Chapter 7 Resource Masters - KTL MATH...
Transcript of Chapter 7 Resource Masters - KTL MATH...
Chapter 7Resource Masters
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ISBN: 0-07-869134-6 Advanced Mathematical ConceptsChapter 7 Resource Masters
1 2 3 4 5 6 7 8 9 10 XXX 11 10 09 08 07 06 05 04
© Glencoe/McGraw-Hill iii Advanced Mathematical Concepts
Vocabulary Builder . . . . . . . . . . . . . . . vii-viii
Lesson 7-1Study Guide . . . . . . . . . . . . . . . . . . . . . . . . . 275Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . . 277
Lesson 7-2Study Guide . . . . . . . . . . . . . . . . . . . . . . . . . 278Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . . 280
Lesson 7-3Study Guide . . . . . . . . . . . . . . . . . . . . . . . . . 281Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . . 283
Lesson 7-4Study Guide . . . . . . . . . . . . . . . . . . . . . . . . . 284Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . . 286
Lesson 7-5Study Guide . . . . . . . . . . . . . . . . . . . . . . . . . 287Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . . 289
Lesson 7-6Study Guide . . . . . . . . . . . . . . . . . . . . . . . . . 290Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . . 292
Lesson 7-7Study Guide . . . . . . . . . . . . . . . . . . . . . . . . . 293Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . . 295
Chapter 7 AssessmentChapter 7 Test, Form 1A . . . . . . . . . . . . 297-298Chapter 7 Test, Form 1B . . . . . . . . . . . . 299-300Chapter 7 Test, Form 1C . . . . . . . . . . . . 301-302Chapter 7 Test, Form 2A . . . . . . . . . . . . 303-304Chapter 7 Test, Form 2B . . . . . . . . . . . . 305-306Chapter 7 Test, Form 2C . . . . . . . . . . . . 307-308Chapter 7 Extended Response
Assessment . . . . . . . . . . . . . . . . . . . . . . . 309Chapter 7 Mid-Chapter Test . . . . . . . . . . . . . 310Chapter 7 Quizzes A & B . . . . . . . . . . . . . . . 311Chapter 7 Quizzes C & D. . . . . . . . . . . . . . . 312Chapter 7 SAT and ACT Practice . . . . . 313-314Chapter 7 Cumulative Review . . . . . . . . . . . 315
SAT and ACT Practice Answer Sheet,10 Questions . . . . . . . . . . . . . . . . . . . . . . . A1
SAT and ACT Practice Answer Sheet,20 Questions . . . . . . . . . . . . . . . . . . . . . . . A2
ANSWERS . . . . . . . . . . . . . . . . . . . . . . A3-A16
Contents
© Glencoe/McGraw-Hill iv Advanced Mathematical Concepts
A Teacher’s Guide to Using theChapter 7 Resource Masters
The Fast File Chapter Resource system allows you to conveniently file theresources you use most often. The Chapter 7 Resource Masters include the corematerials needed for Chapter 7. These materials include worksheets, extensions,and assessment options. The answers for these pages appear at the back of thisbooklet.
All of the materials found in this booklet are included for viewing and printing inthe Advanced Mathematical Concepts TeacherWorks CD-ROM.
Vocabulary Builder Pages vii-viii include a student study tool that presents the key vocabulary terms from the chapter. Students areto record definitions and/or examples for eachterm. You may suggest that students highlight orstar the terms with which they are not familiar.
When to Use Give these pages to studentsbefore beginning Lesson 7-1. Remind them toadd definitions and examples as they completeeach lesson.
Study Guide There is one Study Guide master for each lesson.
When to Use Use these masters as reteaching activities for students who need additional reinforcement. These pages can alsobe used in conjunction with the Student Editionas an instructional tool for those students whohave been absent.
Practice There is one master for each lesson.These problems more closely follow the structure of the Practice section of the StudentEdition exercises. These exercises are ofaverage difficulty.
When to Use These provide additional practice options or may be used as homeworkfor second day teaching of the lesson.
Enrichment There is one master for eachlesson. These activities may extend the conceptsin the lesson, offer a historical or multiculturallook at the concepts, or widen students’perspectives on the mathematics they are learning. These are not written exclusively for honors students, but are accessible for usewith all levels of students.
When to Use These may be used as extracredit, short-term projects, or as activities fordays when class periods are shortened.
© Glencoe/McGraw-Hill v Advanced Mathematical Concepts
Assessment Options
The assessment section of the Chapter 7Resources Masters offers a wide range ofassessment tools for intermediate and finalassessment. The following lists describe eachassessment master and its intended use.
Chapter Assessments
Chapter Tests• Forms 1A, 1B, and 1C Form 1 tests contain
multiple-choice questions. Form 1A isintended for use with honors-level students,Form 1B is intended for use with average-level students, and Form 1C is intended foruse with basic-level students. These testsare similar in format to offer comparabletesting situations.
• Forms 2A, 2B, and 2C Form 2 tests arecomposed of free-response questions. Form2A is intended for use with honors-levelstudents, Form 2B is intended for use withaverage-level students, and Form 2C isintended for use with basic-level students.These tests are similar in format to offercomparable testing situations.
All of the above tests include a challengingBonus question.
• The Extended Response Assessmentincludes performance assessment tasks thatare suitable for all students. A scoringrubric is included for evaluation guidelines.Sample answers are provided for assessment.
Intermediate Assessment• A Mid-Chapter Test provides an option to
assess the first half of the chapter. It iscomposed of free-response questions.
• Four free-response quizzes are included tooffer assessment at appropriate intervals inthe chapter.
Continuing Assessment• The SAT and ACT Practice offers
continuing review of concepts in variousformats, which may appear on standardizedtests that they may encounter. This practiceincludes multiple-choice, quantitative-comparison, and grid-in questions. Bubble-in and grid-in answer sections are providedon the master.
• The Cumulative Review provides studentsan opportunity to reinforce and retain skillsas they proceed through their study ofadvanced mathematics. It can also be usedas a test. The master includes free-responsequestions.
Answers• Page A1 is an answer sheet for the SAT and
ACT Practice questions that appear in theStudent Edition on page 483. Page A2 is ananswer sheet for the SAT and ACT Practicemaster. These improve students’ familiaritywith the answer formats they mayencounter in test taking.
• The answers for the lesson-by-lesson masters are provided as reduced pages withanswers appearing in red.
• Full-size answer keys are provided for theassessment options in this booklet.
primarily skillsprimarily conceptsprimarily applications
BASIC AVERAGE ADVANCED
Study Guide
Vocabulary Builder
Parent and Student Study Guide (online)
Practice
Enrichment
4
5
3
2
Five Different Options to Meet the Needs of Every Student in a Variety of Ways
1
© Glencoe/McGraw-Hill vi Advanced Mathematical Concepts
Chapter 7 Leveled Worksheets
Glencoe’s leveled worksheets are helpful for meeting the needs of everystudent in a variety of ways. These worksheets, many of which are foundin the FAST FILE Chapter Resource Masters, are shown in the chartbelow.
• Study Guide masters provide worked-out examples as well as practiceproblems.
• Each chapter’s Vocabulary Builder master provides students theopportunity to write out key concepts and definitions in their ownwords.
• Practice masters provide average-level problems for students who are moving at a regular pace.
• Enrichment masters offer students the opportunity to extend theirlearning.
Reading to Learn MathematicsVocabulary Builder
NAME _____________________________ DATE _______________ PERIOD ________
This is an alphabetical list of the key vocabulary terms you will learn in Chapter 7.As you study the chapter, complete each term’s definition or description.Remember to add the page number where you found the term.
© Glencoe/McGraw-Hill vii Advanced Mathematical Concepts
Vocabulary Term Foundon Page Definition/Description/Example
counterexample
difference identity
double-angle identity
half-angle identity
identity
normal form
normal line
opposite-angle identity
principal value
Pythagorean identity
(continued on the next page)
Chapter
7
© Glencoe/McGraw-Hill viii Advanced Mathematical Concepts
Reading to Learn MathematicsVocabulary Builder (continued)
NAME _____________________________ DATE _______________ PERIOD ________
Vocabulary Term Foundon Page Definition/Description/Example
quotient identity
reciprocal identity
reduction identity
sum identity
symmetry identity
trigonometric identity
Chapter
7
© Glencoe/McGraw-Hill 275 Advanced Mathematical Concepts
Study GuideNAME _____________________________ DATE _______________ PERIOD ________
7-1
Basic Trigonometric IdentitiesYou can use the trigonometric identities to help find thevalues of trigonometric functions.
Example 1 If sin � � �35�, find tan �.
