titi

es

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

14-4

14-4

1.�

sec2

�

�sin

2

c� o� s2c �os2

��

�se

c2�

� cos1 2

��

�se

c2�

sec2

��

sec2

�

2.�

1

� 1c �o ss i2 n� 2

��

�1

�c co os s2 2

� ��

�1

1 �

1

cos2

��

�1

�si

n2

�

sin

2�

�co

s2�

��

cos2

�

3.(1

�si

n �

)(1

�si

n �

) �

cos2

�

(1 �

sin

�)(

1 �

sin

�)

�co

s2�

1 �

sin

2�

�co

s2�

cos2

��

cos2

�

4.ta

n4

��

2 ta

n2

��

1 �

sec4

�ta

n4

��

2 ta

n2

��

1 �

sec4

�(t

an2

��

1)2

�se

c4�

(sec

2�)

2�

sec4

�se

c4�

�se

c4�

5.co

s2�

cot2

��

cot2

��

cos2

�co

s2�

cot2

��

cot2

��

cos2

�

cos2

�co

t2�

��c so ins 22

��

�co

s2�

cos2

�co

t2�

�

cos2

�co

t2�

�

cos2

�co

t2�

��co

s 12�

��c so ins 22

��

cos2

�co

t2�

�co

s2�

cot2

�

6.(s

in2

�)(

csc2

��

sec2

�)

�se

c2�

(sin

2�)

(csc

2�

�se

c2�)

�se

c2�

(sin

2�)

�� sin1 2

��

�� co

s1 2�

��

sec2

�

1 �

� cs oin s2 2� �

��

sec2

�

1 �

tan

2�

�se

c2�

sec2

��

sec2

�

(co

s2�)

(1 �

sin

2�)

��

�si

n2

�

cos2

��

cos2

�si

n2

��

��

sin

2�

7.PR

OJE

CTI

LES

Th

e sq

uar

e of

th

e in

itia

l ve

loci

ty o

f an

obj

ect

lau

nch

ed f

rom

th

e gr

oun

d is

v2�

,wh

ere

�is

th

e an

gle

betw

een

th

e gr

oun

d an

d th

e in

itia

l pa

th,h

is t

he

max

imu

m h

eigh

t re

ach

ed,a

nd

gis

th

e ac

cele

rati

on d

ue

to g

ravi

ty.V

erif

y th

e id

enti

ty

�.

� s2 ing 2h ��

�� 1

�

2 cg oh s2�

��

��2 sg ech 2s �ec

�2

1��

8.LI

GH

TT

he

inte

nsi

ty o

f a

ligh

t so

urc

e m

easu

red

in c

andl

es i

s gi

ven

by

I�

ER

2se

c �,

wh

ere

Eis

th

e il

lum

inan

ce i

n f

oot

can

dles

on

a s

urf

ace,

Ris

th

e di

stan

ce i

n f

eet

from

th

eli

ght

sou

rce,

and

�is

th

e an

gle

betw

een

th

e li

ght

beam

an

d a

lin

e pe

rpen

dicu

lar

to t

he

surf

ace.

Ver

ify

the

iden

tity

ER

2 (1

�ta

n2

�)

cos

��

ER

2se

c �.

ER

2 (1

�ta

n2

�) c

os

��

ER

2se

c2�

cos

��

ER

2se

c2�

� se

1 c�

��

ER

2se

c �

2gh

��

�sesc e2 c� 2

� �1

�

2gh

��

1 �

� sec1 2

��

2gh

sec2

��

�se

c2�

�1

2gh

� sin

2�

2gh

� sin

2�


An

swer

s


Readin

g t

o L

earn

Math

em

ati

csV

erif

yin

g T

rig

on

om

etri

c Id

enti

ties

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

14-4

14-4

©G

lenc

oe/M

cGra

w-H

ill85

9G

lenc

oe A

lgeb

ra 2

Lesson 14-4

Pre-

Act

ivit

yH

ow c

an y

ou v

erif

y tr

igon

omet

ric

iden

titi

es?

Rea

d th

e in

trod

ucti

on t

o L

esso

n 14

-4 a

t th

e to

p of

pag

e 78

2 in

you

r te

xtbo

ok.

For

��

��

,0,o

r �

,sin

��

sin

2�.D

oes

this

mea

n t

hat

sin

��

sin

2�

is a

nid

enti

ty?

Exp

lain

you

r re

ason

ing.

Sam

ple

an

swer

:N

o;

an id

enti

ty is

an e

qu

atio

n t

hat

is t

rue

for

allv

alu

es o

f a

vari

able

fo

r w

hic

hth

e fu

nct

ion

s in

volv

ed a

re d

efin

ed,n

ot

just

so

me

valu

es.I

f

��

�� 4� ,si

n �

�� 22 � �

,an

d s

in 2

��

1.

Rea

din

g t

he

Less

on

1.D

eter

min

e w

het

her

eac

h e

quat

ion

is

an i

den

tity

or n

ot a

n i

den

tity

.

a.� si

n1 2�

��

� tan1 2

��

�1

iden

tity

b.� si

nc �osta� n

��

no

t an

iden

tity

c.� cs oin s

� ��

��c so ins

��

�co

s �

sin

�n

ot

an id

enti

ty

d.

cos2

�(t

an2

��

1) �

1id

enti

ty

e.� cs oin s2 2

� ��

�si

n �

csc

��

sec2

�id

enti

ty

f.� 1

�1 si

n�

��

� 1�

1 sin

��

�2

cos2

�n

ot

an id

enti

ty

g.ta

n2

�co

s2�

�� cs

c1 2�

�id

enti

ty

h.

� ss ein c� �

��

� ta1 n

��

�� co

1 t�

�n

ot

an id

enti

ty

2.W

hic

h o

f th

e fo

llow

ing

is n

otpe

rmit

ted

wh

en v

erif

yin

g an

ide

nti

ty?

B

A.

sim

plif

yin

g on

e si

de o

f th

e id

enti

ty t

o m

atch

th

e ot

her

sid

e

B.c

ross

mu

ltip

lyin

g if

th

e id

enti

ty i

s a

prop

orti

on

C.

sim

plif

ying

eac

h si

de o

f th

e id

enti

ty s

epar

atel

y to

get

the

sam

e ex

pres

sion

on

both

sid

es

Hel

pin

g Y

ou

Rem

emb

er

3.M

any

stu

den

ts h

ave

trou

ble

know

ing

wh

ere

to s

tart

in

ver

ifyi

ng

a tr

igon

omet

ric

iden

tity

.W

hat

is

a si

mpl

e ru

le t

hat

you

can

rem

embe

r th

at y

ou c

an a

lway

s u

se i

f yo

u d

on’t

see

aqu

icke

r ap

proa

ch?

Sam

ple

an

swer

:Wri

te b

oth

sid

es in

ter

ms

of

sin

es a

nd

cosi

nes

.Th

en s

imp

lify

each

sid

e as

mu

ch a

s p

oss

ible

.

©G

lenc

oe/M

cGra

w-H

ill86

0G

lenc

oe A

lgeb

ra 2

Her

on

’s F

orm

ula

Her

on’s

for

mu

la c

an b

e u

sed

to f

ind

the

area

of

a tr

ian

gle

if y

ou k

now

th

ele

ngt

hs

of t

he

thre

e si

des.

Con

side

r an

y tr

ian

gle

AB

C.L

et K

repr

esen

t th

ear

ea o

f �

AB

C.T

hen

K�

�1 2� bc

sin

A

K2

��b2 c

2s 4in

2A

�S

quar

e bo

th s

ides

.

��b2 c

2 (1

� 4co

s2A

)�

� ��b2 4c2

��1

��b2

�2c b2 c�

a2�

��1 �

�b2�

2c b2 c�a2

��

Use

the

law

of

cosi

nes.

��b

�2c

�a

��b

�2c

�a

��a

�2b

�c

��a

�2b

�c

�S

impl

ify.

Let

s�

�a�

2b�

c�

.Th

en s

�a

��b

�2c

�a

�,s

�b

��a

�2c

�b

�,s

�c

��a

�2b

�c

�.

K2

�s(

s�

a)(s

�b)

(s�

c)S

ubst

itute

.

K�

�s(

s�

�a)

(s�

�b)

(s�

�c)�

Use

Her

on’s

for

mu

la t

o fi

nd

th

e ar

ea o

f �

AB

C.

1.a

�3,

b�

4.4,

c�

72.

a�

8.2,

b�

10.3

,c�

9.5

4.1

36.8

3.a

�31

.3,b

�92

.0,c

�67

.94.

a�

0.54

,b�

1.32

,c�

0.78

782.

9n

o s

uch

tri

ang

le

5.a

�32

1,b

�17

8,c

�29

86.

a�

0.05

,b�

0.08

,c�

0.04

26,1

60.9

0.00

082

7.a

�21

.5,b

�33

.0,c

�41

.78.

a�

2.08

,b�

9.13

,c�

8.99

351.

69.

3

b2 c2 (

1 �

cos

A)(

1 �

cos

A)

��

��

4

AC

B

ca

b

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

14-4

14-4

Her

on

’s F

orm

ula

The

are

a of

�A

BC

is

�,

whe

re s

�.

a�

b�

c

2s(

s�

a)(

s�

b)(s

�c)



Stu

dy G

uid

e a

nd I

nte

rven

tion

Su

m a

nd

Dif

fere

nce

of

An

gle

s F

orm

ula

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

14-5

14-5

©G

lenc

oe/M

cGra

w-H

ill86

1G

lenc

oe A

lgeb

ra 2

Lesson 14-5

Sum

an

d D

iffe

ren

ce F

orm

ula

sT

he

foll

owin

g fo

rmu

las

are

use

ful

for

eval

uat

ing

anex

pres

sion

lik

e si

n 1

5�fr

om t

he

know

n v

alu

es o

f si

ne

and

cosi

ne

of 6

0�an

d 45

�.

Su

m a

nd

T

he f

ollo

win

g id

entit

ies

hold

tru

e fo

r al

l val

ues

of �

and

�.

