Chapter 14 Resource Masters
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
Lesson 14-2Study Guide and Intervention . . . . . . . . 843–844Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 845Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 846Reading to Learn Mathematics . . . . . . . . . . 847Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 848
Lesson 14-3Study Guide and Intervention . . . . . . . . 849–850Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 851Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 852Reading to Learn Mathematics . . . . . . . . . . 853Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 854
Lesson 14-4Study Guide and Intervention . . . . . . . . 855–856Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 857Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 858Reading to Learn Mathematics . . . . . . . . . . 859Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 860
Lesson 14-5Study Guide and Intervention . . . . . . . . 861–862Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 863Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 864Reading to Learn Mathematics . . . . . . . . . . 865Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 866
Lesson 14-6Study Guide and Intervention . . . . . . . . 867–868Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 869Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 870Reading to Learn Mathematics . . . . . . . . . . 871Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 872
Lesson 14-7Study Guide and Intervention . . . . . . . . 873–874Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 875Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 876Reading to Learn Mathematics . . . . . . . . . . 877Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 878
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
© Glencoe/McGraw-Hill 838 Glencoe Algebra 2
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
© Glencoe/McGraw-Hill 839 Glencoe Algebra 2
Less
on
14-
1
Find the amplitude, if it exists, and period of each function. Then graph eachfunction.
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�
© Glencoe/McGraw-Hill 840 Glencoe Algebra 2
Find the amplitude, if it exists, and period of each function. Then graph eachfunction.
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
© Glencoe/McGraw-Hill 841 Glencoe Algebra 2
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�?
© Glencoe/McGraw-Hill 842 Glencoe Algebra 2
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
© Glencoe/McGraw-Hill 843 Glencoe Algebra 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
© Glencoe/McGraw-Hill 844 Glencoe Algebra 2
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�
Study Guide and Intervention (continued)
Translations of Trigonometric Graphs
NAME ______________________________________________ DATE ____________ PERIOD _____
14-214-2
ExampleExample
ExercisesExercises
Skills PracticeTranslations of Trigonometric Graphs
NAME ______________________________________________ DATE ____________ PERIOD _____
14-214-2
© Glencoe/McGraw-Hill 845 Glencoe Algebra 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�
© Glencoe/McGraw-Hill 846 Glencoe Algebra 2
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
© Glencoe/McGraw-Hill 847 Glencoe Algebra 2
Less
on
14-
2
Pre-Activity How can translations of trigonometric graphs be used to showanimal populations?
Read the introduction to Lesson 14-2 at the top of page 769 in your textbook.
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.
© Glencoe/McGraw-Hill 848 Glencoe Algebra 2
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
© Glencoe/McGraw-Hill 849 Glencoe Algebra 2
Less
on
14-
3
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
© Glencoe/McGraw-Hill 850 Glencoe Algebra 2
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 �
Study Guide and Intervention (continued)
Trigonometric Identities
NAME ______________________________________________ DATE ____________ PERIOD _____
14-314-3
Example 1Example 1
Example 2Example 2
ExercisesExercises
Skills PracticeTrigonometric Identities
NAME ______________________________________________ DATE ____________ PERIOD _____
14-314-3
© Glencoe/McGraw-Hill 851 Glencoe Algebra 2
Less
on
14-
3
Find the value of each expression.
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�
Simplify each expression.
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
© Glencoe/McGraw-Hill 852 Glencoe Algebra 2
Find the value of each expression.
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�
Simplify each expression.
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
© Glencoe/McGraw-Hill 853 Glencoe Algebra 2
Less
on
14-
3
Pre-Activity How can trigonometry be used to model the path of a baseball?
Read the introduction to Lesson 14-3 at the top of page 777 in your textbook.
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�
Helping You Remember
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?
© Glencoe/McGraw-Hill 854 Glencoe Algebra 2
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
© Glencoe/McGraw-Hill 855 Glencoe Algebra 2
Less
on
14-
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 �
� cos � � sec �
� 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 �
© Glencoe/McGraw-Hill 856 Glencoe Algebra 2
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
Verify that each of the following is an identity.
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
Study Guide and Intervention (continued)
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
© Glencoe/McGraw-Hill 857 Glencoe Algebra 2
Less
on
14-
4
Verify that each of the following is an identity.
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 �
© Glencoe/McGraw-Hill 858 Glencoe Algebra 2
Verify that each of the following is an identity.
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 _____
14-414-4
© Glencoe/McGraw-Hill 859 Glencoe Algebra 2
Less
on
14-
4
Pre-Activity How can you verify trigonometric identities?
Read the introduction to Lesson 14-4 at the top of page 782 in your textbook.
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
Helping You Remember
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?
© Glencoe/McGraw-Hill 860 Glencoe Algebra 2
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
© Glencoe/McGraw-Hill 861 Glencoe Algebra 2
Less
on
14-
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�
� �
Find the exact value of each expression.