Use two identities to relate sin � and tan �.
sin2 � � cos2 � � 1 Pythagorean identity
��35��2� cos2 � � 1 Substitute �35� for sin �.
cos2 � � �2156�
cos � � ���21�56�� or ��5
4�
To determine the sign of a function value, use the symmetryidentities for sine and cosine. To use these identities with radianmeasure, replace 180� with � and 360� with 2�.
Example 2 Express tan �131�� as a trigonometric function of an angle
in Quadrant I.
The sum of �131�� and ��3�, which is �12
3�� or 4�, is a
multiple of 2�.
�113
�� � 2(2�) � ��3� Case 3, with A � ��3� and k � 2
tan �113
�� � Quotient identity
�
� Symmetry identities
� �tan ��3� Quotient identity
�sin ��3�
�cos �
�3�
sin �2(2�) � ��3����cos �2(2�) � ��3��
sin �131��
��cos �13
1��
Now find tan �.tan � � �c
soins
��
� Quotient identity
tan � �
tan � � ��43�
�35�
���5
4�
Case 1: sin (A � 360k�) � sin A cos (A � 360k�) � cos ACase 2: sin [A � 180�(2k � 1)] � �sin A cos [A � 180�(2k � 1)] � �cos ACase 3: sin (360k� � A) � �sin A cos (360k� � A) � cos ACase 4: sin [180�(2k � 1) � A] � sin A cos [180�(2k � 1) � A] � �cos A
© Glencoe/McGraw-Hill 276 Advanced Mathematical Concepts
Basic Trigonometric Identities
Use the given information to determine the exact trigonometricvalue if 0� � � � 90�.
1. If cos � � �14�, find tan �. 2. If sin � � �23�, find cos �.
3. If tan � � �72�, find sin �. 4. If tan � � 2, find cot �.
Express each value as a trigonometric function of an angle inQuandrant I.
5. cos 892� 6. csc 495� 7. sin �233��
Simplify each expression.
8. cos x � sin x tan x 9. �tcaont A
A� 10. sin2 � cos2 � � cos2 �
11. Kite Flying Brett and Tara are flying a kite. When the string istied to the ground, the height of the kite can be determined by the formula �H
L� � csc �, where L is the length of the string and � is the angle between the string and the level ground. What formulacould Brett and Tara use to find the height of the kite if theyknow the value of sin �?
PracticeNAME _____________________________ DATE _______________ PERIOD ________
7-1
© Glencoe/McGraw-Hill 277 Advanced Mathematical Concepts
EnrichmentNAME _____________________________ DATE _______________ PERIOD ________
7-1
The Physics of SoccerRecall from Lesson 7-1 that the formula for the maximum height hof a projectile is , where � is the measure of the angle of elevation in degrees, v0 is the initial velocity in feet per second, and gis the acceleration due to gravity in feet per second squared.
Solve. Give answers to the nearest tenth.
1. A soccer player kicks a ball at an initial velocity of 60 ft/s and anangle of elevation of 40°. The acceleration due to gravity is 32ft/s2. Find the maximum height reached by the ball.
2. With what initial velocity must you kick a ball at an angle of 35°in order for it to reach a maximum height of 20 ft?
The distance d that a projected object travels is given by the
formula
3. Find the distance traveled by the ball described in Exercise 1.
In order to kick a ball the greatest possible distance at a given
initial velocity, a soccer player must maximizeSince 2, v0, and g are constants, this means the player must maximize sin � cos �.
sin 0°cos 0° � sin 90°cos 90° � 0
sin 10°cos 10° � sin 80°cos 80° � 0.1710
sin 20°cos 20° � sin 70°cos 70° � 0.3214
4. Use the patterns in the table to hypothesize a value of � for whichsin � cos � will be maximal. Use a calculator to check your hypothesis. At what angle should the player kick the ball toachieve the greatest distance?
h �v0
2 sin2 ���2g
d � .2v0
2 sin � cos ���g
d � .2v0
2 sin � cos ���g
© Glencoe/McGraw-Hill 278 Advanced Mathematical Concepts
Verifying Trigonometric IdentitiesWhen verifying trigonometric identities, you cannot add orsubtract quantities from each side of the identity. Anunverified identity is not an equation, so the properties ofequality do not apply.
Example 1 Verify that �sesce2
cx2
�x
1� � sin2 x is an identity.
Since the left side is more complicated,transform it into the expression on the right.
�sesce2
cx2
�x
1�� sin2 x
�(tan2
sxec
�2 x
1) � 1�� sin2 x sec2 x � tan2 x � 1
�tsaenc2
2
xx� � sin2 x Simplify.
� sin2 x tan2 x � �csoins
2
2xx�, sec2 x � �cos
12 x�
�csoins
2
2xx� � cos2 x � sin2 x
sin2 x � sin2 x Multiply.
The techniques that you use to verify trigonometric identitiescan also be used to simplify trigonometric equations.
Example 2 Find a numerical value of one trigonometricfunction of x if cos x csc x � 3.You can simplify the trigonometric epression on the leftside by writing it in terms of sine and cosine.
cos x csc x � 3
cos x � �sin1
x� � 3 csc x � �sin1
x�
�csoins
xx� � 3 Multiply.
cot x � 3 cot x � �csoins
xx�
Therefore, if cos x csc x � 3, then cot x � 3.
�csoins
2
2xx�
��cos
12 x�
Study GuideNAME _____________________________ DATE _______________ PERIOD ________
7-2
© Glencoe/McGraw-Hill 279 Advanced Mathematical Concepts
PracticeNAME _____________________________ DATE _______________ PERIOD ________
Verifying Trigonometric Identities
Verify that each equation is an identity.
1. �cot xcs
�c
txan x� � cos x
2. �sin y1
� 1� � �sin y1
� 1� � �2 sec2 y
3. sin3 x � cos3 x � (1 � sin x cos x)(sin x � cos x)
4. tan � � �1 �cos
si�n �
� � sec �
Find a numerical value of one trigonometric function of x.
5. sin x cot x � 1 6. sin x � 3 cos x 7. cos x � cot x
8. Physics The work done in moving an object is given by the formula W � Fd cos �, where d is the displacement, F is the force exerted, and � is the angle between the displacement and the force. Verify that W � Fd �c
csoct �
�� is an equivalent formula.
7-2
© Glencoe/McGraw-Hill 280 Advanced Mathematical Concepts
EnrichmentNAME _____________________________ DATE _______________ PERIOD ________
7-2
Building from 1 � 1By starting with the most fundamental identity of all, 1 � 1, you cancreate new identities as complex as you would like them to be.
First, think of ways to write 1 using trigonometric identities. Someexamples are the following.
1 � cos A sec A
1 � csc2 A � cot2 A
1 �
Choose two such expressions and write a new identity.
cos A sec A � csc2 A � cot2 A
Now multiply the terms of the identity by the terms of another identityof your choosing, preferably one that will allow some simplificationupon multiplication.
cos A sec A � csc2A � cot2A
� � tan A_________________________________sin A sec A � tan A csc2 A � cot A
Beginning with 1 � 1, create two trigonometric identities.
1. _____________________________________________
2. _____________________________________________
Verify that each of the identities you created is an identity.
3. __________________________ 4. _________________________
__________________________ _________________________
__________________________ _________________________
__________________________ _________________________
__________________________ _________________________
__________________________ _________________________
cos (A � 360°)��cos (360° � A)
sin A�cos A
© Glencoe/McGraw-Hill 281 Advanced Mathematical Concepts
Study GuideNAME _____________________________ DATE _______________ PERIOD ________
Sum and Difference IdentitiesYou can use the sum and difference identities and thevalues of the trigonometric functions of common angles tofind the values of trigonometric functions of other angles.Notice how the addition and subtraction symbols are relatedin the sum and difference identities.
Example 1 Use the sum or difference identity for cosine tofind the exact value of cos 375�.
375� � 360� � 15�cos 375� � cos 15� Symmetry identity, Case 1
cos 15� � cos (60� � 45�) 60� and 45� are two common angles that differ by 15�.
cos 15� � cos 60� cos 45� � sin 60� sin 45� Difference identity for cosine
cos 15� � �12� � ��22�� � ��2
3�� � ��22�� or ��2� �
4�6��
Example 2 Find the value of sin (x � y) if 0 � x � ��2�, 0 � y � ��2�,
sin x � �35�, and sin y � �1327�.