Dif

fere

nce

co

s (�

��

) �

cos

��

cos

��

sin

��

sin

�

of

An

gle

ssi

n (�

��

) �

sin

��

cos

��

cos

��

sin

�

Fin

d t

he

exac

t va

lue

of e

ach

exp

ress

ion

.

a.co

s 34

5�

cos

345�

�co

s (3

00�

�45

�)�

cos

300�

�co

s 45

��

sin

300

��

sin

45�

��1 2�

��

��

�

b.

sin

(�

105�

)

sin

(�

105�

) �

sin

(45

��

150�

)�

sin

45�

�co

s 15

0��

cos

45�

�si

n 1

50�

��

��

�1 2�

��

Fin

d t

he

exac

t va

lue

of e

ach

exp

ress

ion

.

1.si

n 1

05�

2.co

s 28

5�3.

cos

(�75

�)

��2 �

� 4�

6 ��

��6 �

� 4�

2 ��

��6 �

� 4�

2 ��

4.co

s (�

165�

)5.

sin

195

�6.

cos

420�

��

2 �� 4

�6 �

��

2 �� 4

�6 �

��1 2�

7.si

n (

�75

�)8.

cos

135�

9.co

s (�

15�)

��

2 �� 4

�6 �

��

�� 22 � ��

2 �� 4

�6 �

�

10.s

in 3

45�

11.c

os (

�10

5�)

12.s

in 4

95�

��2 �

� 4�

6 ��

��2 �

� 4�

6 ��

�� 22 � �

�2�

��

6��

� 4

�2�

�2

�3�

�2

�2�

�2

�2�

��

6��

� 4

�2�

�2

�3�

�2

�2�

�2

Exam

ple

Exam

ple

Exer

cises

Exer

cises

©G

lenc

oe/M

cGra

w-H

ill86

2G

lenc

oe A

lgeb

ra 2

Ver

ify

Iden

titi

esYo

u c

an a

lso

use

th

e su

m a

nd

diff

eren

ce o

f an

gles

for

mu

las

to v

erif

yid

enti

ties

.

Ver

ify

that

cos

��

�3 2� ��

sin

�is

an

id

enti

ty.

cos ��

��3 2� �

��si

n �

Orig

inal

equ

atio

n

cos

��

cos

�3 2� ��

sin

��

sin

�3 2� ��

sin

�S

um o

f Ang

les

For

mul

a

cos

��

0 �

sin

��

(�1)

�si

n �

Eva

luat

e ea

ch e

xpre

ssio

n.

sin

��

sin

�S

impl

ify.

Ver

ify

that

sin

��

�� 2� ��

cos

(��

�)

��

2 co

s �

is a

n i

den

tity

.

sin

��

�� 2� ��

cos

(��

�)

��

2 co

s �

Orig

inal

equ

atio

n

sin

��

cos

�� 2��

cos

��

sin

�� 2��

cos

��

cos

��

sin

��

sin

��

�2

cos

�S

um a

nd D

iffer

ence

of

Ang

les

For

mul

as

sin

��

0 �

cos

��

1 �

cos

��

(�1)

�si

n �

�0

��

2 co

s �

Eva

luat

e ea

ch e

xpre

ssio

n.

�2

cos

��

�2

cos

�S

impl

ify.

Ver

ify

that

eac

h o

f th

e fo

llow

ing

is a

n i

den

tity

.

1.si

n (

90�

��)

�co

s �

sin

90�

co

s �

�co

s 90

�

sin

��

cos

�1

co

s �

�0

si

n �

�co

s �

cos

��

cos

�

2.co

s (2

70�

��)

�si

n �

cos

270�

co

s �

�si

n 2

70�

si

n �

�si

n �

0

cos

��

(�1)

si

n �

�si

n �

sin

��

sin

�

3.si

n ��2 3� �

��

cos ��

��5 6� �

��si

n �

sin

�2 3� �

cos

��

cos

�2 3� �

sin

��

cos

�

cos

�5 6� ��

sin

�

sin

�5 6� ��

sin

�

�� 23 � �

cos

��

��1 2� �

sin

��

cos

��

�� 23 � ��

sin

�

�1 2��

sin

�si

n �

�si

n �

4.co

s ��3 4� �

��

sin

��

�� 4� ��

��

2�si

n �

cos

�3 4� �

cos

��

sin

�3 4� �

sin

��

�sin

�

cos

�� 4��

cos

�

sin

�� 4� ��

�2�

sin

�

�� 22 � �

�co

s �

�� 22 � �

si

n �

��si

n �

�� 22 � �

�co

s �

�� 22 � �

��

�2�

sin

�

��

2�si

n �

��

�2�

sin

�

Stu

dy G

uid

e a

nd I

nte

rven

tion

(c

onti

nued

)

Su

m a

nd

Dif

fere

nce

of

An

gle

s F

orm

ula

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

14-5

14-5

Exam

ple1

Exam

ple1

Exam

ple2

Exam

ple2

Exer

cises


An

swer

s


Skil

ls P

ract

ice

Su

m a

nd

Dif

fere

nce

of

An

gle

s F

orm

ula

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

14-5

14-5

©G

lenc

oe/M

cGra

w-H

ill86

3G

lenc

oe A

lgeb

ra 2

Lesson 14-5

Fin

d t

he

exac

t va

lue

of e

ach

exp

ress

ion

.

1.si

n 3

30�

��1 2�

2.co

s (�

165�

)��

�6 � 4�

�2 �

�3.

sin

(�

225�

)�� 22 � �

4.co

s 13

5��

�� 22 � �5.

sin

(�

45)�

�� 22 � �

6.co

s 21

0��

�� 23 � �

7.co

s (�

135�

)�

�� 22 � �8.

sin

75�

��6 �

� 4�

2 ��

9.si

n (

�19

5�)��

6 �� 4

�2 �

�

Ver

ify

that

eac

h o

f th

e fo

llow

ing

is a

n i

den

tity

.

10.s

in (

90�

��)

�co

s � si

n (

90�

��)

�co

s �

sin

90�

cos

��

cos

90�

sin

��

cos

�1

cos

��

0 si

n �

�co

s �

cos

��

cos

�

11.

sin

(18

0��

�)

��

sin

� sin

(18

0��

�) �

�si

n �

sin

180

�co

s �

�co

s 18

0�si

n �

��

sin

�0

cos

��

(�1)

sin

��

�si

n �

�si

n �

��

sin

�

12.c

os (

270�

��

) �

�si

n �

cos

(270

��

�)

��

sin

�co

s 27

0�co

s �

�si

n 2

70�

sin

��

�si

n �

0 co

s �

�(�

1) s

in �

��

sin

��

sin

��

�si

n �

13.c

os (

��

90�)

�si

n � co

s (�

�90

�) �

sin

�co

s �

cos

90�

�si

n �

sin

90�

�si

n �

(co

s �)

(0)

�(s

in �

)(1)

�si

n �

sin

��

sin

�

14.s

in ��

�� 2� �

��

cos

�

sin

��

�� 2� ��

�co

s �

sin

�co

s �� 2�

�co

s �

sin

�� 2��

�co

s �

(sin

�)(

0) �

(co

s �)

(1)

��

cos

��

cos

��

�co

s �

15.c

os (

��

�)

��

cos

�co

s (�

��)

��

cos

�co

s �

cos

��

sin

�si

n �

��

cos

��

1 co

s �

�0

sin

��

�co

s �

�co

s �

��

cos

�

©G

lenc

oe/M

cGra

w-H

ill86

4G

lenc

oe A

lgeb

ra 2

Fin

d t

he

exac

t va

lue

of e

ach

exp

ress

ion

.

1.co

s 75

��

6 �� 4

�2 �

�2.

cos

375�

��6 �

� 4�

2 ��

3.si

n (

�16

5�)��

2 �� 4

�6 �

�

4.si

n (

�10

5�)��

�2 � 4�

�6 �

�5.

sin

150

��1 2�

6.co

s 24

0��

�1 2�

7.si

n 2

25�

�� 22 � �

8.si

n (

�75

�)��

�2 � 4�

�6 �

�9.

sin

195

��

2 �� 4

�6 �

�

Ver

ify

that

eac

h o

f th

e fo

llow

ing

is a

n i

den

tity

.

10.c

os (

180�

��)

��

cos

�

cos

(180

��

�)

��

cos

�co

s 18

0�co

s �

�si

n 1

80�

sin

��

�co

s �

�1

cos

��

0 si

n �

��

cos

��

cos

��

�co

s �

11.s

in (

360�

��)

�si

n �si

n (

360�

��)

�si

n �

sin

360

�co

s �

�co

s 36

0�si

n �

�si

n �

0 co

s �

�1

sin

��

sin

�si

n �

�si

n �

12.s

in (

45�

��)

�si

n (

45�

��)

��

2�si

n �

sin

(45

��

�) �

sin

(45

��

�)�

sin

45�

cos

��

cos

45�

sin

��

(sin

45�

cos

��

cos

45�

sin

�)

�2

co

s 45

�si

n �

�2

�� 22 � �

si

n �

��

2�si

n �

13.c

os �x

�� 6� �

�si

n �x

�� 3� �

�si

n x

cos

�x�

�� 6� ��

sin

�x�

�� 3� ��

cos

xco

s �� 6�

�si

n x

sin

�� 6��

sin

xco

s �� 3�

�co

s x

sin

�� 3�

�� 23 � �

cos

x�

�1 2�si

n x

��1 2�

sin

x�

�� 23 � �co

s x

�si

n x

14.S

OLA

R E

NER

GY

On

Mar

ch 2

1,th

e m

axim

um

am

oun

t of

sol

ar e

ner

gy t

hat

fal

ls o

n a

squ

are

foot

of

grou

nd

at a

cer

tain

loc

atio

n i

s gi

ven

by

Esi

n (

90�

��

),w

her

e �

is t

he

lati

tude

of

the

loca

tion

an

d E

is a

con

stan

t.U

se t

he

diff

eren

ce o

f an

gles

for

mu

la t

o fi

nd

the

amou

nt

of s

olar

en

ergy

,in

ter

ms

of c

os �

,for

a l

ocat

ion

th

at h

as a

lat

itu

de o

f �

.E

cos

�

ELEC

TRIC

ITY

In E

xerc

ises

15

and

16,

use

th

e fo

llow

ing

info

rmat

ion

.In

a c

erta

in c

ircu

it c

arry

ing

alte

rnat

ing

curr

ent,

the

form

ula

i�

2 si

n (

120t

) ca

n b

e u

sed

tofi

nd

the

curr

ent

iin

am

pere

s af

ter

tse

con

ds.