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
© Glencoe/McGraw-Hill 862 Glencoe Algebra 2
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.
Verify that each of the following is an identity.
1. sin (90� � �) � cos �
2. cos (270� � �) � sin �
3. sin ��23�� � �� � cos �� � �
56��� � sin �
4. cos ��34�� � �� � sin �� � �
�4�� � ��2� sin �
Study Guide and Intervention (continued)
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
© Glencoe/McGraw-Hill 863 Glencoe Algebra 2
Less
on
14-
5
Find the exact value of each expression.
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�)
Verify that each of the following is an identity.
10. sin (90� � �) � cos �
11. sin (180� � �) � �sin �
12. cos (270� � � ) � �sin �
13. cos (� � 90�) � sin �
14. sin �� � ��2�� � �cos �
15. cos (� � �) � �cos �
© Glencoe/McGraw-Hill 864 Glencoe Algebra 2
Find the exact value of each expression.
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�
Verify that each of the following is an identity.
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
© Glencoe/McGraw-Hill 865 Glencoe Algebra 2
Less
on
14-
5
Pre-Activity How are the sum and difference formulas used to describecommunication interference?
Read the introduction to Lesson 14-5 at the top of page 786 in your textbook.
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�
Helping You Remember
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?
© Glencoe/McGraw-Hill 866 Glencoe Algebra 2
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
NAME ______________________________________________ DATE ____________ PERIOD _____
14-514-5
ExampleExample
Study Guide and InterventionDouble-Angle and Half-Angle Formulas
NAME ______________________________________________ DATE ____________ PERIOD _____
14-614-6
© Glencoe/McGraw-Hill 867 Glencoe Algebra 2
Less
on
14-
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
© Glencoe/McGraw-Hill 868 Glencoe Algebra 2
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
Study Guide and Intervention (continued)
Double-Angle and Half-Angle Formulas
NAME ______________________________________________ DATE ____________ PERIOD _____
14-614-6
ExampleExample
ExercisesExercises
Skills PracticeDouble-Angle and Half-Angle Formulas
NAME ______________________________________________ DATE ____________ PERIOD _____
14-614-6
© Glencoe/McGraw-Hill 869 Glencoe Algebra 2
Less
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�
Find the exact value of each expression by using the half-angle formulas.
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.
© Glencoe/McGraw-Hill 870 Glencoe Algebra 2
Find the exact values of sin 2�, cos 2�, sin �2�
�, and cos �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�
Find the exact value of each expression by using the half-angle formulas.
5. tan 105� 6. tan 15� 7. cos 67.5� 8. sin ����8��
Verify that each of the following is an identity.
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
NAME ______________________________________________ DATE ____________ PERIOD _____
14-614-6
Reading to Learn MathematicsDouble-Angle and Half-Angle Formulas
NAME ______________________________________________ DATE ____________ PERIOD _____
14-614-6
© Glencoe/McGraw-Hill 871 Glencoe Algebra 2
Less
on
14-
6Pre-Activity How can trigonometric functions be used to describe music?
Read the introduction to Lesson 14-6 at the top of page 791 in your textbook.
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
Helping You Remember
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?
© Glencoe/McGraw-Hill 872 Glencoe Algebra 2
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
NAME ______________________________________________ DATE ____________ PERIOD _____
14-614-6
ExampleExample
Study Guide and InterventionSolving Trigonometric Equations
NAME ______________________________________________ DATE ____________ PERIOD _____
14-714-7
© Glencoe/McGraw-Hill 873 Glencoe Algebra 2
Less
on
14-
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
© Glencoe/McGraw-Hill 874 Glencoe Algebra 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
Study Guide and Intervention (continued)
Solving Trigonometric Equations
NAME ______________________________________________ DATE ____________ PERIOD _____
14-714-7
ExampleExample
ExercisesExercises
Skills PracticeSolving Trigonometric Equations
NAME ______________________________________________ DATE ____________ PERIOD _____
14-714-7
© Glencoe/McGraw-Hill 875 Glencoe Algebra 2
Less
on
14-
7
Find all solutions of each equation for the given interval.
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 � � �
Solve each equation for all values of � if � is measured in radians.
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�
Solve each equation for all values of � if � is measured in degrees.
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
© Glencoe/McGraw-Hill 876 Glencoe Algebra 2
Find all solutions of each equation for the given interval.
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� � � �
Solve each equation for all values of � if � is measured in radians.
7. cos2 � � sin2 � 8. cot � � cot3 �
9. �2� sin3 � � sin2 � 10. cos2 � sin � � sin �
11. 2 cos 2� � 1 � 2 sin2 � 12. sec2 � � 2
Solve each equation for all values of � if � is measured in degrees.