In order to use the sum identity for sine, youneed to know cos x and cos y. Use a Pythagoreanidentity to determine the necessary values.
sin2 � � cos2 � � 1 ⇒ cos2 � � 1 � sin2 �. Pythagorean identity
Since it is given that the angles are in Quadrant I,the values of sine and cosine are positive. Therefore,cos � � �1� �� s�in�2���.
cos x � �1� �� ���35���2� cos y � �1� �� ���13�2
7���2�
� ��12�65�� or �45� � ��11�2
3�26�59�� or �33
57�
Now substitute these values into the sumidentity for sine.
sin (x � y) � sin x cos y � cos x sin y
� ��35����3357�� � ��45����13
27�� or �1
18553�
7-3
Sum and Difference Identities
Cosine function cos (� � �) � cos � cos � � sin � sin �
Sine function sin (� � �) � sin � cos � � cos � sin �
Tangent function tan (� � �) ��1ta�
nt�
an�
�
tatann�
��
�
© Glencoe/McGraw-Hill 282 Advanced Mathematical Concepts
Sum and Difference Identities
Use sum or difference identities to find the exact value of each trigonometricfunction.
1. cos �51�2� 2. sin (�165�) 3. tan 345�
4. csc 915� 5. tan ���71�2�� 6. sec �1
�2�
Find each exact value if 0 � x � ��2
� and 0 � y � ��2
� .
7. cos (x � y) if sin x � �153� and sin y � �5
4�
8. sin (x � y) if cos x � �187� and cos y � �5
3�
9. tan (x � y) if csc x � �153� and cot y � �3
4�
Verify that each equation is an identity.
10. cos (180� � �) � �cos � 11. sin (360� � �) � sin �
12. Physics Sound waves can be modeled by equations of the form y � 20 sin (3t � �). Determine what type of interferenceresults when sound waves modeled by the equations y � 20 sin (3t � 90�) and y � 20 sin (3t � 270�) are combined.(Hint: Refer to the application in Lesson 7-3.)
PracticeNAME _____________________________ DATE _______________ PERIOD ________
7-3
© Glencoe/McGraw-Hill 283 Advanced Mathematical Concepts
EnrichmentNAME _____________________________ DATE _______________ PERIOD ________
7-3
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.
Example Write sin 3� cos � as a sum.
In the right side of identity (i) let � � 3� and � � � sothat 2 sin 3� cos � �sin (3� � �) � sin (3� � �).Thus, sin 3� cos� � sin 4� � sin 2�.
By subtracting the identities for sin (� � �) and sin (� � �), you obtain a similar identity for expressing aproduct as a difference.
(ii) sin (� � �) � sin (� � �) � 2 cos � sin �
Example Verify the identity �csions
22xx
csoins
xx� � .
In the right sides of identities (i) and (ii) let � � 2x and �� x. Then write the following quotient.
�
By simplifying and multiplying by the conjugate, theidentity is verified.
� .
�
Complete.
1. Use the identities for cos (� � �) and cos (� � �) to find identities for expressing theproducts 2 cos � cos � and 2 sin � sin � as a sum or difference.
2. Find the value of sin 105° cos 75° by using the identity above.
(sin 3x � sin x)2��sin2 3x � sin2 x
sin 3x � sin x��sin 3x � sin x
sin 3x � sin x��sin 3x � sin x
cos 2x sin x��sin 2x cos x
sin (2x � x) � sin (2x � x)����sin (2x � x) � sin (2x � x)
2 cos 2x sin x��2 sin 2x cos x
(sin 3x � sin x)2���sin2 3x � sin2 x
1�2
1�2
© Glencoe/McGraw-Hill 284 Advanced Mathematical Concepts
Double-Angle and Half-Angle Identities
Example 1 If sin � � �14� and � has its terminal side in the firstquadrant, find the exact value of sin 2�.
To use the double-angle identity for sin 2�, we mustfirst find cos �.
sin2 � � cos2 � � 1
��14��2
� cos2 � � 1 sin � � �14�
cos2 � � �1165�
cos � � ��41�5��
Now find sin 2�.
sin 2� � 2 sin � cos � Double-angle identity for sine
� 2��14����41�5�� sin � � �14�, cos � � ��4
1�5��
� ��81�5��
Example 2 Use a half-angle identity to find the exact valueof sin �1
�2�.
sin �1�2� � sin
� �� Use sin ��2� � ���
1� �� 2c�o�s�a��. Since �1
�2� is in
Quadrant I, choose the positive sine value.
� ��� ��2� ��4
��3���� �
�2�2�� ��3���
1���23��
�2
1 � cos �6��
��2
�6���2
Study GuideNAME _____________________________ DATE _______________ PERIOD ________
7-4
© Glencoe/McGraw-Hill 285 Advanced Mathematical Concepts
PracticeNAME _____________________________ DATE _______________ PERIOD ________
Double-Angle and Half-Angle Identities
Use a half-angle identity to find the exact value of each function.
1. sin 105� 2. tan ��8� 3. cos �58��
Use the given information to find sin 2�, cos 2�, and tan 2�.
4. sin � � �1123�, 0� � 90� 5. tan � � �12�, � � �32
��
6. sec � � ��52�, ��2� � � 7. sin � � �35�, 0 � �2��
Verify that each equation is an identity.
8. 1 � sin 2x � (sin x � cos x)2
9. cos x sin x � �sin22x�
10. Baseball A batter hits a ball with an initial velocity v0 of 100 feet per second at an angle � to the horizontal. An outfieldercatches the ball 200 feet from home plate. Find � if the range of a projectile is given by the formula R � �3
12�v0
2 sin 2�.
7-4
© Glencoe/McGraw-Hill 286 Advanced Mathematical Concepts
EnrichmentNAME _____________________________ DATE _______________ PERIOD ________
7-4
Reading Mathematics: Using ExamplesMost mathematics books, including this one, use examples to illustratethe material of each lesson. Examples are chosen by the authors toshow how to apply the methods of the lesson and to point out placeswhere possible errors can arise.
1. Explain the purpose of Example 1c in Lesson 7-4.
2. Explain the purpose of Example 3 in Lesson 7-4.
3. Explain the purpose of Example 4 in Lesson 7-4.
To make the best use of the examples in a lesson, try following thisprocedure:
a. When you come to an example, stop. Think about what you havejust read. If you don’t understand it, reread the previous section.
b. Read the example problem. Then instead of reading the solution,try solving the problem yourself.
c. After you have solved the problem or gone as far as you can go,study the solution given in the text. Compare your method andsolution with those of the authors. If necessary, find out whereyou went wrong. If you don’t understand the solution, reread thetext or ask your teacher for help.
4. Explain the advantage of working an example yourself over simply reading the solution given in the text.
© Glencoe/McGraw-Hill 287 Advanced Mathematical Concepts
Study GuideNAME _____________________________ DATE _______________ PERIOD ________
7-5
Solving Trigonometric EquationsWhen you solve trigonometric equations for principal values ofx, x is in the interval �90� x 90� for sin x and tan x. Forcos x, x is in the interval 0� x 180�. If an equation cannotbe solved easily by factoring, try writing the expressions interms of only one trigonometric function.
Example 1 Solve tan x cos x � cos x � 0 for principal valuesof x. Express solutions in degrees.
tan x cos x � cos x � 0cos x (tan x � 1) � 0 Factor.
cos x � 0 or tan x � 1 � 0 Set each factor equal to 0.x � 90� tan x � 1
x � 45�
When x � 90°, tan x is undefined, so the only principal value is 45�.
Example 2 Solve 2 tan2 x � sec2 x � 3 � 1 � 2 tan x for 0 � x � 2�.
This equation can be written in terms of tan x only.
2 tan2 x � sec2 x � 3 � 1 � 2 tan x2 tan2 x � (tan2 x � 1) � 3 � 1 � 2 tan x sec2 x � tan2 x � 1
tan2 x � 2 � 1 � 2 tan x Simplify.tan2 x � 2 tan x � 1 � 0
(tan x � 1)2 � 0 Factor.tan x � 1 � 0 Take the square root of each side.
tan x � �1x � �34
�� or x � �74��
When you solve for all values of x, the solution should berepresented as x � 360�k or x � 2�k for sin x and cos x andx � 180�k or x � �k for tan x, where k is any integer.
Example 3 Solve sin x � �3� � �sin x for all real values of x.sin x � �3� � �sin x
2 sin x � �3� � 02 sin x � ��3�
sin x � ���23��
x � �43�� � 2�k or x � �53
�� � 2�k, where k is any integer
The solutions are �43�� � 2�k and �53
�� � 2�k.
© Glencoe/McGraw-Hill 288 Advanced Mathematical Concepts
Solving Trigonometric Equations
Solve each equation for principal values of x. Expresssolutions in degrees.