Sam

ple

an

swer

:15

.Rew

rite

th

e fo

rmu

la u

sin

g th

e su

m o

f tw

o an

gles

.i

�2

sin

(90

t�

30t)

16.U

se t

he

sum

of

angl

es f

orm

ula

to

fin

d th

e ex

act

curr

ent

at t

�1

seco

nd.

�3�

amp

eres

Pra

ctic

e (

Ave

rag

e)

Su

m a

nd

Dif

fere

nce

of

An

gle

s F

orm

ula

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

14-5

14-5



Readin

g t

o L

earn

Math

em

ati

csS

um

an

d D

iffe

ren

ce o

f A

ng

les

Fo

rmu

las

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

14-5

14-5

©G

lenc

oe/M

cGra

w-H

ill86

5G

lenc

oe A

lgeb

ra 2

Lesson 14-5

Pre-

Act

ivit

yH

ow a

re t

he

sum

an

d d

iffe

ren

ce f

orm

ula

s u

sed

to

des

crib

eco

mm

un

icat

ion

in

terf

eren

ce?

Rea

d th

e in

trod

ucti

on t

o L

esso

n 14

-5 a

t th

e to

p of

pag

e 78

6 in

you

r te

xtbo

ok.

Con

side

r th

e fu

nct

ion

s y

�si

n x

and

y�

2 si

n x

.Do

the

grap

hs

of t

hes

e tw

ofu

nct

ion

s h

ave

con

stru

ctiv

ein

terf

eren

ce o

r d

estr

uct

ive

inte

rfer

ence

?co

nst

ruct

ive

Rea

din

g t

he

Less

on

1.M

atch

eac

h e

xpre

ssio

n f

rom

th

e li

st o

n t

he

left

wit

h a

n e

xpre

ssio

n f

rom

th

e li

st o

n t

he

righ

t th

at i

s eq

ual

to

it f

or a

ll v

alu

es o

f th

e va

riab

les.

(Som

e of

th

e ex

pres

sion

s fr

om t

he

list

on

th

e ri

ght

may

be

use

d m

ore

than

on

ce o

r n

ot a

t al

l.)

a.si

n (

��

�)

vi.

sin

�

b.

cos

(��

�)

viii

.si

n �

cos

��

cos

�si

n �

c.si

n (

180�

��

)vi

iii

i.�

cos

�

d.

sin

(18

0��

�)

iiv

.co

s �

cos

��

sin

�si

n �

e.co

s (1

80�

��

)iii

v.si

n �

cos

��

cos

�si

n �

f.si

n (

��

�)

iivi

.co

s �

cos

��

sin

�si

n �

g.co

s (9

0��

�)

ivi

i.�

sin

�

h.

cos

(��

�)

ivvi

ii.

cos

�

2.W

hic

h e

xpre

ssio

ns

are

equ

al t

o si

n 1

5�?

(Th

ere

may

be

mor

e th

an o

ne

corr

ect

choi

ce.)

A.

sin

45�

cos

30�

�co

s 45

�si

n 3

0�B

.si

n 4

5�co

s 30

��

cos

45�

sin

30�

B a

nd

C

C.

sin

60�

cos

45�

�co

s 60

�si

n 4

5�D

.co

s 60

�co

s 45

��

sin

60�

sin

45�

Hel

pin

g Y

ou

Rem

emb

er

3.S

ome

stu

den

ts h

ave

trou

ble

rem

embe

rin

g w

hic

h s

ign

s to

use

on

th

e ri

ght-

han

d si

des

ofth

e su

m a

nd

diff

eren

ce o

f an

gle

form

ula

s.W

hat

is

an e

asy

way

to

rem

embe

r th

is?

Sam

ple

an

swer

:In

th

e si

ne

iden

titi

es,t

he

sig

ns

are

the

sam

eo

n b

oth

sid

es.I

n t

he

cosi

ne

iden

titi

es,t

he

sig

ns

are

op

po

site

on

th

e tw

o s

ides

.

©G

lenc

oe/M

cGra

w-H

ill86

6G

lenc

oe A

lgeb

ra 2

Iden

titi

es f

or

the

Pro

du

cts

of

Sin

es a

nd

Co

sin

esB

y ad

din

g th

e id

enti

ties

for

th

e si

nes

of

the

sum

an

d di

ffer

ence

of

the

mea

sure

s of

tw

o an

gles

,a n

ew i

den

tity

is

obta

ined

.

sin

(�

��

) �

sin

�co

s �

�co

s �

sin

�si

n (

��

�)

�si

n �

cos

��

cos

�si

n �

(i)

sin

(�

��

) �

sin

(�

��

) �

2 si

n �

cos

�

Th

is n

ew i

den

tity

is

use

ful

for

expr

essi

ng

cert

ain

pro

duct

s as

su

ms.

Wri

te s

in 3

�co

s �

as a

su

m.

In t

he

iden

tity

let

��

3�an

d �

��

so t

hat

2

sin

3�

cos

��

sin

(3�

��)

�si

n (

3��

�).

Th

us,

sin

3�

cos

��

�1 2�si

n 4

��

�1 2�si

n 2

�.

By

subt

ract

ing

the

iden

titi

es f

or s

in (

��

�)

and

sin

(�

��

),a

sim

ilar

ide

nti

ty f

or e

xpre

ssin

g a

prod

uct

as

a di

ffer

ence

is

obta

ined

.

(ii)

sin

(�

��

) �

sin

(�

��

) �

2 co

s �

sin

�

Sol

ve.

1.U

se t

he

iden

titi

es f

or c

os (

��

�)

and

cos

(��

�)

to f

ind

iden

titi

es

for

expr

essi

ng

the

prod

uct

s 2

cos

�co

s �

and

2 si

n �

sin

�as

a s

um

or

dif

fere

nce

.2

cos

�co

s �

�co

s (�

��

) �

cos

(��

�)

2 si

n �

sin

��

cos

(��

�)

�co

s (�

��

)

2.F

ind

the

valu

e of

sin

105

�co

s 75

�w

ith

out

usi

ng

tabl

es.

�1 2�[s

in (

105�

�75

�) �

sin

(10

5��

75�)

];

�1 2��0

��1 2� �;

�1 2�

�1 2��

�1 4�

3.E

xpre

ss c

os �

sin

� 2� �as

a d

iffe

ren

ce.

2 co

s �

sin

� 2� ��

sin

��

� 2� � ��si

n ��

�� 2� � �

cos

�si

n � 2� �

��1 2�

sin

�3 2� ��

�1 2�si

n � 2� �

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

14-5

14-5

Exam

ple

Exam

ple


An

swer

s


Stu

dy G

uid

e a

nd I

nte

rven

tion

Do

ub

le-A

ng

le a

nd

Hal

f-A

ng

le F

orm

ula

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

14-6

14-6

©G

lenc

oe/M

cGra

w-H

ill86

7G

lenc

oe A

lgeb

ra 2

Lesson 14-6

Do

ub

le-A

ng

le F

orm

ula

s

The

fol

low

ing

iden

titie

s ho

ld t

rue

for

all v

alue

s of

�.

Do

ub

le-A

ng

lesi

n 2�

�2

sin

��

cos

�co

s 2�

�co

s2�

�si

n2�

Fo

rmu

las

cos

2��

1 �

2 si

n2�

cos

2��

2 co

s2�

�1

Fin

d t

he

exac

t va

lues

of

sin

2�

and

cos

2�

if

sin

��

�� 19 0�

and

180

�

�

270�

.

Fir

st,f

ind

the

valu

e of

cos

�.

cos2

��

1 �

sin

2�

cos2

��

sin2

��

1

cos2

��

1 �

�� 19 0�

�2si

n �

��

� 19 0�

cos2

��

� 11 09 0�

cos

��

�

Sin

ce �

is i

n t

he

thir

d qu

adra

nt,

cos

�is

neg

ativ

e.T

hu

s co

s �

��

.

To

fin

d si

n 2

�,u

se t

he

iden

tity

sin

2�

�2

sin

��

cos

�.

sin

2�

�2

sin

��

cos

�

�2 ��

� 19 0��

��

Th

e va

lue

of s

in 2

�is

.

To

fin

d co

s 2�

,use

th

e id

enti

ty c

os 2

��

1 �

2 si

n2

�.

cos

2��

1 �

2 si

n2

�

�1

�2 ��

� 19 0��2

��

�3 51 0�.

Th

e va

lue

of c

os 2

�is

��3 51 0�

.

Fin

d t

he

exac

t va

lues

of

sin

2�

and

cos

2�

for

each

of

the

foll

owin

g.

1.si

n �

��1 4� ,

0�

�

90�

��815 � �

,�7 8�

2.si

n �

��

�1 8� ,27

0�

�

360�

��3 3� 27 �

�,�

3 31 2�

3.co

s �

��

�3 5� ,18

0�

�

270�

�2 24 5�,�

� 27 5�4.

cos

��

��4 5� ,

90�

�

18

0��

�2 24 5�,�

27 5�

5.si

n �

��

�3 5� ,27

0�

�

360�

6.co

s �

��

�2 3� ,90

�

�

180�

��2 24 5�

,�27 5�

��4 �

95 �

�,�

�1 9�

9�19�

�50

9�19�

�50

�19�

�10

�19�

�10

�19�

�10

Exam

ple

Exam

ple

Exer

cises

Exer

cises

©G

lenc

oe/M

cGra

w-H

ill86

8G

lenc

oe A

lgeb

ra 2

Hal

f-A

ng

le F

orm

ula

s

Hal

f-A

ng

leT

he f

ollo

win

g id

entit

ies

hold

tru

e fo

r al

l val

ues

of �

.

Fo

rmu

las

sin

�� 2��

��1

�2co

s�

�co

s �� 2�

��1

�2co

s�

�

Fin

d t

he

exac

t va

lue

of s

in �� 2�

if s

in �

��2 3�

and

90�

�

18

0�.

Fir

st f

ind

cos

�.

cos2

��

1 �

sin

2�

cos2

��

sin2

��

1

cos2

��

1 �

��2 3� �2si

n �

��2 3�

cos2

��

�5 9�S

impl

ify.

cos

��

�Ta

ke t

he s

quar

e ro

ot o

f ea

ch s

ide.