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
NAME ______________________________________________ DATE ____________ PERIOD _____
14-714-7
Reading to Learn MathematicsSolving Trigonometric Equations
NAME ______________________________________________ DATE ____________ PERIOD _____
14-714-7
© Glencoe/McGraw-Hill 877 Glencoe Algebra 2
Less
on
14-
7
Pre-Activity How can trigonometric equations be used to predict temperature?
Read the introduction to Lesson 14-7 at the top of page 799 in your textbook.
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
Helping You Remember
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.
© Glencoe/McGraw-Hill 878 Glencoe Algebra 2
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
NAME ______________________________________________ DATE ____________ PERIOD _____
14-714-7
Chapter 14 Test, Form 1
NAME DATE PERIOD
SCORE
© Glencoe/McGraw-Hill 879 Glencoe Algebra 2
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
© Glencoe/McGraw-Hill 880 Glencoe Algebra 2
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
© Glencoe/McGraw-Hill 881 Glencoe Algebra 2
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 �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.
4. Which equation is graphed?
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
© Glencoe/McGraw-Hill 882 Glencoe Algebra 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
© Glencoe/McGraw-Hill 883 Glencoe Algebra 2
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 � 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.
4. Which equation is graphed?
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
© Glencoe/McGraw-Hill 884 Glencoe Algebra 2
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
© Glencoe/McGraw-Hill 885 Glencoe Algebra 2
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
© Glencoe/McGraw-Hill 886 Glencoe Algebra 2
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�.
16. Find the exact value of cos �2�
� 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
© Glencoe/McGraw-Hill 887 Glencoe Algebra 2
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
© Glencoe/McGraw-Hill 888 Glencoe Algebra 2
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�.
16. Find the exact value of sin �2�
� 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
© Glencoe/McGraw-Hill 889 Glencoe Algebra 2
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
© Glencoe/McGraw-Hill 890 Glencoe Algebra 2
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�.
16. Find the exact value of cos �2�
� 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
© Glencoe/McGraw-Hill 891 Glencoe Algebra 2
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�
�, and cos �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��
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Ass
essm
ent
© Glencoe/McGraw-Hill 892 Glencoe Algebra 2
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
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Chapter 14 Quiz (Lessons 14–1 and 14–2)
1414
© Glencoe/McGraw-Hill 893 Glencoe Algebra 2
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
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2
�2
����
23��4
��4
y
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1
90� 180� 270� 360��1
�
NAME DATE PERIOD
SCORE
Chapter 14 Quiz (Lessons 14–3 and 14–4)
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
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© Glencoe/McGraw-Hill 894 Glencoe Algebra 2
Find the exact value of each expression.
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
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Chapter 14 Quiz (Lessons 14–5 and 14–6)
1414
NAME DATE PERIOD
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1414
1.
2.
3.
Chapter 14 Mid-Chapter Test (Lessons 14–1 through 14–4)
© Glencoe/McGraw-Hill 895 Glencoe Algebra 2
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
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Ass
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ent
y
O
2
4
�2
�4
�
2��
y
O
2
4
�2
�
y � 3
y � 1
y � �1
2��
Part II
© Glencoe/McGraw-Hill 896 Glencoe Algebra 2
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.
13. Find the exact value of sin �2�
� 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
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Standardized Test Practice (Chapters 1–14)
© Glencoe/McGraw-Hill 897 Glencoe Algebra 2
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
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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.
© Glencoe/McGraw-Hill 898 Glencoe Algebra 2
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)
© Glencoe/McGraw-Hill 899 Glencoe Algebra 2
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
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7.
8.
9.
10.
11.
12.
Ass
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ent
© Glencoe/McGraw-Hill 900 Glencoe Algebra 2
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
© Glencoe/McGraw-Hill 901 Glencoe Algebra 2
Ass
essm
ent
For Questions 1–20, write the letter for the correct answer in the blank at the right of each question.
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
© Glencoe/McGraw-Hill 902 Glencoe Algebra 2
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
Second Semester Test (continued)(Chapters 8–14)
© Glencoe/McGraw-Hill 903 Glencoe Algebra 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
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ent
© Glencoe/McGraw-Hill 904 Glencoe Algebra 2
Second Semester Test (continued)(Chapters 8–14)
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
© Glencoe/McGraw-Hill 905 Glencoe Algebra 2
Ass
essm
ent
For Questions 1–28, write the letter for the correct answer in the blank at the right of each question.
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
© Glencoe/McGraw-Hill 906 Glencoe Algebra 2
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
Final Test (continued)(Chapters 1–14)
NAME DATE PERIOD
© Glencoe/McGraw-Hill 907 Glencoe Algebra 2
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
© Glencoe/McGraw-Hill 908 Glencoe Algebra 2
Final Test (continued)(Chapters 1–14)
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
Final Test (continued)(Chapters 1–14)
© Glencoe/McGraw-Hill 909 Glencoe Algebra 2
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
© Glencoe/McGraw-Hill 910 Glencoe Algebra 2
Final Test (continued)(Chapters 1–14)
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
DCBADCBADCBA
0 0 0
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.