1. cos x � 3 cos x � 2 2. 2 sin2 x � 1 � 0
Solve each equation for 0� � x � 360�.3. sec2 x � tan x � 1 � 0 4. cos 2x � 3 cos x � 1 � 0
Solve each equation for 0 � x � 2�.5. 4 sin2 x � 4 sin x � 1 � 0 6. cos 2x � sin x � 1
Solve each equation for all real values of x.7. 3 cos 2x � 5 cos x � 1 8. 2 sin2 x � 5 sin x � 2 � 0
9. 3 sec2 x � 4 � 0 10. tan x (tan x � 1) � 0
11. Aviation An airplane takes off from the ground and reaches a height of 500 feet after f lying 2 miles. Given the formula H � d tan �, where H is the height of the plane and d is the distance (along the ground) the plane has flown, find the angle of ascent � at which the plane took off.
PracticeNAME _____________________________ DATE _______________ PERIOD ________
7-5
The SpectrumIn some ways, light behaves as though it were composed of waves. Thewavelength of visible light ranges from about 4 � 10�5 cm for violetlight to about 7 � 10�5 cm for red light.
As light passes through a medium, its velocity depends upon the wavelength of the light. The greater the wavelength, the greater thevelocity. Since white light, including sunlight, is composed of light ofvarying wavelengths, waves will pass through the medium at an infinite number of different speeds. The index of refraction n of the medium is defined by n � , where c is the velocity of light in avacuum (3 � 1010 cm/s), and v is the velocity of light in the medium.As you can see, the index of refraction of a medium is not a constant. Itdepends on the wavelength and the velocity of light passing through it.(The index of refraction of diamond given in the lesson is an average.)
1. For all media, n > 1. Is the speed of light in a medium greater thanor less than c? Explain.
2. A beam of violet light travels through water at a speed of 2.234 � 1010 cm/s. Find the index of refraction of water for violetlight.
The diagram shows why a prism splits white lightinto a spectrum. Because they travel at differentvelocities in the prism, waves of light of different colors are refracted different amounts.
3. Beams of red and violet light strike crown glass at an angle of 20°. Use Snell’s Law to find the difference between the angles ofrefraction of the two beams.
violet light: n � 1.531 red light: n � 1.513
© Glencoe/McGraw-Hill 289 Advanced Mathematical Concepts
NAME _____________________________ DATE _______________ PERIOD ________
Enrichment7-5
c�v
© Glencoe/McGraw-Hill 290 Advanced Mathematical Concepts
Normal Form of a Linear Equation
You can write the standard form of a linear equation if youare given the values of � and p.
Example 1 Write the standard form of the equation of a linefor which the length of the normal segment tothe origin is 5 and the normal makes an angle of135� with the positive x-axis.
x cos � � y sin � � p � 0 Normal formx cos 135� � y sin 135� � 5 � 0 � � 135° and p � 5
���22��x � ��2
2��y � 5 � 0
�2�x � �2�y � 10 � 0 Multiply each side by �2.
The standard form of the equation is �2�x � �2�y � 10 � 0.
The standard form of a linear equation, Ax � By � C � 0, can bechanged to the normal form by dividing each term of the equationby ��A�2��� B�2�. The sign is chosen opposite the sign of C. You canthen find the length of the normal, p units, and the angle �.
Example 2 Write 3x � 4y � 10 � 0 in normal form. Then findthe length of the normal and the angle it makeswith the positive x-axis.
Since C is negative, use �A�2��� B�2� to determinethe normal form.�A�2��� B�2� � �3�2��� 4�2� or 5
The normal form is �35�x � �45� y � �150� � 0 or �35�x � �45�y � 2 � 0.
Therefore, cos � � �35�, sin � � �45�, and p � 2.
Since cos � and sin � are both positive, � mustlie in Quadrant I.
tan � � or �43� tan � � �scions �
��
� 53�
The normal segment has length 2 units andmakes an angle of 53� with the positive x-axis.
�45���35�
Study GuideNAME _____________________________ DATE _______________ PERIOD ________
7-6
Normal FormThe normal form of a linear equation is x cos � � y sin � � p � 0,where p is the length of the normal from the line to the origin and �is the positive angle formed by the positive x-axis and the normal.
© Glencoe/McGraw-Hill 291 Advanced Mathematical Concepts
PracticeNAME _____________________________ DATE _______________ PERIOD ________
Normal Form of a Linear Equation
Write the standard form of the equation of each line, givenp, the length of the normal segment, and �, the angle thenormal segment makes with the positive x-axis.
1. p � 4, � � 30� 2. p � 2�2�, � � �4��
3. p � 3, � � 60� 4. p � 8, � � �56��
5. p � 2�3�, � � �74�� 6. p � 15, � � 225�
Write each equation in normal form. Then find the length ofthe normal and the angle it makes with the positive x-axis.
7. 3x � 2y � 1 � 0
8. 5x � y � 12 � 0
9. 4x � 3y � 4 � 0
10. y � x � 5
11. 2x � y � 1 � 0
12. x � y � 5 � 0
7-6
© Glencoe/McGraw-Hill 292 Advanced Mathematical Concepts
Slopes of Perpendicular LinesThe derivation of the normal form of a linear equation uses this familiar theorem, first stated in Lesson 1-6: Two nonvertical lines areperpendicular if and only if the slope of one is the negative reciprocalof the slope of the other.
You can use trigonometric identities to prove that iflines are perpendicular, then their slopes are negative reciprocals of each other.
�1 and �2 are perpendicular lines.�1 and �2 are the angles that �1 and �2,respectively, make with the horizontal.Let m1 � slope of �1
m2 � slope of �2
Complete the following exercises to prove that m1 � � .
1. Explain why m1 � tan �1 and m2 � tan �2.
2. According to the difference identity for the cosine function,cos (�2 � �1) � cos �2 cos �1 � sin �2 sin �1. Explain why the left side of the equation is equal to zero.
3. cos �2 cos �1 � sin �2 sin �1 � 0sin �2 sin �1 � � cos �2 cos �1
� �
Complete using the tangent function. ____________ � ____________
Complete, using m1 and m2. ____________ � ____________
cos �2�sin �2
sin �1�cos �1
1�m2
EnrichmentNAME _____________________________ DATE _______________ PERIOD ________
7-6
© Glencoe/McGraw-Hill 293 Advanced Mathematical Concepts
Study GuideNAME _____________________________ DATE _______________ PERIOD ________
Distance from a Point to a LineThe distance from a point at (x1, y1) to a line with equationAx � By � C � 0 can be determined by using the formula
d � �A
�
x1
�
�
A�
B2�
y
��1 �
B�2�
C�. The sign of the radical is chosen opposite
the sign of C.
Example 1 Find the distance between P(3, 4) and the linewith equation 4x � 2y � 10.
First, rewrite the equation of the line in standard form.
4x � 2y � 10 � 0
Then, use the formula for the distance from a point to a line.
d �
d � A � 4, B � 2, C � �10, x1 � 3, y1 � 4
d � �21�05�
� or �5� Since C is negative, use ��A�2��� B�2�.
d 2.24 units
Therefore, P is approximately 2.24 units from the linewith equation 4x � 2y � 10. Since d is positive, P is onthe opposite side of the line from the origin.
You can also use the formula to find the distancebetween two parallel lines. To do this, choose any pointon one of the lines and use the formula to find thedistance from that point to the other line.
Example 2 Find the distance between the lines withequations 2x � 2y � 5 and y � x � 1.
Since y � x � 1 is in slope-intercept form, youcan see that it passes through the point at (0, �1). Use this point to find the distance to the other line.
The standard form of the other equation is 2x � 2y � 5 � 0.
d �
d � A � 2, B � �2, C � �5, x1 � 0, y1 � �1
d � ��2�
32�
� or ��3�4
2�� Since C is negative, use ��A�2��� B�2�.
�1.06
The distance between the lines is about 1.06 units.
2(0) � 2(�1) � 5����2�2��� (���2�)2�
Ax1 � By1 � C��
��A�2��� B�2�
4(3) � 2(4) � 10��
��4�2��� 2�2�
Ax1 � By1 � C��
��A�2��� B�2�
7-7
© Glencoe/McGraw-Hill 294 Advanced Mathematical Concepts
Distance From a Point to a Line
Find the distance between the point with the given coordinatesand the line with the given equation.
1. (�1, 5), 3x � 4y � 1 � 0 2. (2, 5), 5x � 12y � 1 � 0
3. (1, �4), 12x � 5y � 3 � 0 4. (�1, �3), 6x � 8y � 3 � 0
Find the distance between the parallel lines with the givenequations.