Sin

ce �

is i

n t

he

seco

nd

quad

ran

t,co

s �

��

.

sin

�� 2��

��1

�2co

s�

�H

alf-

Ang

le f

orm

ula

��

cos

��

�

��

Sim

plify

.

��

Rat

iona

lize.

Sin

ce �

is b

etw

een

90�

and

180�

,�� 2�

is b

etw

een

45�

and

90�.

Th

us

sin

�� 2�is

pos

itiv

e an

d

equ

als

.

Fin

d t

he

exac

t va

lue

of s

in �� 2�

and

cos

�� 2�fo

r ea

ch o

f th

e fo

llow

ing.

1.co

s �

��

�3 5� ,18

0�

�

270�

2.co

s �

��

�4 5� ,90

�

�

180�

�2 �5

5 ��

,�� 55 � �

�3 �10

10 � �,�

� 11 00 � �

3.si

n �

��

�3 5� ,27

0�

�

360�

4.co

s �

��

�2 3� ,90

�

�

180�

�� 11 00 � �,�

�3 �10

10 � ��

630 � �,�

� 66 � �

Fin

d t

he

exac

t va

lue

of e

ach

exp

ress

ion

by

usi

ng

the

hal

f-an

gle

form

ula

s.

5.co

s 22

�1 2� �6.

sin

67.

5�7.

cos

�7 8� �

��

2 �

��

2��

2�

2 �

��

2��

2�

2 �

��

2��

2�18

��

6�5�

��

� 6

�18

��

6�5�

��

� 6

3 �

�5�

�6

�5�

�3

1 �

�� 35 � �

��

� 2

�5�

�3

�5�

�3Stu

dy G

uid

e a

nd I

nte

rven

tion

(c

onti

nued

)

Do

ub

le-A

ng

le a

nd

Hal

f-A

ng

le F

orm

ula

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

14-6

14-6

Exam

ple

Exam

ple

Exer

cises

Exer

cises



Skil

ls P

ract

ice

Do

ub

le-A

ng

le a

nd

Hal

f-A

ng

le F

orm

ula

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

14-6

14-6

©G

lenc

oe/M

cGra

w-H

ill86

9G

lenc

oe A

lgeb

ra 2

Lesson 14-6

Fin

d t

he

exac

t va

lues

of

sin

2�,

cos

2�,s

in � 2� � ,

and

cos

� 2� �fo

r ea

ch o

f th

e fo

llow

ing.

1.co

s �

�� 27 5�

,0�

�

90

�2.

sin

��

��4 5� ,

180�

�

27

0�

�3 63 26 5�,�

�5 62 27 5�,�

3 5� ,�4 5�

��2 24 5�

,�� 27 5�

,�� 55 � �

,��2 �

55 �

�

3.si

n �

��4 40 1�

,90�

�

18

0�4.

cos

��

�3 7� ,27

0�

�

360�

�� 17 62 80 1

�,�

�1 15 61 89 1�

,�5 �

4141 � �

,�4 �

4141 � �

��12

4� 910 ��

,��3 41 9�

,��

714 � �,�

��735 � �

5.co

s �

��

�3 5� ,90

�

�

180�

6.si

n �

�� 15 3�

,0�

�

90

�

��2 24 5�

,�� 27 5�

,�2 �

55 �

�,�

� 55 � ��1 12 60 9�

,�1 11 69 9�

,�� 22 66 � �

,�5 �

2626 � �

Fin

d t

he

exac

t va

lue

of e

ach

exp

ress

ion

by

usi

ng

the

hal

f-an

gle

form

ula

s.

7.co

s 22

�1 2� �8.

sin

165

�

9.co

s 10

5��

10.s

in �� 8�

11.s

in �15

8� ��

12.c

os 7

5��

2 �

��

3� ��

2�

2 �

��

2� ��

2

�2

��

�2� �

�2

�2

��

�3� �

�2

�2

��

�3� �

�2

�2

��

�2� �

�2

Ver

ify

that

eac

h o

f th

e fo

llow

ing

is a

nid

enti

ty.

13.s

in 2

��

� 12 �

t ta an n� 2�

�

sin

2�

�� 1

2 �ta tan n� 2

��

2 si

n �

cos

��

�2 seta cn 2

��

2 si

n �

cos

��

2� cs oin s

� ��

co

s2�

2 si

n �

cos

��

2 si

n �

cos

�

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lenc

oe/M

cGra

w-H

ill87

0G

lenc

oe A

lgeb

ra 2

Fin

d t

he

exac

t va

lues

of

sin

2�,

cos

2�,s

in � 2� � ,

and

cos

� 2� �fo

r ea

ch o

f th

e fo

llow

ing.

1.co

s �

�� 15 3�

,0�

�

90

�2.

sin

��

� 18 7�,9

0�

�

180�

�1 12 60 9�,�

�1 11 69 9�,�

2� 1313� �

,�3� 13

13� ��

�2 24 80 9�,�

1 26 81 9�,�

4 �17

17 � �,�

� 11 77 � �

3.co

s �

��1 4� ,

270�

�

36

0�4.

sin

��

��2 3� ,

180�

�

27

0�

�� 815� �

,��7 8� ,

�� 46� �,�

� 410� ��4� 9

5� �,�

1 9� ,,�

Fin

d t

he

exac

t va

lue

of e

ach

exp

ress

ion

by

usi

ng

the

hal

f-an

gle

form

ula

s.

5.ta

n 1

05�

6.ta

n 1

5�7.

cos

67.5

�8.

sin

�� 8� �

�2

��

3�2

��

3��

Ver

ify

that

eac

h o

f th

e fo

llow

ing

is a

n i

den

tity

.

9.si

n2

� 2� ��ta

n 2� ta�n

s �in�

�

10.s

in 4

��

4 co

s 2�

sin

�co

s �

sin

4�

�4

cos

2�si

n �

cos

�si

n 2

(2�)

�4

cos

2�si

n �

cos

�2

sin

2�

cos

2��

4 co

s 2�

sin

�co

s �

2(2

sin

�co

s �)

(co

s 2�

) �

4 co

s 2�

sin

�co

s �

4 co

s 2�

sin

�co

s �

�4

cos

2�si

n �

cos

�

11.A

ERIA

L PH

OTO

GR

APH

YIn

aer

ial p

hoto

grap

hy,t

here

is a

red

ucti

on in

fil

m e

xpos

ure

for

any

poin

t X

not

dir

ectl

y be

low

th

e ca

mer

a.T

he

redu

ctio

n E

�is

giv

en b

y E

��

E0

cos4

�,

whe

re �

is t

he a

ngle

bet

wee

n th

e pe

rpen

dicu

lar

line

from

the

cam

era

to t

he g

roun

d an

d th

eli

ne

from

th

e ca

mer

a to

poi

nt

X,a

nd

E0

is t

he

expo

sure

for

th

e po

int

dire

ctly

bel

ow t

he

cam

era.

Usi

ng

the

iden

tity

2 s

in2

��

1 �

cos

2�,v

erif

y th

at E

0co

s4�

�E

0��1 2��

�cos 22� �

�2 .

E0

cos4

��

E0(

cos2

�)2

�E

0(1

�si

n2

�)2

�E

0�1

��2

si2n

2�

��2

�

E0�1

��1

�c 2o

s2�

��2

�E

0��1 2��

�cos 2

2� ��2

12.I

MA

GIN

GA

sca

nn

er t

akes

th

erm

al i

mag

es f

rom

alt

itu

des

of 3

00 t

o 12

,000

met

ers.

Th

ew

idth

Wof

the

sw

ath

cove

red

by t

he i

mag

e is

giv

en b

y W

�2H

�ta

n �,w

her

e H

�is

th

e

hei

ght

and

�is

hal

f th

e sc

ann

er’s

fie

ld o

f vi

ew.V

erif

y th

at �2 1H �

�cs oin s

22 ��

�2H

�ta

n �

.

� 12H�

� cs oin s2 2� �

��

��4H

� 2s cin os� 2c �o

s�

��

�2Hco

�s sin ��

��

2H�t

an �

4H�s

in �

cos

��

��

1 �

(2 c

os2

��

1)

�2

��

�2 ��

�2

�2

��

�2 ��

�2

�18

�6

��

5��

�� 6

�18

�6

��

5��

�� 6

Pra

ctic

e (

Ave

rag

e)

Do

ub

le-A

ng

le a

nd

Hal

f-A

ng

le F

orm

ula

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

14-6

14-6

��

��1

�2co

s�

��2

��ta

n 2� ta�ns �in

��

;

�1�

2cos

��

�;�1

�2co

s�

��

�1�

2cos

��

�t ta an n� �

��

� ts ain n� �

�

��

2�t ta an n

� ��


An

swer

s


Readin

g t

o L

earn

Math

em

ati

csD

ou

ble

-An

gle

an

d H

alf-

An

gle

Fo

rmu

las

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

14-6

14-6

©G

lenc

oe/M

cGra

w-H

ill87

1G

lenc

oe A

lgeb

ra 2

Lesson 14-6

Pre-

Act

ivit

yH

ow c

an t

rigo

nom

etri

c fu

nct

ion

s b

e u

sed

to

des

crib

e m

usi

c?

Rea

d th

e in

trod

ucti

on t

o L

esso

n 14

-6 a

t th

e to

p of

pag

e 79

1 in

you

r te

xtbo

ok.

Su

ppos

e th

at t

he

equ

atio

n f

or t

he

seco

nd

har

mon

ic i

s y

�si

n a

�.T

hen

wh

atw

ould

be

the

equ

atio

ns

for

the

fun

dam

enta

l to

ne

(fir

st h

arm

onic

),th

ird

har

mon

ic,f

ourt

h h

arm

onic

,an

d fi

fth

har

mon

ic?

y�

sin

0.5

a�;

y�

sin

1.5

a�;

y�

sin

2a

�;

y�

sin

2.5

a�

Rea

din

g t

he

Less

on

1.M

atch

eac

h e

xpre

ssio

n f

rom

th

e li

st o

n t

he

left

wit

h a

llex

pres

sion

s fr

om t

he

list

on

th

eri

ght

that

are

equ

al t

o it

for

all

val

ues

of

�.

a.si

n �� 2�

vi.

2 si

n �

cos

�

b.

cos

2�ii

and

iii

ii.