99 9 987654321
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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
© Glencoe/McGraw-Hill A2 Glencoe Algebra 2
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
____
____
____
____
____
____
____
____
____
____
____
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AT
E__
____
____
__P
ER
IOD
____
_
14-1
14-1
©G
lenc
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cGra
w-H
ill83
7G
lenc
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lgeb
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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
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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
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ous
curv
e.T
he
peri
odof
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e si
ne,
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ne,
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and
cose
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nct
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s is
360
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plit
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nct
ion
The
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plit
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the
gra
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f a
perio
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func
tion
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he a
bsol
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valu
e of
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f th
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ffere
nce
betw
een
its m
axim
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r �
360�
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irst
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e a
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e of
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.
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llow
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ctio
ns
for
the
give
n d
omai
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1.co
s �,�
360�
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2.ta
n �
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��
�0
Wh
at i
s th
e am
pli
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h f
un
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3.4
4.8
x
y
O2
2
x
y
O
y
O
�2
�44 2
�
y �
tan
�
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y
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y O �0.
5
�1.
0
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0.5
��
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sin
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��
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°
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5°�
120°
�90
°�
60°
�45
°�
30°
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sin
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Exam
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Exam
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cises
©G
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cGra
w-H
ill83
8G
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Var
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f Tr
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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
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0�54
0�10
80�
900�
360�
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� 4
cos
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360°
�
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y�
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tan
2�
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e am
plit
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ined
,an
d th
e pe
riod
is
�� 2� .
y O� 4
2 –2 –44
� 23� 4
��
Exer
cises
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cises
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d t
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litu
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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
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.
1.y
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n �
2.y
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�o
r 36
0�n
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litu
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per
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2�
or
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y
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y
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© Glencoe/McGraw-Hill A3 Glencoe Algebra 2
An
swer
s
Answers (Lesson 14-1)
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
lenc
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cGra
w-H
ill83
9G
lenc
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lgeb
ra 2
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
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0�
4.y
��1 2�
tan
�5.
y�
sin
3�
6.y
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c 3�
no
am
plit
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e;18
0�1;
120�
no
am
plit
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n 2
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2�9.
y�
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n �1 2� �
no
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�1;
180�
4;72
0� y
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d t
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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
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s 5�
4;36
0�n
o a
mp
litu
de;
360�
1;72
�
4.y
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c �3 4� �
5.y
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tan
�1 2� �6.
2y�
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360�
no
am
plit
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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
© Glencoe/McGraw-Hill A4 Glencoe Algebra 2
Answers (Lesson 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
lenc
oe/M
cGra
w-H
ill84
1G
lenc
oe A
lgeb
ra 2
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.
©G
lenc
oe/M
cGra
w-H
ill84
2G
lenc
oe A
lgeb
ra 2
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��
A�
40�
x�
5.72
��
D�
26.5
7��
B�
50�
�A
�40
�t
�9.
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
© Glencoe/McGraw-Hill A5 Glencoe Algebra 2
An
swer
s
Answers (Lesson 14-2)
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
© Glencoe/McGraw-Hill A6 Glencoe Algebra 2
Answers (Lesson 14-2)
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
�2O
�2�
3� 2�
� 2
y
�2
�44 2
y
O6 4 2
�2
�36
0�27
0�18
0�90
�
y
O6 4 2
�2
�36
0�27
0�18
0�90
�
y
O6 4 2
�2
�36
0�27
0�18
0�90
�
y
O2 1
�1
�72
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
�36
0�27
0�18
0�90
�
y
O2 1
�1
�2
�36
0�27
0�18
0�90
�
©G
lenc
oe/M
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
© Glencoe/McGraw-Hill A7 Glencoe Algebra 2
An
swer
s
Answers (Lesson 14-2)
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� ��
4am
plit
ud
e ch
ang
e,p
erio
d c
han
ge,
ph
ase
shif
t,an
d v
erti
cal s
hif
t
Hel
pin
g Y
ou
Rem
emb
er
3.M
any
stu
den
ts h
ave
trou
ble
rem
embe
rin
g w
hic
h o
f th
e fu
nct
ion
s y
�si
n (
��
�)
and
y�
sin
(�
��
) re
pres
ents
a s
hif
t to
th
e le
ft a
nd
wh
ich
rep
rese
nts
a s
hif
t to
th
e ri
ght.
Usi
ng
��
45�,
expl
ain
a g
ood
way
to
rem
embe
r w
hic
h i
s w
hic
h.