5. 2x � 3y � 4 � 0 6. 4x � y � 1 � 0y � �23�x � 5 4x � y � 8 � 0
Find equations of the lines that bisect the acute and obtuseangles formed by the lines with the given equations.
7. x � 2y � 3 � 0x � y � 4 � 0
8. 9x � 12y � 10 � 03x � 2y � 6 � 0
PracticeNAME _____________________________ DATE _______________ PERIOD ________
7-7
© Glencoe/McGraw-Hill 295 Advanced Mathematical Concepts
EnrichmentNAME _____________________________ DATE _______________ PERIOD ________
Deriving the Point-Line DistanceLine � has the equation Ax � By � C � 0.Answer these questions to derive the formula given in Lesson 7-7 for the distance from P(x1, y1) to � .
1. Use the equation of the line to find the coordinates of J and K, the x- and y-intercepts of � .
2. P�Q� is a vertical segment from P to �. Find the x-coordinate of Q.
3. Since Q is on �, its coordinates must satisfy the equation of �.Use your answer to Exercise 2 to find the y-coordinate of Q.
4. Find PQ by finding the difference between the y-coordinates of Pand Q. Write your answer as a fraction.
5. Triangle KJO is a right triangle. Use your answers to Exercise 1 and the Pythagorean Theorem to find KJ. Simplify.
6. Since �Q �K , �JKO ~ �PQR .
�
Use your answers to Exercises 1, 4, and 5 to find PR, the distancefrom P to �. Simplify.
PQ�KJ
PR�OJ
7-7
BLANK
© Glencoe/McGraw-Hill 297 Advanced Mathematical Concepts
Chapter 7 Test, Form 1A
NAME _____________________________ DATE _______________ PERIOD ________Chapter
7Write the letter for the correct answer in the blank at the right ofeach problem.
1. Find an expression equivalent to �secs�in
ta�n ��. 1. ________
A. sec2 � B. cot � C. tan2 � D. cos2 �
2. If csc � � ��54� and 180� � 270�, find tan �. 2. ________
A. ��43� B. �34� C. �43� D. ��54�
3. Simplify �tan2 �ta
cns
2c2
�� � 1�. 3. ________
A. csc2 � B. �1 C. tan2 � D. 1
4. Simplify �seccoxs
�x
1� � �seccoxs
�x
1�. 4. ________
A. 2 tan2 x B. 2 cos x C. 2 cos2 x � 1 D. 2 cot2 x
5. Find a numerical value of one trigonometric function of x if 5. ________�tcaont x
x� � �scoesc x
x� � �cs2c x�.
A. csc x � 1 B. sin x � ��12� C. csc x � �1 D. sin x � �21�
6. Use a sum or difference identity to find the exact value of sin 255�. 6. ________
A. ���2�4� �6�� B. �
�6� �4
�2�� C. �
�6� �4
�2�� D. �
�2� �4
�6��
7. Find the value of tan (� � �) if cos � � ��35�, sin � � �153�, 7. ________
90� � 180�, and 90� � 180�.
A. �6536� B. ��65
36� C. ��35
36� D. �5
363�
8. Which expression is equivalent to cos (� � �)? 8. ________A. �cos � B. cos � C. �sin � D. sin �
9. Which expression is not equivalent to cos 2�? 9. ________A. cos2 � � sin2 � B. 2 cos2 � � 1C. 1 � 2 sin2 � D. 2 sin � cos �
10. If cos � � 0.8 and 270� � 360�, find the exact value of sin 2�. 10. ________A. �0.96 B. �0.6 C. 0.96 D. 0.28
11. If csc � � ��53� and � has its terminal side in Quadrant III, find the 11. ________exact value of tan 2�.
A. �2245� B. �2
75� C. �27
4� D. ��275�
© Glencoe/McGraw-Hill 298 Advanced Mathematical Concepts
12. Use a half-angle identity to find the exact value of cos 165�. 12. ________A. �12� �2� �� ��3�� B. ��12� �2� �� ��3��
C. �12� �2� �� ��2�� D. ��12� �1� �� ��3��
13. Solve 4 sin2 x � 4�2� cos x � 6 � 0 for all real values of x. 13. ________A. �34
�� � 2�k, �54�� � 2�k B. ��4� � 2�k, �74
�� � 2�k
C. ��4� � 2�k, �54�� � 2�k D. �34
�� � 2�k, �74�� � 2�k
14. Solve 2 cos2 x � 5 cos x � 2 � 0 for principal values of x. 14. ________A. 0� and 30� B. 30� C. 60� D. 60� and 300�
15. Solve 2 sin x � �3� 0 for 0 x 2�. 15. ________A. �43
�� x �53�� B. �23
�� x �43��
C. �76�� x �11
6�� D. �56
�� x �76��
16. Write the equation 2x � 3y � 5 � 0 in normal form. 16. ________
A. �2�13
1�3��x � �3�13
1�3��y � �5�13
1�3�� � 0 B. ��2�13
1�3��x � �3�13
1�3��y � �5�13
1�3�� � 0
C. ���2�13
1�3��x � �3�13
1�3��y � �5�13
1�3�� � 0 D. �2�13
1�3��x � �3�13
1�3��y � �5�13
1�3�� � 0
17. Write the standard form of the equation of a line for which the 17. ________length of the normal is 6 and the normal makes an angle of 120�with the positive x-axis.A. x � �3� y � 12 � 0 B. x � �3� y � 12 � 0C. �3� x � y � 12 � 0 D. �3� x � y � 12 � 0
18. Find the distance between P(�4, 3) and the line with equation 18. ________2x � 5y � �7.
A. �14
2�92�9�
� B. 0 C. ��162�92�9�� D. �16
2�92�9��
19. Find the distance between the lines with equations 3x � y � 9 and 19. ________y � 3x � 4.
A. �54� B. �5�
21�0�� C. ��2
1�0�� D. �13�2
1�0��
20. Find an equation of the line that bisects the obtuse angles formed by 20. ________the lines with equations 3x � y � 1 and x � y � �2.A. (3�2� � �1�0�) x � (�1�0� � �2�) y � 2�1�0� � �2� � 0B. (3�2� � �1�0�) x � (�1�0� � �2�) y � 2�1�0� � �2� � 0C. (3�2� � �1�0�) x � (�1�0� � �2�) y � 2�1�0� � �2� � 0D. (3�2� � �1�0�) x � (�1�0� � �2�) y � 2�1�0� � �2� � 0
Bonus If 90� � 180� and cos � � ��45�, find sin 4�. Bonus: ________
A. ��4285� B. �42
85� C. �36
3265� D. ��36
3265�
Chapter 7 Test, Form 1A (continued)
NAME _____________________________ DATE _______________ PERIOD ________Chapter
7
© Glencoe/McGraw-Hill 299 Advanced Mathematical Concepts
Chapter 7 Test, Form 1B
NAME _____________________________ DATE _______________ PERIOD ________
Write the letter for the correct answer in the blank at the right ofeach problem.
1. Find an expression equivalent to sec � sin � cot � csc �. 1. ________A. tan � B. csc � C. sec � D. sin �
2. If sec � � ��54� and 180� � 270�, find tan �. 2. ________
A. ��35� B. ��45� C. �34� D. �53�
3. Simplify �tatna
2
n�2�
�
1�. 3. ________
A. csc2 � B. �1 C. tan2 � D. 1
4. Simplify �tsainn
xx� � �co
1s x�. 4. ________
A. 2 tan2 x B. 2 cos x C. 2 cos x � 1 D. 2 sec x
5. Find a numerical value of one trigonometric function of 5. ________x if sec x cot x � 4.
A. csc x � �14� B. sec x � 4 C. sec x � �14� D. csc x � 4
6. Use a sum or difference identity to find the exact value of sin 105�. 6. ________
A. ���2�4� �6�� B. ��6� �
4�2�� C. ��6� �
4�2�� D. ��2� �
4�6��
7. Find the value of tan (� ��) if cos � � �45�, sin � � ��153�, 7. ________
270� � 360�, and 270� � 360�.