1 �

2 si

n2

�

c.co

s �� 2�

ivii

i.co

s2�

�si

n2

�

d.

sin

2�

iiv

.��1

�

2cos

��

v.��1

�

2cos

��

2.D

eter

min

e w

het

her

you

wou

ld u

se t

he

posi

tive

or n

egat

ive

squ

are

root

in

th

e h

alf-

angl

e

iden

titi

es f

or s

in �� 2�

and

cos

�� 2�in

eac

h o

f th

e fo

llow

ing

situ

atio

ns.

(Do

not

act

ual

ly

calc

ula

te s

in �� 2�

and

cos

�� 2� .)

a.si

n �� 2� ,

if c

os �

��2 5�

and

�is

in

Qu

adra

nt

Ip

osi

tive

b.

cos

�� 2� ,if

cos

��

�0.

9 an

d �

is i

n Q

uad

ran

t II

po

siti

ve

c.co

s �� 2� ,

if s

in �

��

0.75

an

d �

is i

n Q

uad

ran

t II

In

egat

ive

d.

sin

�� 2� ,if

sin

��

�0.

8 an

d �

is i

n Q

uad

ran

t IV

po

siti

ve

Hel

pin

g Y

ou

Rem

emb

er

3.M

any

stu

den

ts f

ind

it d

iffi

cult

to

rem

embe

r a

larg

e n

um

ber

of i

den

titi

es.H

ow c

an y

ouob

tain

all

th

ree

of t

he

iden

titi

es f

or c

os 2

�by

rem

embe

rin

g on

ly o

ne

of t

hem

an

d u

sin

g a

Pyt

hag

orea

n i

den

tity

?S

amp

le a

nsw

er:

Just

rem

emb

er t

he

iden

tity

co

s 2�

�co

s2�

�si

n2

�.U

sin

g t

he

Pyt

hag

ore

an id

enti

ty c

os2

��

sin

2�

�1,

you

can

su

bst

itu

teei

ther

1 �

sin

2�

for

cos2

�o

r 1

�co

s2�

for

sin

2�

to g

et t

he

oth

er t

wo

iden

titi

es f

or

cos

2�.

©G

lenc

oe/M

cGra

w-H

ill87

2G

lenc

oe A

lgeb

ra 2

Alt

ern

atin

g C

urr

ent

Th

e fi

gure

at

the

righ

t re

pres

ents

an

alt

ern

atin

g cu

rren

t ge

ner

ator

.A r

ecta

ngu

lar

coil

of

wir

e is

su

spen

ded

betw

een

th

e po

les

of a

mag

net

.As

the

coil

of

wir

e is

rot

ated

,it

pass

es t

hro

ugh

th

e m

agn

etic

fie

ldan

d ge

ner

ates

cu

rren

t.

As

poin

t X

on t

he

coil

pas

ses

thro

ugh

th

e po

ints

Aan

d C

,its

mot

ion

is

alon

g th

e di

rect

ion

of

the

mag

net

ic

fiel

d be

twee

n t

he

pole

s.T

her

efor

e,n

o cu

rren

t is

ge

ner

ated

.How

ever

,th

rou

gh p

oin

ts B

and

D,t

he

mot

ion

of

Xis

per

pen

dicu

lar

to t

he

mag

net

ic f

ield

.T

he m

axim

um c

urre

nt m

ay h

ave

a po

sitiv

e

Th

is i

ndu

ces

max

imu

m c

urr

ent

in t

he

coil

.Bet

wee

n A

or n

egat

ive

valu

e.

and

B,B

and

C,C

and

D,a

nd

Dan

d A

,th

e cu

rren

t in

th

e co

il w

ill

hav

e an

in

term

edia

te v

alu

e.T

hu

s,th

e gr

aph

of

the

curr

ent

of a

n a

lter

nat

ing

curr

ent

gen

erat

or i

s cl

osel

y re

late

d to

th

e si

ne

curv

e.

Th

e ac

tual

cu

rren

t,i,

in a

hou

seh

old

curr

ent

is g

iven

by

i�

I Msi

n(1

20�

t�

�)

wh

ere

I Mis

th

e m

axim

um

va

lue

of t

he

curr

ent,

tis

th

e el

apse

d ti

me

in s

econ

ds,

and

�is

th

e an

gle

dete

rmin

ed b

y th

e po

siti

on o

f th

e co

il a

t ti

me

t n.

If �

�� 2� ,

fin

d a

val

ue

of t

for

wh

ich

i�

0.

If i

�0,

then

IM

sin

(12

0�t

��

) �

0.i�

I Msi

n(12

0�t

��

)

Sin

ce I

M�

0,si

n(1

20�

t�

�)

�0.

If ab

�0

and

a�

0, t

hen

b�

0.

Let

120

�t

��

�s.

Th

us,

sin

s�

0.s

��

beca

use

sin

��

0.12

0�t

��

��

Sub

stitu

te 1

20�

t�

�fo

r s.

120�

t�

�� 2��

�S

ubst

itute

�� 2�fo

r �

.

�� 21 40�

Sol

ve f

or t

.

Th

is s

olu

tion

is

the

firs

t po

siti

ve v

alu

e of

tth

at s

atis

fies

th

e pr

oble

m.

Usi

ng

the

equ

atio

n f

or t

he

actu

al c

urr

ent

in a

hou

seh

old

cir

cuit

,i

�I M

sin

(120

�t

��

),so

lve

each

pro

ble

m.F

or e

ach

pro

ble

m,f

ind

th

efi

rst

pos

itiv

e va

lue

of t

.

1.If

��

0,fi

nd

a va

lue

of t

for

2.If

��

0,fi

nd

a va

lue

of t

for

wh

ich

wh

ich

i�

0.t

�� 11 20�

i�

�I M

.t

�� 21 40�

3.If

��

�� 2� ,fi

nd

a va

lue

of t

for

wh

ich

4.If

��

�� 4� ,fi

nd

a va

lue

of t

for

wh

ich

i�

�I M

.t

�� 11 20�

i�

0.t

�� 11 60�OA

B

C

D

i(am

pere

s)

t(se

cond

s)

XA

BD

C

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

14-6

14-6

Exam

ple

Exam

ple



Stu

dy G

uid

e a

nd I

nte

rven

tion

So

lvin

g T

rig

on

om

etri

c E

qu

atio

ns

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

14-7

14-7

©G

lenc

oe/M

cGra

w-H

ill87

3G

lenc

oe A

lgeb

ra 2

Lesson 14-7

Solv

e Tr

igo

no

met

ric

Equ

atio

ns

You

can

use

tri

gon

omet

ric

iden

titi

es t

o so

lve

trig

onom

etri

c eq

uat

ion

s,w

hic

h a

re t

rue

for

only

cer

tain

val

ues

of

the

vari

able

.

Fin

d a

ll s

olu

tion

s of

4

sin

2�

�1

�0

for

the

inte

rval

0�

�

36

0�.

4 si

n2

��

1 �

04

sin

2�

�1

sin

2�

��1 4�

sin

��

��1 2�

��

30�,

150�

,210

�,33

0�

Sol

ve s

in 2

��

cos

��

0fo

r al

l va

lues

of

�.G

ive

you

r an

swer

in

bot

h r

adia

ns

and

deg

rees

.si

n 2

��

cos

��

02

sin

�co

s �

�co

s �

�0

cos

�(2

sin

��

1) �

0co

s �

�0

or2

sin

��

1 �

0

sin

��

��1 2�

��

90�

�k

�18

0�;

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210�

�k

�36

0�,

��

�� 2��

k�

�33

0��

k�

360�

;�

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�k

�2�

,

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�k

�2�

Exam

ple1

Exam

ple1

Exam

ple2

Exam

ple2

Exer

cises

Exer

cises

Fin

d a

ll s

olu

tion

s of

eac

h e

qu

atio

n f

or t

he

give

n i

nte

rval

.

1.2

cos2

��

cos

��

1,0

��

2�

2.si

n2

�co

s2�

�0,

0 �

�

2�

�� 3� ,�

,�5 3� �

0,�� 2� ,

�,�

3 2� �

3.co

s 2�

�,0

��

�

360�

4.2

sin

��

�3�

�0,

0 �

�

2�

15�,

165�

,195

�,34

5�� 3� ,

�2 3� �

Sol

ve e

ach

eq

uat

ion

for

all

val

ues

of

�if

�is

mea

sure

d i

n r

adia

ns.

5.4

sin

2�

�3

�0

6.2

cos

�si

n �

�co

s �

�0

�� 3��

k

�,�

2 3� ��

k

�� 2�

�k

2�

,�3 2� �

�k

2�

,

�7 6� ��

k

2�,�

11 6� ��

k

2�

Sol

ve e

ach

eq

uat

ion

for

all

val

ues

of

�if

�is

mea

sure

d i

n d

egre

es.

7.co

s 2�

�si

n2

��

�1 2�8.

tan

2�

��

1

45�

�k

90

�67

.5�

�k

36

0�,1

57.5

��

k

360�

�3�

�2

©G

lenc

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w-H

ill87

4G

lenc

oe A

lgeb

ra 2

Use

Tri

go

no

met

ric

Equ

atio

ns

LIG

HT

Sn

ell’s

law

say

s th

at s

in �

�1.

33 s

in �

,wh

ere

�is

th

e an

gle

at w

hic

h a

bea

m o

f li

ght

ente

rs w

ater

an

d �

is t

he

angl

e at

wh

ich

th

e b

eam

tra

vels

thro

ugh

th

e w

ater

.If

a b

eam

of

ligh

t en

ters

wat

er a

t 42

�,at

wh

at a

ngl

e d

oes

the

ligh

t tr

avel

th

rou

gh t

he

wat

er?

sin

��

1.33

sin

�O

rigin

al e

quat

ion

sin

42�

�1.

33 s

in �

��

42�

sin

��

�si 1n .34 32��

Div

ide

each

sid

e by

1.3

3.

sin

��

0.50

31U

se a

cal

cula

tor.

��

30.2

�Ta

ke t

he a

rcsi

n of

eac

h si

de.

Th

e li

ght

trav

els

thro

ugh

th

e w

ater

at

an a

ngl

e of

app

roxi

mat

ely

30.2

�.

1.A

6-f

oot

pipe

is

prop

ped

on a

3-f

oot

tall

pac

kin

g cr

ate

that

sit

s on

lev

el g

rou

nd.