Sam
ple
an
swer
:A
lth
ou
gh
sin
e cu
rves
are
infi
nit
ely
rep
eati
ng
per
iod
icg
rap
hs,
thin
k o
f y
�si
n x
star
tin
g a
per
iod
or
cycl
e at
(0,
0).T
hen
y
�si
n (
��
45�)
“st
arts
ear
ly”
at (
�45
�),a
sh
ift
of
45�
to t
he
left
,wh
ile
y�
sin
(�
�45
�) “
star
ts la
te”
at 4
5�,a
sh
ift
of
45�
to t
he
rig
ht.
©G
lenc
oe/M
cGra
w-H
ill84
8G
lenc
oe A
lgeb
ra 2
Tran
slat
ing
Gra
ph
s o
f Tri
go
no
met
ric
Fu
nct
ion
sT
hre
e gr
aph
s ar
e sh
own
at
the
righ
t:y
�3
sin
2�
y�
3 si
n 2
(��
30�)
y�
4 �
3 si
n 2
�
Rep
laci
ng
�w
ith
(�
�30
�) t
ran
slat
esth
e gr
aph
to
the
righ
t.R
epla
cin
g y
wit
h y
�4
tran
slat
es t
he
grap
h
4 u
nit
s do
wn
.
Gra
ph
on
e cy
cle
of y
�6
cos
(5�
�80
�) �
2.
Ste
p 1
Tra
nsf
orm
th
e eq
uat
ion
in
to
the
form
y�
k�
aco
s b(
��
h).
y�
2 �
6 co
s 5(
��
16�)
Ste
p 2
Ske
tch
y�
6 co
s 5�
.
Ste
p 3
Tra
nsl
ate
y�
6 co
s 5�
to
obta
in t
he
desi
red
grap
h.
Sk
etch
th
ese
grap
hs
on t
he
sam
e co
ord
inat
e sy
stem
.S
ee s
tud
ents
’gra
ph
s.1.
y�
3 si
n 2
(��
45�)
2.y
�1
�3
sin
2�
3.y
�5
�3
sin
2(�
�90
�)
On
an
oth
er p
iece
of
pap
er,g
rap
h o
ne
cycl
e of
eac
h c
urv
e.S
ee s
tud
ents
’gra
ph
s.4.
y�
2 si
n 4
(��
50�)
5.y
�5
sin
(3�
�90
�)
6.y
�6
cos
(4�
�36
0�)
�3
7.y
�6
cos
4��
3
8.T
he
grap
hs
for
prob
lem
s 6
and
7 sh
ould
be
the
sam
e.U
se t
he
sum
fo
rmu
la f
or c
osin
e of
a s
um
to
show
th
at t
he
equ
atio
ns
are
equ
ival
ent.
cos
(4�
�36
0�)
�(c
os
4�)(
cos
360�
) �
(sin
4�)
(sin
360
�)�
(co
s 4�
)(1)
�(s
in 4
�)(0
)�
cos
4�S
o,y
�6
cos
(4�
�36
0�)
�3
and
y�
6 co
s 4�
�3
are
equ
ival
ent.
Oy
56°
y
� 2
= 6
cos
5(
+ 1
6°)
6 –6
y =
6 c
os 5
( +
16°
)
Oy
72
°
y
= 6
cos
56 –6
O
y
90
°18
0°
y
= 3
sin
2
y
= 3
sin
2(
– 3
0°)
y
+ 4
= 3
sin
2
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
14-2
14-2
Ste
p 2
Ste
p 3
Exam
ple
Exam
ple
© Glencoe/McGraw-Hill A8 Glencoe Algebra 2
Answers (Lesson 14-3)
Stu
dy G
uid
e a
nd I
nte
rven
tion
Trig
on
om
etri
c Id
enti
ties
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
14-3
14-3
©G
lenc
oe/M
cGra
w-H
ill84
9G
lenc
oe A
lgeb
ra 2
Lesson 14-3
Fin
d T
rig
on
om
etri
c V
alu
esA
tri
gon
omet
ric
iden
tity
is a
n e
quat
ion
in
volv
ing
trig
onom
etri
c fu
nct
ion
s th
at i
s tr
ue
for
all
valu
es f
or w
hic
h e
very
exp
ress
ion
in
th
e eq
uat
ion
is d
efin
ed.
Bas
icQ
uo
tien
t Id
enti
ties
tan
��
� cs oin s� �
�co
t �
��c so ins
���
Trig
on
om
etri
cR
ecip
roca
l Id
enti
ties
csc
��
� sin1
��
sec
��
� co1 s
��
cot
��
� ta1 n
��
Iden
titi
esP
yth
ago
rean
Iden
titi
esco
s2�
�si
n2�
�1
tan2
��
1 �
sec2
�co
t2�
�1
�cs
c2�
Fin
d t
he
valu
e of
cot
�if
csc
��
��1 51 �
;180
�
�
270�
.co
t2�
�1
�cs
c2�
Trig
onom
etric
iden
tity
cot2
��
1 �
���1 51 �
�2S
ubst
itute
��1 51 �
for
csc
�.
cot2
��
1 �
�1 22 51 �S
quar
e �
�1 51 �.
cot2
��
�9 26 5�S
ubtr
act
1 fr
om e
ach
side
.
cot
��
��4 �
56 �
�Ta
ke t
he s
quar
e ro
ot o
f ea
ch s
ide.