A. �1663� B. ��16
63� C. ��53
63� D. �53
63�
8. Which expression is equivalent to cos (� � �)? 8. ________A. �cos � B. cos � C. �sin � D. sin �
9. Which expression is not equivalent to cos 2�? 9. ________A. 2 cos2 � �1 B. 1 �2 sin2 � C. cos2 � � sin2 � D. cos2 � �sin2 �
10. If sin � � 0.6 and 90� � 180�, find the exact value of sin 2�. 10. ________A. �0.6 B. �0.96 C. 0.96 D. 0.28
11. If cos � � ��35� and � has its terminal side in Quadrant II, find 11. ________the exact value of tan 2�.A. �22
45� B. �2
75� C. �27
4� D. ��275�
12. Use a half-angle identity to find the exact value of cos 75�. 12. ________
A. �12��2� �� ��3�� B. �12��2� �� ��3�� C. �12��2� �� ��2�� D. ��12��1� �� ��3��
Chapter
7
© Glencoe/McGraw-Hill 300 Advanced Mathematical Concepts
13. Solve csc x � 2 � 0 for 0 x 2�. 13. ________
A. ��6� and �56�� B. ��6� and �76
�� C. �43�� and �53
�� D. �76�� and �16
1��
14. Solve 2 cos2 x � cos x � 1 � 0 for principal values of x. 14. ________A. 0� and 120� B. 30� C. 60� D. 0� and 60�
15. Solve 4 sin2 x � 4 sin x � 1 � 0 for all real values of x. 15. ________A. ��6� � 2�k, �76
�� � 2�k B. �76�� � 2�k, �11
6�� � 2�k
C. �56�� � 2�k, �11
6�� � 2�k D. ��6� � 2�k, �56
�� � 2�k
16. Write the equation 3x�2y � 7 � 0 in normal form. 16. ________
A. �3�13
1�3��x � �2�13
1�3��y � �7�13
1�3�� � 0 B. ��3�13
1�3��x � �2�13
1�3��y � �7�13
1�3�� � 0
C. ��3�13
1�3��x � �2�13
1�3��y � �7�13
1�3�� � 0 D. �3�13
1�3��x � �2�13
1�3��y � �7�13
1�3�� � 0
17. Write the standard form of the equation of a line for which the 17. ________length of the normal is 3 and the normal makes an angle of 135�with the positive x-axis.A. �2�x ��2�y � 6 � 0 B. �2�x � �2�y � 6 � 0C. �2�x ��2�y � 6 � 0 D. �2�x � �2�y � 6 � 0
18. Find the distance between P(�2, 5) and the line with 18. ________equation x �3y � 4 � 0.
A. �171�01�0�
� B. 0 C. ��171�01�0�
� D. �131�01�0�
�
19. Find the distance between the lines with equations 19. ________5x � 12y � 12 and y � ��1
52� x � 3.
A. �2143� B. �41
83� C. ��41
83� D. �1
478�
20. Find an equation of the line that bisects the acute angles formed 20. ________by the lines with equations 2x � y � 5 � 0 and 3x � 2y � 6 � 0.A. (2�1�3� � 3�5�)x � (�1�3� � 2�5�)y � 5�1�3� � 6�5� � 0B. (�2�1�3� � 3�5�)x � (��1�3� � 2�5�)y � 5�1�3� � 6�5� � 0C. (�2�1�3� � 3�5�)x � (��1�3� � 2�5�)y � 5�1�3� � 6�5� � 0D. (�2�1�3� � 3�5�)x � (��1�3� � 2�5�)y � 5�1�3� � 6�5� � 0
Bonus If 90� � 180�, express cos � in terms of tan �. Bonus: ________
A. ���1� �� t1�a�n�2���� B. ��1� �� t
1�a�n�2���� C. �1� �� t�a�n�2��� D. ��1� �� t�a�n�2���
Chapter 7 Test, Form 1B (continued)
NAME _____________________________ DATE _______________ PERIOD ________Chapter
7
© Glencoe/McGraw-Hill 301 Advanced Mathematical Concepts
Chapter 7 Test, Form 1C
NAME _____________________________ DATE _______________ PERIOD ________
Write the letter for the correct answer in the blank at the right of each problem.
1. Find an expression equivalent to �csoins
���. 1. ________
A. tan � B. cot � C. sec � D. csc �
2. If sec � � �54� and 0� � 90�, find sin �. 2. ________
A. ��35� B. ��45� C. �34� D. �53�
3. Simplify �1 �
tanse
2c�
2 ��. 3. ________
A. csc2 � B. �1 C. tan2 � D. 1
4. Simplify �sec
12 x� � �
csc12 x�. 4. ________
A. 2 tan2 x B. 2 cos x C. 1 D. 2 cot2 x
5. Find a numerical value of one trigonometric function of x if 5. ________sin x cot x � �4
1�.
A. cos x � �41� B. sec x � �4
1� C. csc x � 4 D. cos x � 4
6. Use a sum or difference identity to find the exact value of sin 15�. 6. ________
A. ���2�4��6�� B. �
�6� �
4�2�� C. �
�6� �
4�2�
� D. ��2� �
4�6�
�
7. Find the value of tan (� � �) if cos � � �35�, sin � � �153�, 0� � 90�, 7. ________
and 0� � 90�.
A. �6536� B. �61
36� C. �31
36� D. �35
36�
8. Which expression is equivalent to cos (� � 2�)? 8. ________A. �cos � B. sin � C. cos � D. �sin �
9. Which expression is equivalent to cos 2� for all values of �? 9. ________A. cos2 � � sin2 � B. cos2 � � 1C. 1 � sin2 � D. 2 sin � cos �
10. If cos � � 0.8 and 0� � 90�, find the exact value of sin 2�. 10. ________A. 9.6 B. 2.8 C. 0.96 D. 0.28
11. If sin � � �35� and � has its terminal side in Quadrant II, find the exact 11. ________value of tan 2�.
A. �2245� B. ��22
45� C. �27
4� D. ��274�
Chapter
7
© Glencoe/McGraw-Hill 302 Advanced Mathematical Concepts
12. Use a half-angle identity to find the exact value of sin 105�. 12. ________
A. ��12��2� �� ��3�� B. �12��2� �� ��3��C. �12��2� �� ��2�� D. ��12��1� �� ��3��
13. Solve 2 cos x � 1 � 0 for 0 x 2�. 13. ________A. ��6� and �56
�� B. ��3� and �53�� C. ��3� and �23
�� D. �76�� and �16
1��
14. Solve 2 sin2 x � sin x � 0 for principal values of x. 14. ________A. 60� and 120� B. 0� and 150� C. 0� and 30� D. 60�
15. Solve cos x tan x � sin2 x � 0 for all real values of x. 15. ________A. �k, ��2� � 2�k B. ��2� � �k, 2�k
C. ��2� � 2�k, �32�� � 2�k D. �k, ��4� � 2�k
16. Write the equation 3x � 4y � 7 � 0 in normal form. 16. ________A. �35�x � �45�y � �7
5� � 0 B. ��35�x � �45�y � �7
5� � 0
C. ��35�x � �45�y � �75
� � 0 D. �35�x � �45�y � �75
� � 0
17. Write the standard form of the equation of a line for which the length 17. ________of the normal is 4 and the normal makes an angle of 45� with the positive x-axis.A. �2�x � �2�y � 8 � 0 B. 2x � 2y � 8 � 0C. �2�x � �2�y � 8 � 0 D. 2x � 2y � 8 � 0
18. Find the distance between P(�2, 1) and the line with equation 18. ________x � 2y � 4 � 0.
A. �4�
55�
� B. 0 C. � �4�
55�
� D. �45
�
19. Find the distance between the lines with equations 3x � 4y � 8 and 19. ________y � �34�x � 4.
A. �85� B. � �274� C. �27
4� D. �254�
20. Find an equation of the line that bisects the acute angles formed by 20. ________the lines with equations 4x � y � 3 � 0 and x � y � 2 � 0.A. (4�2� � �1�7�)x � (�2� � �1�7�)y � 3�2� � 2�1�7� � 0B. (�2� � �1�7�)x � (4�2� � �1�7�)y � 3�2� � 2�1�7� � 0C. (4�2� � �1�7�)x � (�2� � �1�7�)y � 3�2� � 2�1�7� � 0D. (�2� � �1�7�)x � (4�2� � �1�7�)y � 3�2� � 2�1�7� � 0
Bonus If 90� � 180�, express sin � in terms of cos �. Bonus: ________A. ��1� �� c�o�s2� �� B. ��1� �� c�o�s2� ��C. �1� �� c�o�s2� �� D. �1� �� c�o�s2� ��
Chapter 7 Test, Form 1C (continued)
NAME _____________________________ DATE _______________ PERIOD ________Chapter
7
© Glencoe/McGraw-Hill 303 Advanced Mathematical Concepts
Chapter 7 Test, Form 2A
NAME _____________________________ DATE _______________ PERIOD ________
1. Simplify (sec � � tan �)(1 � sin �). 1. __________________
2. If tan � � ��34� and 90� � 180�, find sec �. 2. __________________
3. Simplify . 3. __________________
4. Simplify sin � � cos � tan �. 4. __________________
5. If �1 �
setcan
x
2 x� � sin2 x � �sec
12 x�, find the value of cos x. 5. __________________
6. Use a sum or difference identity to find the exact 6. __________________value of sin 285�.
7. Find the value of sin (� � �) if cos � � �1157�, cot � � �27
4�, 7. __________________________0� � 90�, and 0� � 90�.