On

e fo

otof

th

e pi

pe e

xten

ds a

bove

th

e to

p of

th

e cr

ate

and

the

oth

er e

nd

rest

s on

th

e gr

oun

d.W

hat

an

gle

does

th

e pi

pe f

orm

wit

h t

he

grou

nd?

36.9

�

2.A

t 1:

00 P

.M.o

ne

afte

rnoo

n a

180

-foo

t st

atu

e ca

sts

a sh

adow

th

at i

s 85

fee

t lo

ng.

Wri

te a

neq

uat

ion

to

fin

d th

e an

gle

of e

leva

tion

of

the

Su

n a

t th

at t

ime.

Fin

d th

e an

gle

ofel

evat

ion

.ta

n �

��1 88 50 �

;64

.7�

3.A

con

veyo

r be

lt i

s se

t u

p to

car

ry p

acka

ges

from

th

e gr

oun

d in

to a

win

dow

28

feet

abo

veth

e gr

oun

d.T

he

angl

e th

at t

he

con

veyo

r be

lt f

orm

s w

ith

th

e gr

oun

d is

35�

.How

lon

g is

the

con

veyo

r be

lt f

rom

th

e gr

oun

d to

th

e w

indo

w s

ill?

48.8

ft

SPO

RTS

Th

e d

ista

nce

a g

olf

bal

l tr

avel

s ca

n b

e fo

un

d u

sin

g th

e fo

rmu

la

d�

sin

2�,

wh

ere

v 0is

th

e in

itia

l ve

loci

ty o

f th

e b

all,

gis

th

e ac

cele

rati

on d

ue

to g

ravi

ty (

wh

ich

is

32 f

eet

per

sec

ond

sq

uar

ed),

and

�is

th

e an

gle

that

th

e p

ath

of

the

bal

l m

akes

wit

h t

he

grou

nd

.

4.H

ow f

ar w

ill

a ba

ll t

rave

l h

it 9

0 fe

et p

er s

econ

d at

an

an

gle

of 5

5�?

237.

9 ft

5.If

a b

all

that

tra

vele

d 30

0 fe

et h

ad a

n i

nit

ial

velo

city

of

110

feet

per

sec

ond,

wh

at a

ngl

edi

d th

e pa

th o

f th

e ba

ll m

ake

wit

h t

he

grou

nd?

26.3

�o

r 63

.7�

6.S

ome

chil

dren

set

up

a te

epee

in

th

e w

oods

.Th

e po

les

are

7 fe

et l

ong

from

th

eir

inte

rsec

tion

to

thei

r ba

ses,

and

the

chil

dren

wan

t th

e di

stan

ce b

etw

een

th

e po

les

to b

e 4

feet

at

the

base

.How

wid

e m

ust

th

e an

gle

be b

etw

een

th

e po

les?

33.2

�

v 02

�g

Stu

dy G

uid

e a

nd I

nte

rven

tion

(c

onti

nued

)

So

lvin

g T

rig

on

om

etri

c E

qu

atio

ns

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

14-7

14-7

Exam

ple

Exam

ple

Exer

cises

Exer

cises


An

swer

s


Skil

ls P

ract

ice

So

lvin

g T

rig

on

om

etri

c E

qu

atio

ns

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

14-7

14-7

©G

lenc

oe/M

cGra

w-H

ill87

5G

lenc

oe A

lgeb

ra 2

Lesson 14-7

Fin

d a

ll s

olu

tion

s of

eac

h e

qu

atio

n f

or t

he

give

n i

nte

rval

.

1.si

n �

�,0

��

�

360�

45�,

135�

2.2

cos

��

��

3�,90

�

�

180�

150�

3.ta

n2

��

1,18

0�

�

360�

225�

,315

�4.

2 si

n �

�1,

0 �

�

�� 6� ,

�5 6� �

5.si

n2

��

sin

��

0,�

��

2�

�,�

3 2� �6.

2 co

s2�

�co

s �

�0,

0 �

�

�� 2� ,

�2 3� �

Sol

ve e

ach

eq

uat

ion

for

all

val

ues

of

�if

�is

mea

sure

d i

n r

adia

ns.

7.2

cos2

��

cos

��

18.

sin

2�

�2

sin

��

1 �

0

0 �

2k�

,�2 3� �

�2k

�,a

nd

�4 3� ��

2k�

�� 2��

2k�

9.si

n �

�si

n �

cos

��

010

.sin

2�

�1

k�� 2�

�k�

11.4

cos

��

�1

�2

cos

�12

.tan

�co

s �

��1 2�

�2 3� ��

2k�

,�4 3� �

�2k

�� 6�

�2k

�,�

5 6� ��

2k�

Sol

ve e

ach

eq

uat

ion

for

all

val

ues

of

�if

�is

mea

sure

d i

n d

egre

es.

13.2

sin

��

1 �

014

.2 c

os �

��

3��

0

210�

�k

36

0 an

d 3

30�

�k

36

0�15

0��

k

360

and

210

��

k

360�

15.�

2�si

n �

�1

�0

16.2

cos

2�

�1

225�

�k

36

0�an

d 3

15�

�k

36

0�45

��

k

90�

17.4

sin

2�

�3

18.c

os 2

��

�1

60�

�k

18

0�an

d 1

20�

�k

18

0�90

��

k

180�

Sol

ve e

ach

eq

uat

ion

for

all

val

ues

of

�.

19.3

cos

2�

�si

n2

��

020

.sin

��

sin

2�

�0

�� 3��

k�an

d �2 3� �

�k�

,or

k�an

d �2 3� �

�2k

�,o

r

60�

�k

�18

0�an

d 1

20�

�k

�18

0�k

�18

0�an

d 1

20�

�k

�36

0�

21.2

sin

2�

�si

n �

�1

22.c

os �

�se

c �

�2

�� 2��

k�2 3� �

,or

90�

�k

�12

0�2k

�,o

r k

�36

0�

�2�

�2

©G

lenc

oe/M

cGra

w-H

ill87

6G

lenc

oe A

lgeb

ra 2

Fin

d a

ll s

olu

tion

s of

eac

h e

qu

atio

n f

or t

he

give

n i

nte

rval

.

1.si

n 2

��

cos

�,9

0��

�

180�

2.�

2�co

s �

�si

n 2

�,0

��

�

360�

90�,

150�

45�,

90�,

135�

,270

�

3.co

s 4�

�co

s 2�

,180

��

�

360�

4.co

s �

�co

s (9

0 �

�)

�0,

0 �

�

2�

180�

,240

�,30

0��3 4� �

,�7 4� �

5.2

�co

s �

�2

sin

2�,�

��

��3 2� �

6.ta

n2

��

sec

��

1,�� 2�

��

�

�4 3� �,�

3 2� ��2 3� �

Sol

ve e

ach

eq

uat

ion

for

all

val

ues

of

�if

�is

mea

sure

d i

n r

adia

ns.

7.co

s2�

�si

n2

�8.

cot

��

cot3

�

�� 4��

k�� 2�

�� 2��

k�an

d �� 4�

�k

�� 2�

9.�

2�si

n3

��

sin

2�

10.c

os2

�si

n �

�si

n �

k�,�

� 4��

2k�

,an

d �3 4� �

�2k

�k�

11.2

cos

2�

�1

�2

sin

2�

12.s

ec2

��

2

�� 4��

k�� 2�

�� 4��

k�� 2�

Sol

ve e

ach

eq

uat

ion

for

all

val

ues

of

�if

�is

mea

sure

d i

n d

egre

es.

13.s

in2

�co

s �

�co

s �

14.c

sc2

��

3 cs

c �

�2

�0

90�

�k

18

0�30

��

k

360�

,90�

�k

36

0�,a

nd

150�

�k

36

0�

15.�

1�

3 cos

��

�4(

1 �

cos

�)

16.�

2�co

s2�

�co

s2�

60�

�k

18

0�an

d 1

20�

�k

18

0�90

��

k

180�

and

450

��

k

360�

Sol

ve e

ach

eq

uat

ion

for

all

val

ues

of

�.

17.4

sin

2�

�3

�� 3��

k�an

d �2 3� �

�k�

,18

.4 s

in2

��

1 �

0�� 6�

�k�

and

�5 6� ��

k�,

or

60�

�k

18

0�an

d 1

20�

�k

18

0�o

r 30

��

k

180�

and

150

��

k

180�

19.2

sin

2�

�3

sin

��

�1

�� 6��

�� 3k �,

20.c

os 2

��

sin

��

1 �

0k�

and

�� 6��

2k�

,o

r 30

��

k

60�

or

k

180�

and

30�

�k

36

0�

21.W

AV

ESW

aves

are

cau

sin

g a

buoy

to

floa

t in

a r

egu

lar

patt

ern

in

th

e w

ater

.Th

e ve

rtic

alpo

siti

on o

f th

e bu

oy c

an b

e de

scri

bed

by t

he

equ

atio

n h

�2

sin

x.W

rite

an

exp

ress

ion

that

des

crib

es t

he

posi

tion

of

the

buoy

wh

en i

ts h

eigh

t is

at

its

mid

lin

e.k�

or

k

180�

22.E

LEC

TRIC

ITY

Th

e el

ectr

ic c

urr

ent

in a

cer

tain

cir

cuit

wit

h a

n a

lter

nat

ing

curr

ent

can

be d

escr

ibed

by

the

form

ula

i�

3 si

n 2

40t,

wh

ere

iis

th

e cu

rren

t in

am

pere

s an

d t

is t

he

tim

e in

sec

onds

.Wri

te a

n e

xpre

ssio

n t

hat

des

crib

es t

he

tim

es a

t w

hic

h t

her

e is

no

curr

ent.

0.75

kt

Pra

ctic

e (

Ave

rag

e)

So

lvin

g T

rig

on

om

etri

c E

qu

atio

ns

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

14-7

14-7



Readin

g t

o L

earn

Math

em

ati

csS

olv

ing

Tri

go

no

met

ric

Eq

uat

ion

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

14-7

14-7

©G

lenc

oe/M

cGra

w-H

ill87

7G

lenc

oe A

lgeb

ra 2

Lesson 14-7

Pre-

Act

ivit

yH

ow c

an t

rigo

nom

etri

c eq

uat

ion

s b

e u

sed

to

pre

dic

t te

mp

erat

ure

?