Sin
ce �
is i
n t
he
thir
d qu
adra
nt,
cot
�is
pos
itiv
e,T
hu
s co
t �
��4 �
56 �
�.
Fin
d t
he
valu
e of
eac
h e
xpre
ssio
n.
1.ta
n �
,if
cot
��
4;18
0�
�
270�
�1 4�2.
csc
�,i
f co
s �
��� 23 � �
;0�
��
90
�2
3.co
s �,i
f si
n �
��3 5� ;
0��
�
90�
�4 5�4.
sec
�,i
f si
n �
��1 3� ;
0��
�
90�
�3 �4
2 ��
5.co
s �,i
f ta
n �
��
�4 3� ;90
�
�
180�
��3 5�
6.ta
n �
,if
sin
��
�3 7� ;0�
��
90
��3 �
2010 � �
7.se
c �,i
f co
s �
��
�7 8� ;90
�
�
180�
��8 7�
8.si
n �
,if
cos
��
�6 7� ;27
0��
�
360�
���
713 � �
9.co
t �,i
f cs
c �
��1 52 �
;90�
�
18
0�10
.sin
�,i
f cs
c �
��
�9 4� ;27
0�
�
360�
���
1 519 � ��
�4 9�
Exam
ple
Exam
ple
Exer
cises
Exer
cises
©G
lenc
oe/M
cGra
w-H
ill85
0G
lenc
oe A
lgeb
ra 2
Sim
plif
y Ex
pre
ssio
ns
Th
e si
mpl
ifie
d fo
rm o
f a
trig
onom
etri
c ex
pres
sion
is
wri
tten
as
an
um
eric
al v
alu
e or
in
ter
ms
of a
sin
gle
trig
onom
etri
c fu
nct
ion
,if
poss
ible
.An
y of
th
etr
igon
omet
ric
iden
titi
es o
n p
age
849
can
be
use
d to
sim
plif
y ex
pres
sion
s co
nta
inin
gtr
igon
omet
ric
fun
ctio
ns.
Sim
pli
fy (
1 �
cos2
�) s
ec �
cot
��
tan
�se
c �
cos2
�.
(1 �
cos2
�)
sec
�co
t �
�ta
n �
sec
�co
s2�
�si
n2
��
� co1 s
��
��c so ins
� ��
�� cs oin s
� ��
�� co
1 s�
��
cos2
�
�si
n �
�si
n �
�2
sin
�
Sim
pli
fy
�.
�se 1c ��� sic not
��
��
� 1�cs
c si� n
��
��
� � �
Sim
pli
fy e
ach
exp
ress
ion
.
1.1
2.co
s �
3.�
cos
�4.
1 �
sin
�
5.�
cot
��
sin
��
tan
��
csc
�2
6.cs
c �
7.3
tan
��
cot
��
4 si
n �
�cs
c �
�2
cos
��
sec
�9
8.co
s �
1 �
cos2
��
�ta
n �
�si
n �
csc2
��
cot2
��
�ta
n �
�co
s �
tan
��
cos
��
�si
n �
cos
��
�se
c �
�ta
n �
sin
2�
�co
t �
�ta
n �
��
�co
t �
�si
n �
sin
��
cot
��
�se
c2�
�ta
n2
�
tan
��
csc
��
�se
c �
2� co
s2�
� sin1
��
�1
�� si
n1�
��
1�
��
1 �
sin
2�
� sin1
��
(1 �
sin
�)
�� si
n1�
�(1
�si
n �
)�
��
�(1
�si
n �
)(1
�si
n �
)
� sin1
��
��
1�
sin
�
� co1 s
��
��c so ins
���
��
1�
sin
�
csc
��
�1
�si
n �
sec
�
cot
��
�1
�si
n �
Stu
dy G
uid
e a
nd I
nte
rven
tion
(c
onti
nued
)
Trig
on
om
etri
c Id
enti
ties
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
14-3
14-3
Exam
ple1
Exam
ple1
Exam
ple2
Exam
ple2
Exer
cises
Exer
cises
© Glencoe/McGraw-Hill A9 Glencoe Algebra 2
An
swer
s
Answers (Lesson 14-3)
Skil
ls P
ract
ice
Trig
on
om
etri
c Id
enti
ties
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
14-3
14-3
©G
lenc
oe/M
cGra
w-H
ill85
1G
lenc
oe A
lgeb
ra 2
Lesson 14-3
Fin
d t
he
valu
e of
eac
h e
xpre
ssio
n.