8. Simplify cos ���2� � ��. 8. __________________
9. If sec � � 4, find the exact value of cos 2�. 9. __________________
10. If cos � � 0.6 and 270� � 360�, find the exact 10. __________________value of sin 2�.
11. If sin � � � �45� and � has its terminal side in Quadrant III, 11. __________________find the exact value of tan 2�.
sec2 ����tan � � cot2 � tan �
Chapter
7
© Glencoe/McGraw-Hill 304 Advanced Mathematical Concepts
12. Use a half-angle identity to find the exact value 12. __________________of cos 67.5�.
13. Solve 2 cos x � sin2 x � 2 � 0 for all real values of x. 13. __________________
14. Solve 2 cos2 x � �3� cos x for principal values of x. Express 14. __________________the solution(s) in degrees.
15. Solve 2 sin x � 1 0 for 0 x 2�. 15. __________________
16. Write the equation 3x � 2y � 4 � 0 in normal form. 16. __________________
17. Write the standard form of the equation of a line for which 17. __________________the length of the normal is 5 and the normal makes an angle of 120� with the positive x-axis.
18. Find the distance between P(3, �2) and the line with 18. __________________equation x � 2y � 3 � 0.
19. Find the distance between the lines with equations 19. __________________3x � y � 7 and y � �3x � 4.
20. Find an equation of the line that bisects the obtuse angles 20. __________________formed by the lines with equations 2x � y � 5 � 0 and 3x � 2y � 6 � 0.
Bonus If 180� � 270� and cos � � ��45�, find sin 4�. Bonus: __________________
Chapter 7 Test, Form 2A (continued)
NAME _____________________________ DATE _______________ PERIOD ________Chapter
7
© Glencoe/McGraw-Hill 305 Advanced Mathematical Concepts
Chapter 7 Test, Form 2B
NAME _____________________________ DATE _______________ PERIOD ________
1. Simplify cos � tan2 � � cos �. 1. __________________
2. If cot � � ��34� and 90� � 180�, find sin �. 2. __________________
3. Simplify csc � � cot � cos �. 3. __________________
4. Simplify �11
��
csoins
2
2��
�. 4. __________________
5. If sin2 x sec x cot x � 3, find the value of csc x. 5. __________________
6. Use a sum or difference identity to find the exact value 6. __________________of cos 255�.
7. Find the value of sin (� � �) if tan � � �43�, cot � � �152�, 7. __________________
0� � 90�, and 0� � 90�.
8. Simplify sin ���2� � ��. 8. __________________
9. If � is an angle in the first quadrant and csc � � 3, find 9. __________________the exact value of cos 2�.
10. If sin � � �0.6 and 180� � 270�, find the exact 10. __________________value of sin 2�.
11. If cos � � �45� and � has its terminal side in Quadrant IV, 11. __________________find the exact value of tan 2�.
Chapter
7
© Glencoe/McGraw-Hill 306 Advanced Mathematical Concepts
12. Use a half-angle identity to find the exact value of 12. __________________cos 105�.
13. Solve tan x � �3� � 0 for 0 x 2�. 13. __________________
14. Solve 4 sin2 x � 1 � 0 for principal values of x. Express 14. __________________the solution(s) in degrees.
15. Solve cos4 x � 1 � 0 for all real values of x. 15. __________________
16. Write the equation 2x � 5y � 3 � 0 in normal form. 16. __________________
17. Write the standard form of the equation of a line for 17. __________________which the length of the normal is 7 and the normal makes an angle of 150� with the positive x-axis.
18. Find the distance between P(�1, 4) and the line with 18. __________________equation 4x � 2y � 3 � 0.
19. Find the distance between the lines with equations 19. __________________x � 2y � 3 and y � �12�x � 2.
20. Find an equation of the line that bisects the acute angles 20. __________________formed by the lines with equations 3x � y � 6 � 0 and 2x � y � 1 � 0.
Bonus Express ��� in terms of sin �. Bonus: __________________tan2 ����sec2 � + cot2 � sec2 �
Chapter 7 Test, Form 2B (continued)
NAME _____________________________ DATE _______________ PERIOD ________Chapter
7
© Glencoe/McGraw-Hill 307 Advanced Mathematical Concepts
Chapter 7 Test, Form 2C
NAME _____________________________ DATE _______________ PERIOD ________
1. Simplify �tsainn
��
�. 1. __________________
2. If cos � � ��45� and 90� � 180�, find cot �. 2. __________________
3. Simplify sec2 � � tan2 �. 3. __________________
4. Simplify �sinta
2
n�
2�� �
cos12 ��. 4. __________________
5. If tan x cos x � �21�, find the value of sin x. 5. __________________
6. Use a sum or difference identity to find the exact value 6. __________________of cos 15�.
7. Find the value of tan (� � �) if cos � � �153�, sin � � �35�, 7. __________________
0� � 90�, and 0� � 90�.
8. Simplify sin (� � �). 8. __________________
9. If � is an angle in the first quadrant and cos � � �12�, find 9. __________________the exact value of cos 2�.
10. If cos � � 0.6 and 0� � 90�, find the exact 10. __________________value of sin 2�.
11. If cos � � � �45� and � has its terminal side in Quadrant II, 11. __________________find the exact value of tan 2�.
Chapter
7
© Glencoe/McGraw-Hill 308 Advanced Mathematical Concepts
12. Use a half-angle identity to find the exact value of cos 22.5�. 12. __________________
13. Solve 2 sin x � 1 � 0 for 0 x 2�. 13. __________________
14. Solve tan x � �3� � 0 for principal values of x. Express 14. __________________the solution(s) in degrees.
15. Solve �scescc
xx� � 1 � 0 for all real values of x. 15. __________________
16. Write the equation 3x � 2y � 6 � 0 in normal form. 16. __________________
17. Write the standard form of the equation of a line for 17. __________________which the length of the normal is 9 and the normal makes an angle of 60� with the positive x-axis.
18. Find the distance between P(2, 3) and the line with 18. __________________equation 2x � 5y � 4 � 0.
19. Find the distance between the lines with equations 19. __________________2x � 2y � 5 and y � x � 1.
20. Find an equation of the line that bisects the acute angles 20. __________________formed by the lines with equations 3x � 4y � 5 � 0 and 5x � 12y � 3 � 0.
Bonus How are the lines that bisect the angles formed Bonus: __________________by the graphs of the equations 3x � y � 6 and x � 3y � 1 related to each other?
Chapter 7 Test, Form 2C (continued)
NAME _____________________________ DATE _______________ PERIOD ________Chapter
7
© Glencoe/McGraw-Hill 309 Advanced Mathematical Concepts
Chapter 7 Open-Ended Assessment
NAME _____________________________ DATE _______________ PERIOD ________
Instructions: Demonstrate your knowledge by giving a clear, concise solution to each problem. Be sure to include all relevant drawings and justify your answer. You may show your solution in more than one way or investigate beyond the requirements of the problem.
1. a. Verify that �1 �cos
si�n �
� � �1 �cos
si�n �� � 0 is an identity.
b. Why is it usually easier to transform the more complicated side of the equation into the simpler side rather than the other way around?
c. Is the following method for verifying an identity correct? Why or why not? If not, write a correct verification.
sec A sin A � tan A
sec A sin A � �csoins A
A�
cos A sec A sin A � sin A
cos A �co1s A� sin A � sin A
sin A � sin A
2. a. Write the equation 2y � 3x � 6 in normal form. Then,find the length of the normal and the angle it makes with the positive x-axis. Explain how you determined the angle.
b. Find the distance from a point on the line in part a to the line with equation 6x � 4y � 16 � 0. Tell what the sign of the distance d means.
c. Will the sign of the distance from a point on the line with equation 6x � 4y � 16 � 0 to the line described in part abe the same as in part b? Why or why not?
d. When will the sign of the distance between two parallel lines be the same regardless of which line it is measured from?
Chapter
7
© Glencoe/McGraw-Hill 310 Advanced Mathematical Concepts
1. If csc A � 2, find the value of sin A. 1. __________________
2. If tan � � ��34� and 90� � 180�, find cos �. 2. __________________
3. Simplify csc x � cos x cot x. 3. __________________
4. Simplify �c1sc�
�
tatann2 �
��. 4. __________________
5. If tan x csc x � 3, find the value of cos x. 5. __________________
6. Use a sum or difference identity to find the exact value 6. __________________of sin 285�.
7. Find the value of tan (� � �) if csc � � �153�, tan � � �34�, 7. __________________
0� � 90�, and 0� � 90�.