Rea

d th

e in

trod

ucti

on t

o L

esso

n 14

-7 a

t th

e to

p of

pag

e 79

9 in

you

r te

xtbo

ok.

Des

crib

e ho

w y

ou c

ould

use

a g

raph

ing

calc

ulat

or t

o de

term

ine

the

mon

ths

inw

hich

the

ave

rage

dai

ly h

igh

tem

pera

ture

is

abov

e 80

�F.(

Ass

ume

that

x�

1re

pres

ents

Jan

uar

y.)

Spe

cify

th

e gr

aph

ing

win

dow

th

at y

ou w

ould

use

.S

amp

le a

nsw

er:

Gra

ph

th

e fu

nct

ion

s y

�11

.56

sin

(0.

4516

x�

1.64

1) �

80.8

9 (u

sin

g r

adia

n m

od

e)an

d y

�80

on

th

e sa

me

scre

en.U

se t

he

win

do

w [

1,12

] by

[6

0,10

0] w

ith

Xsc

l �1

and

Ysc

l �4.

No

te t

he

xva

lues

fo

rw

hic

h t

he

curv

e is

ab

ove

the

ho

rizo

nta

l lin

e.

Rea

din

g t

he

Less

on

1.Id

enti

fy w

hic

h e

quat

ion

s h

ave

no

solu

tion

.C,E

,an

d G

A.

sin

��

1B

.tan

��

0.00

1C

.sec

��

�1 2�

D.c

sc �

��

3E

.cos

��

1.01

F.co

t �

��

1000

G.c

os �

�2

��

1H

.sec

��

1.5

�0

I.si

n �

�0.

009

�0.

99

2.U

se a

tri

gon

omet

ric

iden

tity

to

wri

te t

he

firs

t st

ep i

n t

he

solu

tion

of

each

tri

gon

omet

ric

equ

atio

n.(

Do

not

com

plet

e th

e so

luti

on.)

a.ta

n �

�co

s2�

�si

n2

�,0

��

2�

tan

��

1

b.

sin

2�

�2

sin

��

1 �

0,0�

��

36

0�(s

in �

�1)

2�

0

c.co

s 2�

�si

n �

,0�

��

36

0�1

�2

sin

2�

�si

n �

d.

sin

2�

�co

s �,0

��

2�

2 si

n �

cos

��

cos

�

e.2

cos

2��

3 co

s �

��

1,0�

��

36

0�2(

2 co

s2�

�1)

�3

cos

��

�1

f.3

tan

2�

�5

tan

��

2 �

0(3

tan

��

1)(t

an �

�2)

�0

Hel

pin

g Y

ou

Rem

emb

er

3.A

goo

d w

ay t

o re

mem

ber

som

eth

ing

is t

o ex

plai

n i

t to

som

eon

e el

se.H

ow w

ould

you

expl

ain

to

a fr

ien

d th

e di

ffer

ence

bet

wee

n v

erif

yin

g a

trig

onom

etri

c id

enti

ty a

nd

solv

ing

a tr

igon

omet

ric

equ

atio

n.

Sam

ple

an

swer

:Ver

ifyi

ng

a t

rig

on

om

etri

c id

enti

tym

ean

s sh

ow

ing

th

at t

he

two

sid

es a

re e

qu

al f

or

allv

alu

es o

f th

e va

riab

lefo

r w

hic

h t

he

fun

ctio

ns

invo

lved

are

def

ined

.Th

is is

do

ne

bytr

ansf

orm

ing

on

e o

r b

oth

sid

es u

nti

l th

e sa

me

exp

ress

ion

is o

bta

ined

on

bo

th s

ides

.So

lvin

g a

tri

go

no

met

ric

equ

atio

n m

ean

s fi

nd

ing

th

e va

lues

of

the

vari

able

fo

r w

hic

h b

oth

sid

es a

re e

qu

al.T

his

pro

cess

may

req

uir

esi

mp

lifyi

ng

tri

go

no

met

ric

exp

ress

ion

s,bu

t it

als

o r

equ

ires

fin

din

g t

he

ang

les

for

wh

ich

a t

rig

on

om

etri

c fu

nct

ion

has

a p

arti

cula

r va

lue.

©G

lenc

oe/M

cGra

w-H

ill87

8G

lenc

oe A

lgeb

ra 2

Fam

ilies

of

Cu

rves

Use

th

ese

grap

hs

for

the

pro

ble

ms

bel

ow.

1.U

se t

he

grap

h o

n t

he

left

to

desc

ribe

th

e re

lati

onsh

ip a

mon

g th

e cu

rves

y�

x�1 2� ,y�

x1,a

nd

y�

x2.

Fo

r n

��1 2�

and

n�

2,th

e g

rap

hs

are

refl

ecti

on

s o

f o

ne

ano

ther

in t

he

line

wit

h e

qu

atio

n y

�x1

.

2.G

raph

y�

xnfo

r n

�� 11 0�

,�1 4� ,

4,an

d 10

on

th

e gr

id w

ith

y�

x�1 2� ,y�

x1,a

nd

y�

x2.

See

stu

den

ts’g

rap

hs.

3.W

hic

h t

wo

regi

ons

in t

he

firs

t qu

adra

nt

con

tain

no

poin

ts o

f th

e gr

aph

sof

th

e fa

mil

y fo

r y

�xn

?

{(x,

y)

x>

1 an

d 0

< y

<1}

an

d {

(x,y

) 0

< x

<1

and

y>

1}

4.O

n t

he

righ

t gr

id,g

raph

th

e m

embe

rs o

f th

e fa

mil

y y

�em

xfo

r w

hic

h

m�

1 an

d m

��

1.

See

stu

den

ts’g

rap

hs.

5.D

escr

ibe

the

rela

tion

ship

am

ong

thes

e tw

o cu

rves

an

d th

e y-

axis

.

the

gra

ph

s fo

r m

�1

and

m�

�1

are

refl

ecti

on

s in

th

e y-

axis

.

6.G

raph

y�

emx

for

m�

0,�

�1 4� ,�

�1 2� ,�

2,an

d �

4.

See

stu

den

ts’g

rap

hs.

m =

–1 – 4

m =

0

m =

1 – 4m =

1 – 2m

= 1

m =

2m

= 4

m =

– 2

m =

– 1

m =

–1 – 2

O

y

x

234

–2–3

–11

23

m =

– 4

Th

e F

amil

y y

� e

mx

n =

4

n =

10

n =

1 – 4

n =

1 –– 10

O

y

x

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

n =

1

n =

1 – 2

Th

e F

amil

y y

� x

n

n =

2

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

14-7

14-7


1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

B:

1.

2.

3.

4.

5.

6.

7.

8.

9. B

D

B

C

D

A

B

C

A

See students’ answers.

C

A

B

A

C

D

C

D

B

A

C

C

D

B

A

A

C

D

A

B

An

swer

s


Chapter 14 Assessment Answer Key Form 1 Form 2APage 879 Page 880 Page 881


10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

B:

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

B: See students’ answers.

D

A

B

C

A

D

C

C

B

D

A

B

D

C

A

A

B

D

B

C


A

C

B

C

D

D

B

A

C

A

C

Chapter 14 Assessment Answer Key Form 2A (continued) Form 2BPage 882 Page 883 Page 884


1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

B:�12

�

about 15 weeks


��22��3�

��18 �6

�12�2��


��2� �4

�6��

��6� �4

�2��



tan2 �

1

��221��

�54

�

y

O

2

�1

�3�4

�6

�

2��

y � 1

y � �5

y � �2

y

O

2

�2

�

� 2�

��23��

none; 900� or 5�

3; 90� or ��2

�

y

O

1

2

�1

�2

�

� 2�

An

swer

s

Chapter 14 Assessment Answer Key Form 2CPage 885 Page 886


1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

B: 0

about 10 weeks

0� � k � 180�, 60� � k � 360�, 300� � k � 360�


��22��3�

��18 �6

�12�2��

��78

�


��2� �4

�6��

��6� �4

�2��



�sec2 �

1

��221��

�3�4

2��

y

O

2

5

�3

�1

�

2��

y � 4

y � 1

y � �2

y

O

2

�2

�� 2�

�23��

none; 720� or 4�

2; 120� or �23��

y

O

2

�2

�

� 2�

Chapter 14 Assessment Answer Key Form 2DPage 887 Page 888


1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

B:�17

7�510��

0.42 sec

��3

� � �2k

3��


��3�32

55��



��6� �4

�2��



1

y

O

1

6

�2�3

�

2��

y � 92

y � 32

y � � 32

�32

�; 3; �; ��4

�

y

O

2

4

6

8

�

90° 180° 270° 360°

y � 3

3; none; 90�; 45�

none; 900� or 5�

�125�; 90� or �

�2

�

y

O

1

2

3

�1

�2

�

� 3�2� 4�

An

swer

s

Chapter 14 Assessment Answer Key Form 3Page 889 Page 890

��32 ��2�87��

��2 � ��3��


Chapter 14 Assessment Answer KeyPage 891, Open-Ended Assessment

Scoring Rubric

Score General Description Specific Criteria

• Shows thorough understanding of the concepts oftrigonometric functions and their translations; using andverifying trigonometric identities; finding values of sine andcosine involving sum and difference, double-angle, andhalf-angle formulas; and solving trigonometric equations.

• Uses appropriate strategies to solve problems.• Computations are correct.• Written explanations are exemplary.• Graphs are accurate and appropriate.• Goes beyond requirements of some or all problems.

• Shows an understanding of the concepts of trigonometricfunctions and their translations; using and verifyingtrigonometric identities; finding values of sine and cosineinvolving sum and difference, double-angle, and half-angleformulas; and solving trigonometric equations.

• Uses appropriate strategies to solve problems.• Computations are mostly correct.• Written explanations are effective.• Graphs are mostly accurate and appropriate.• Satisfies all requirements of problems.

• Shows an understanding of most of the concepts oftrigonometric functions and their translations; using andverifying trigonometric identities; finding values of sine andcosine involving sum and difference, double-angle, andhalf-angle formulas; and solving trigonometric equations.

• May not use appropriate strategies to solve problems.• Computations are mostly correct.• Written explanations are satisfactory.• Graphs are mostly accurate.• Satisfies the requirements of most of the problems.

• Final computation is correct.• No written explanations or work is shown to substantiate

the final computation.• Satisfies minimal requirements of some of the problems.