1.si
n �
,if
cos
��
��4 5�
and
90�
�
18
0�2.
cos
�,i
f ta
n �
�1
and
180�
�
27
0�
�3 5��
�� 22 � �
3.se
c �,i
f ta
n �
�1
and
0��
�
90�
4.co
s �,i
f ta
n �
��1 2�
and
0��
�
90�
�2�
�2 �5
5 ��
5.ta
n �
,if
sin
��
�an
d 18
0�
�
270�
6.co
s �,i
f se
c �
�2
and
270�
�
36
0�
1�1 2�
7.co
s �,i
f cs
c �
��
2 an
d 18
0�
�
270�
8.ta
n �,i
f co
s �
��
and
180�
�
27
0�
��� 23 � �
�1 2�
9.co
s �,i
f co
t �
��
�3 2�an
d 90
�
�
180�
10.c
sc �
,if
cos
��
� 18 7�an
d 0�
�
90
�
��1 17 5�
11.c
ot �
,if
csc
��
�2
and
180�
�
27
0�12
.tan
�,i
f si
n �
��
� 15 3�an
d 18
0�
�
270�
�3�
� 15 2�
Sim
pli
fy e
ach
exp
ress
ion
.
13.s
in �
sec
�ta
n �
14.c
sc �
sin
�1
15.c
ot �
sec
�cs
c �
16.�
c so es c� �
�co
s2�
17.t
an �
�co
t �� co
s�1 si
n�
�18
.csc
�ta
n �
�ta
n �
sin
�co
s �
19.
1 �
sin
�20
.csc
��
cot
��1
� sinco
�s�
�
21.
csc2
�22
.1 �
sec
�ta
n2
��
�1
�se
c �
sin
2�
�co
s2�
��
1 �
cos2
�
1 �
sin
2�
��
sin
��
1
3�13�
�13
2�5�
�5
�2�
�2
©G
lenc
oe/M
cGra
w-H
ill85
2G
lenc
oe A
lgeb
ra 2
Fin
d t
he
valu
e of
eac
h e
xpre
ssio
n.
1.si
n �
,if
cos
��
� 15 3�an
d 0�
��
90
�2.
sec
�,i
f si
n �
��
�1 15 7�an
d 18
0�
�
270�
�1 12 3��
�1 87 �
3.co
t �,i
f co
s �
�� 13 0�
and
270�
�
36
0�4.
sin
�,i
f co
t �
��1 2�
and
0��
�
90�
��3 �
9191 � �
�2 �5
5 ��
5.co
t �,i
f cs
c �
��
�3 2�an
d 18
0�
�
270�
6.se
c �,i
f cs
c �
��
8 an
d 27
0�
�
360�
�� 25 � ��8 2� 17 �
�
7.se
c �,i
f ta
n �
�4
and
180�
�
27
0�8.
sin
�,i
f ta
n �
��
�1 2�an
d 27
0�
�
360�
��
17��
�� 55 � �
9.co
t �,i
f ta
n �
��2 5�
and
0��
�
90�
10.c
ot �
,if
cos
��
�1 3�an
d 27
0�
�
360�
�5 2��
�� 42 � �
Sim
pli
fy e
ach
exp
ress
ion
.
11.c
sc �
tan
�se
c x
12.
cos2
�13
.sin
2�
cot2
�co
s2�
14.c
ot2
��
1cs
c2�
15.
csc2
�16
.�cs
c� co
� s�si
n�
�co
t �
17.s
in �
�co
s �
cot
�18
.�
19.s
ec2
�co
s2�
�ta
n2
�
csc
�2
tan
�se
c2�
20.A
ERIA
L PH
OTO
GR
APH
YT
he
illu
stra
tion
sh
ows
a pl
ane
taki
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
ow t
he
plan
e,h
owev
er,t
he
incr
ease
in
dis
tan
ce c
reat
esdi
stor
tion
in
th
e ph
otog
raph
.Th
is i
s be
cau
se a
s th
e di
stan
ce f
rom
the
cam
era
to t
he
poin
t be
ing
phot
ogra
phed
in
crea
ses,
the
expo
sure
of
the
film
red
uce
s by
(si
n �
)(cs
c �
�si
n �
).E
xpre
ss
(sin
�)(
csc
��
sin
�)
in t
erm
s of
cos
�on
ly.
cos2
�
21.T
SUN
AM
IST
he
equ
atio
n y
�a
sin
�t
repr
esen
ts t
he
hei
ght
of t
he
wav
es p
assi
ng
abu
oy a
t a
tim
e t
in s
econ
ds.E
xpre
ss a
in t
erm
s of
csc
�t.