• Shows little or no understanding of most of the concepts oftrigonometric functions and their translations; using andverifying trigonometric identities; finding values of sine andcosine involving sum and difference, double-angle, andhalf-angle formulas; and solving trigonometric equations.

• Does not use appropriate strategies to solve problems.• Computations are incorrect.• Written explanations are unsatisfactory.• Does not satisfy requirements of problems.• Graphs are inaccurate or inappropriate.• No answer may be given.

0 UnsatisfactoryAn incorrect solutionindicating no mathematicalunderstanding of theconcept or task, or nosolution is given

1 Nearly Unsatisfactory A correct solution with nosupporting evidence orexplanation

2 Nearly SatisfactoryA partially correctinterpretation and/orsolution to the problem

3 SatisfactoryA generally correct solution,but may contain minor flawsin reasoning or computation

4 SuperiorA correct solution that is supported by well-developed, accurateexplanations


1. Students should explain that theanswers given by Groups A and B areincorrect. For � � 0�, cot � is undefined,so this solution is extraneous. While theanswer given by Group D is correct,giving all angles coterminal with 90�and 270�, the Group C answer includesall of these same values for � in a singleexpression, so is the most efficient wayin which to express the solution.

2. Student responses must have one of thefour forms: y � a csc 4(� � h) � k,y � a sec 4(� � h) � k,y � a tan 2(� � h) � k, or y � a cot 2(� � h) � k, where a is anyreal number, h � 0, and k � 0.

Sample answer: y � 3 tan 2�� 4�� 1

3. Ideally, students should verify theidentity by transforming one side of theequation into the form of the other side(as in 14-4A), and by transforming bothsides of the equation separately into acommon form (as covered in 14-4B).Sample answer by method in 14-4A:

�1 � s

1in2 �� tan2 � � 1

�1 � (1 �

1cos2 �)� � tan2 � � 1

�cos

12 �� tan2 � � 1

sec2 � � tan2 � � 1tan2 � � 1 � tan2 � � 1

Sample answer by method in 14-4B:

�1 � s

1in2 �� tan2 � � 1

�1 � (1 �

1cos2 �)� � �

csoins

2

2�

�� 1

�cos

12 ��

scions

2

2�

��

ccooss

2

2�

��

�cos

12 ��

sin2

c�

o�

s2c�

os2 ��

�cos

12 ��

cos12 ��

4. For sin � to exist, students must select pand q so that � p � � � q �. Signs of p and qmust be consistent with the quadrantselected and the sign of the sinefunction in that quadrant. Then, usingappropriate values and signs for p andq, students should apply the necessaryidentities and formulas to evaluate eachfunction.Sample answer: For p � �3 and q � 5,and the terminal side of � in Quadrant

III, sin � � ��35�. Therefore, cos � � ��

45�,

tan � � �34�, csc � � ��

53�, sec � � ��

54�,

cot � � �43�, sin 2� � �

2245�

, cos 2� � �275�

,

sin �2�

� � �3�

1010��, and cos �2

��

�1100�

�.

5. Sample answers:5a. sin 240� � sin (180� � 60�)

� sin 180� cos 60� � cos 180� sin 60�

� ��23�

�

5b. sin 240� � sin (270� � 30�)� sin 270� cos 30� � cos 270� sin 30�

� ��23�

�

5c. sin 240� � sin (2 120�)

� 2 sin 120� cos 120� � ��23�

�

5d. sin 240� � sin �4820��

� ��1 � co�2s 480��

�23�

�

y

O

2345

�3

�1�2

�

y � 1

2��

An

swer

s

Chapter 14 Assessment Answer Key Page 891, Open-Ended Assessment

Sample Answers

In addition to the scoring rubric found on page A28, the following sample answers may be used as guidance in evaluating open-ended assessment items.


1. false; amplitude

2. false; vertical shift

3. false; midline

4. true

5. false; half-angleformula

6. true

7. true

8. Sample answer: Aphase shift is ahorizontaltranslation of thegraph of atrigonometricfunction.

1.

2.

3.

4.

Quiz (Lessons 14–3 and 14–4)

Page 893

1.

2.

3.

4.

5.

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

Quiz (Lesson 14–7)

Page 894

1.

2.

3.

4.

5.s � �

ta4n0

��; about 53�

��6

� � 2k�, �56�� 2k�

0� � k � 120�

��3

�, �23��, �

43��, �

53��

30�, 150�, 270�




��33��

��6� �4

�2��

A

tan2 �

�4

�12

�

2; y � 2

��4

�

y

O

2

�2

��

23�4

�4

none; ��2

�

y

O

1

90� 180� 270� 360��1

�

�12

�; 360�

Chapter 14 Assessment Answer Key Vocabulary Test/Review Quiz (Lessons 14–1 and 14–2) Quiz (Lessons 14–5 and 14–6)

Page 892 Page 893 Page 894

��50 ��10�2�1��

��2 � ��2��


1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

��98 ��14�28�1�0��

��152�

2; 6�

sin � � ��12

�, cos � � ��23��

��23��

Sample answers:

�1151��, ��

2191��

�3121�

�5161�

21

�lolo

gg

280

�; 1.4406

13(n2)2 � 52(n2) � 0; �2, 0, 2

x2 � 3x � 9 � �x

1�6

1�

�1�2�3�4 0 1 2 43

x � �2 � x � 3


1

cos �

��47��

none; 45� or ��4

�

y

O

1

�1

�

� 2�

C

A

B

A

C

An

swer

s

Chapter 14 Assessment Answer Key Mid-Chapter Test Cumulative ReviewPage 895 Page 896


1.

2.

3.

4.

5.

6.

7.

8.

9. 10.

11. 12.

13.

14.

15. DCBA

DCBA

DCBA

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

1 . 7 0

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

4 5

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

1 5

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

3 0

HGFE

DCBA

HGFE

DCBA

HGFE

DCBA

HGFE

DCBA

Chapter 14 Assessment Answer KeyStandardized Test Practice

Page 897 Page 898


1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

21.

22.

23.

24.

25.

26.

27.

28.

29.

30. about 12 weeks

��621��

��1385��


��2� �4

�6��

��22��

1

�53

�

�1; y � �1

y

O

2

�2

�

� 2�

��3

�

none; 720� or 4�

1; 120� or �23��

y

O

2

4

�2

�4

�� 2�

See students’ work.

��3

� or �60�

sin � � ��187�; cos � � ��

1157�

Law of Sines; C � 86�, b � 9.7, c � 15.6

Law of Cosines; A � 87.1�,B � 54.2�, C � 38.6�

one; B � 129.5�, C � 15.5�, b � 57.8

99.5 ft

��3�2� 1�

0

��22��

��3�

O

y

x��3�� 2��

3� � �

sin � � ��4�

4141��;

cos � � ��5�

4141��;

tan � � �45

�; csc � � ��441��;

sec � � ��451�; cot � � �

54

�

Sample answers: 50�, �670�

324�

��356�

�

A � 70�, a � 27.5, c � 29.2

Chapter 14 Assessment Answer Key Unit 5 TestPage 899 Page 900

An

swer

s


1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20. A

B

D

C

C

B

A

D

B

D

C

B

D

A

A

D

C

C

A

B

Chapter 14 Assessment Answer Key Second Semester TestPage 901 Page 902



21.

22.

23.

24.

25.

26.

27.

28.

29.

30.

31.

32.

33.

34.

35.

36.

37.

38.

39.

40.

41.

42.

43.

44.

45.

46.��

178�


one; B � 22�, C � 123�, c � 29.2

sin � � ��1157�; cos � � ��

187�;

tan � � �185�; csc � � ��

1175�;

sec � � ��187�; cot � � �

185�

�65

2,5536

�

positively skewed

�2312�

�118�

35

Sample answer: n � 2

243x5 � 405x4y � 270x3y2 �

90x2y3 � 15xy4 � y5

�181�

192

1, 4, 7

about 0.00012; y � ae�0.00012t;about 32,600

years ago

�lolo

gg

372

� � 1.7810

27

3.1945

��43

�

�92

�

inverse; 3.1

(x � 1)2 � (y � 1)2 � 25; circle

y

xO

(0, �1); (0, ��10�);

y � ��13

�x

(x � 10)2 � (y � 3)2 � �215�

y � �18

�(x � 2)2 � 3

Chapter 14 Assessment Answer Key Second Semester Test (continued)Page 903 Page 904

An

swer

s


1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

17. A

D

D

B

A

B

C

D

A

D

B

D

D

B

A

C

A

Chapter 14 Assessment Answer Key Final TestPage 905 Page 906



18.

19.

20.

21.

22.

23.

24.

25.

26.

27.

28.

29.

30.

31.

32.

33.

34.

35.

36.

37. �40

(�1, 2, �3)

500

t 0; b 100;

consistent and independent

Sample answer using (2, 100) and (3, 150):

y � 50x; 300 mi

d

tO

75

150

225

1 2 3 4Time (h)

Dis

tan

ce (

mi)

0 1�3 �2 �1�4

� 72 � 5

2 � 32

12

32

�a � ��72

� � a � ��32

��

A

B

C

A

D

B

B

B

A

C

C

An

swer

s

Chapter 14 Assessment Answer Key Final Test (continued)Page 907 Page 908

� ��3 60 6



38.

39.

40.

41.

42.

43.

44.

45.

46.

47.

48.

49.

50.

51.

52.

53.

54.

55.

56.

57.

58.

59.

60.

61.

62.

63.

64. 1

one; B � 14�, C � 141�, c � 10.4

47.5%

�16

�

2520

�3, �24, �171

2400

45

y � 5000e0.0087t

24

��12

�

asymptote: x � �4; hole: x � 3

y � 3(x � 2)2 � 7;parabola

(x � 2)2 � (y � 1)2 � 25

y

xO

g�1(x) � �x �

21

�

�1, �2, �3, �4, �6,

�12, ��13

�, ��23

�, ��43

�

�228

y � 4(x � 2)2 � 9

3x2 � 7x � 6 � 0

3t�83

�u2

�15 �

35�6��

18x6 � 45x4 �2x3 � 5x

Chapter 14 Assessment Answer Key Final Test (continued)Page 909 Page 910

�110�� 0 �5

2 1

chapter 14 resource masters - ktl math classes

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