a�
ycs
c �t
AB
�
cos
��
�1
�si
n �
cos
��
�1
�si
n �
csc2
��
cot2
��
�1
�co
s2�
sin
2�
� tan
2�
Pra
ctic
e (
Ave
rag
e)
Trig
on
om
etri
c Id
enti
ties
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
14-3
14-3
© Glencoe/McGraw-Hill A10 Glencoe Algebra 2
Answers (Lesson 14-3)
Readin
g t
o L
earn
Math
em
ati
csTr
igo
no
met
ric
Iden
titi
es
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
14-3
14-3
©G
lenc
oe/M
cGra
w-H
ill85
3G
lenc
oe A
lgeb
ra 2
Lesson 14-3
Pre-
Act
ivit
yH
ow c
an t
rigo
nom
etry
be
use
d t
o m
odel
th
e p
ath
of
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
ebal
l is
hit
fro
m h
ome
plat
e w
ith
an
in
itia
l ve
loci
ty o
f 58
fee
t pe
r se
con
d at
an
an
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
oe/M
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
© Glencoe/McGraw-Hill A11 Glencoe Algebra 2
An
swer
s
Answers (Lesson 14-4)
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
oe/M
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 �
© Glencoe/McGraw-Hill A12 Glencoe Algebra 2
Answers (Lesson 14-4)
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�
©G
lenc
oe/M
cGra
w-H
ill85
8G
lenc
oe A
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�
© Glencoe/McGraw-Hill A13 Glencoe Algebra 2
An
swer
s
Answers (Lesson 14-4)
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)
© Glencoe/McGraw-Hill A14 Glencoe Algebra 2
Answers (Lesson 14-5)
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
© Glencoe/McGraw-Hill A15 Glencoe Algebra 2
An
swer
s
Answers (Lesson 14-5)
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
© Glencoe/McGraw-Hill A16 Glencoe Algebra 2
Answers (Lesson 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
© Glencoe/McGraw-Hill A17 Glencoe Algebra 2
An
swer
s
Answers (Lesson 14-6)
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
© Glencoe/McGraw-Hill A18 Glencoe Algebra 2
Answers (Lesson 14-6)
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
�
©G
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
� ��
© Glencoe/McGraw-Hill A19 Glencoe Algebra 2
An
swer
s
Answers (Lesson 14-6)
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
© Glencoe/McGraw-Hill A20 Glencoe Algebra 2
Answers (Lesson 14-7)
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�;
��
210�
�k
�36
0�,
��
�� 2��
k�
�33
0��
k�
360�
;�
��7 6� �
�k
�2�
,
�116� �
�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
oe/M
cGra
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
© Glencoe/McGraw-Hill A21 Glencoe Algebra 2
An
swer
s
Answers (Lesson 14-7)
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
© Glencoe/McGraw-Hill A22 Glencoe Algebra 2
Answers (Lesson 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
© Glencoe/McGraw-Hill A23 Glencoe Algebra 2
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
(continued on the next page)
Chapter 14 Assessment Answer Key Form 1 Form 2APage 879 Page 880 Page 881
© Glencoe/McGraw-Hill A24 Glencoe Algebra 2
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
See students’ answers.
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
© Glencoe/McGraw-Hill A25 Glencoe Algebra 2
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
See students’ answers.
���22����3�
��18 �6
�12�2���
See students’ answers.
��2� �4
�6��
��6� �4
�2��
See students’ answers.
See students’ answers.
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
© Glencoe/McGraw-Hill A26 Glencoe Algebra 2
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�
See students’ answers.
���22����3�
��18 �6
�12�2���
��78
�
See students’ answers.
��2� �4
�6��
��6� �4
�2��
See students’ answers.
See students’ answers.
�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
© Glencoe/McGraw-Hill A27 Glencoe Algebra 2
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��
See students’ answers.
��3�32
55��
See students’ answers.
See students’ answers.
��6� �4
�2��
See students’ answers.
See students’ answers.
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����
© Glencoe/McGraw-Hill A28 Glencoe Algebra 2
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
© Glencoe/McGraw-Hill A29 Glencoe Algebra 2
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.
© Glencoe/McGraw-Hill A30 Glencoe Algebra 2
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�
See students’ answers.
See students’ answers.
See students’ answers.
��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����
© Glencoe/McGraw-Hill A31 Glencoe Algebra 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
See students’ answers.
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
© Glencoe/McGraw-Hill A32 Glencoe Algebra 2
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
© Glencoe/McGraw-Hill A33 Glencoe Algebra 2
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��
See students’ answers.
��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
© Glencoe/McGraw-Hill A34 Glencoe Algebra 2
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
(continued on the next page)
© Glencoe/McGraw-Hill A35 Glencoe Algebra 2
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�
See students’ answers.
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
© Glencoe/McGraw-Hill A36 Glencoe Algebra 2
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
(continued on the next page)
© Glencoe/McGraw-Hill A37 Glencoe Algebra 2
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
(continued on the next page)
© Glencoe/McGraw-Hill A38 Glencoe Algebra 2
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