8. If tan � � �34� and 180� � 270�, find the exact value 8. __________________of sin 2�.
9. If � is an angle in the first quadrant and csc � � 4, find the 9. __________________exact value of cos 2�.
10. Use a half-angle identity to find the exact value of sin 22.5�. 10. __________________
Chapter 7 Mid-Chapter Test (Lessons 7-1 through 7-4)
NAME _____________________________ DATE _______________ PERIOD ________Chapter
7
1. If sec � � 3, find the value of cos �. 1. __________________
2. If cot � � �43� and 180� � 270�, find csc �. 2. __________________
3. Simplify cot2 x sec2 x. 3. __________________
4. Simplify �11
�
�
ccoost2
2
�
��. 4. __________________
5. If sec � sin � � 2, find the value of cot �. 5. __________________
Chapter 7, Quiz B (Lessons 7-3 and 7-4)
NAME _____________________________ DATE _______________ PERIOD ________
Chapter 7, Quiz A (Lessons 7-1 and 7-2)
NAME _____________________________ DATE _______________ PERIOD ________
© Glencoe/McGraw-Hill 311 Advanced Mathematical Concepts
Chapter
7
Chapter
71. Use a sum or difference identity to find the exact value 1. __________________
of cos 345�.
2. Find the value of tan (� � �) if sin � � � �153�, cos � � �45�, 2. __________________
270� � 360�, and 270� � 360�.
3. If sec � � ��153� and 90� � 180�, find the exact 3. __________________
value of sin 2�.
4. If cos � � �45� and � has its terminal side in Quadrant IV, find 4. __________________the exact value of tan 2�.
5. Use a half-angle identity to find the exact value of sin 165�. 5. __________________
1. Solve 2 sin x � 2 � 0 for 0 x 2�. 1. __________________
2. Solve 4 cos2 x � 3 � 0 for principal values of x. Express 2. __________________the solution(s) in degrees.
3. Solve tan x � 1 � 0 for all real values of x. 3. __________________
4. Write the equation x � 5y � 8 � 0 in normal form. 4. __________________
5. Write the standard form of the equation of a line for which 5. __________________the length of the normal is 3 and the normal makes an angle of 240� with the positive x-axis.
Chapter 7, Quiz D (Lesson 7-7)
NAME _____________________________ DATE _______________ PERIOD ________
Chapter 7, Quiz C (Lessons 7-5 and 7-6)
NAME _____________________________ DATE _______________ PERIOD ________
© Glencoe/McGraw-Hill 312 Advanced Mathematical Concepts
Chapter
7
Chapter
71. Find the distance between P(1, 4) and the line with 1. __________________
equation x � 2y � 5 � 0.
2. Find the distance between P(3, 1) and the line with 2. __________________equation 2x � 3y � 3 � 0.
3. Find the distance between the lines with equations 3. __________________2x � y � 5 and y � 2x � 3.
4. Find the distance between the lines with equations 4. __________________3x � 4y � 18 � 0 and y � ��34�x � 3.
5. Find an equation of the line that bisects the acute angles 5. __________________formed by the lines with equations x � 3y � 3 � 0 and x � 2y � 2 � 0.
© Glencoe/McGraw-Hill 313 Advanced Mathematical Concepts
Chapter 7 SAT and ACT Practice
NAME _____________________________ DATE _______________ PERIOD ________Chapter
7After working each problem, record thecorrect answer on the answer sheetprovided or use your own paper.
Multiple Choice1. In the figure below, the measure of
�A is 65�. If the measure of �C is�45� the measure of �A, what is the measure of �B?A 50�B 52�C 63�D 65�E 68�
2. In the figure below, three lines intersect to form a triangle. Find thesum of the measures of the markedangles.A 90�B 180�C 360�D 540�E It cannot be determined from the
information given.
3. If x � y � z, and x � y � 24 and x � 10,then z �A �4 B 0C 4 D 8E 16
4. What are the roots of x2 � 169 � 0?A 0, 169B 0, 13C 0, �13D 169, �169E 13, �13
5. If �ABC is equilateral, what is thevalue of x � ( y � z) � w?
A �60B 0C 20D 60E It cannot be determined from the
information given.
6. In the figure below, A�D� is parallel toB�C�. Find the value of x.
A 20B 40C 60D 80E It cannot be determined from the
information given.
7. Sin�1 ���23��� � Cos�1 ���2
3��� �
A 0B �6
��
C �3��
D �2��
E �
8. The lengths of the sides of a rectangleare 6 inches and 8 inches. Which ofthe following can be used to find �,the angle that a diagonal makes witha longer side?A sin � � �4
3�
B cos � � �43�
C tan � � �43�
D tan � � �34�
E cos � � �53�
9. Points A(�1, �2), B(2, 1), and C(4, �2)are vertices of parallelogram ABCD.What are the coordinates of D?A (0, �4)B (1, �5)C (�1, �5)D (2, �5)E (2, �4)
© Glencoe/McGraw-Hill 314 Advanced Mathematical Concepts
Chapter 7 SAT and ACT Practice (continued)
NAME _____________________________ DATE _______________ PERIOD ________Chapter
710. The vertices of a triangle are (2, 4),
(7, 9), and (8, 2). Which of the follow-ing best describes this triangle?A scaleneB equilateralC rightD isoscelesE right, isosceles
11. In right �ABD, C�A� bisects �DAB.What is the value of x?A 20B 40C 70D 80E None of these
12. In the figure below, what is the valueof x in terms of y?A yB 2yC 180 � 2yD 180 � yE 360 � 2y
13. What is the greatest common factor of the terms in the expansion of 2(6x2y � 9xy3)(15a3x � 10ay2)?A 2B 6yC 10aD 30ayE None of these
14. If 5x � 4y � xy � 8 � 0 and x � 3 � 9,then 3 � y � A �19B �16C 8D 19E 22
15. Which of the following could belengths of the sides of a triangle?A 7, 8, 14B 8, 8, 16C 8, 9, 20D 9, 10, 100E 1, 50, 55
16. In the rectangle ABDC below, what isthe measure of � ACB?A 63�
B 53�
C 37�
D 45�
E It cannot be determined from theinformation given.
17–18. Quantitative ComparisonA if the quantity in Column A is
greaterB if the quantity in Column B is
greaterC if the two quantities are equalD if the relationship cannot be
determined from the information given
Column A Column B
17. Side A�B� of triangle ABC is extendedbeyond B to point D.
18. Angles P, Q, and R are the angles of aright triangle.
19–20. Refer to the figure below.
19. Grid-In What is the value of x?
20. Grid-In What is the value of y?
The measure of �ABC
180� �m �P 90�
The measure of �DBC
© Glencoe/McGraw-Hill 315 Advanced Mathematical Concepts
Chapter 7 Cumulative Review (Chapters 1-7)
NAME _____________________________ DATE _______________ PERIOD ________
1. Find the standard form of the equation of the line that 1. __________________passes through (�1, 2) and has a slope of 3.
2. If A � � � and B � � �, find AB. 2. __________________
3. Given ƒ(x) � (x � 2)2 � 5, find ƒ�1(x). Then state whether 3. __________________ƒ�1(x) is a function.
4. If y varies inversely as the square of x and y � 18 when 4. __________________x � 3, find y when x � �9.
5. Write a polynomial equation of least degree with roots 3, 5. __________________2i, and �2i.
6. Use the Remainder Theorem to find the remainder when 6. __________________x2 � 5x � 2 is divided by x � 5. State whether the binomial is a factor of the polynomial.
7. Given the triangle at the 7. __________________right, find m� A to the nearest tenth of a degree if b � 12 and c � 16.
8. If a � 8, b � 11, and c � 13, find the area of �ABC to the 8. __________________nearest tenth.
9. State the amplitude, period, and phase shift for the graph 9. __________________of y � 3 sin(2x � 4�).
10. Find the value of Cos�1�tan ��4��. 10. __________________
11. Solve 4 sin2 x � 3 � 0 for principal values of x. 11. __________________Express the solution(s) in degrees.
12. Find the distance between P(2, 4) and the line with 12. __________________equation 2x � y � 5 � 0.
045
312
14
03
2�2
Chapter
7
BLANK
© Glencoe/McGraw-Hill A1 Advanced Mathematical Concepts
SAT and ACT Practice Answer Sheet(10 Questions)
NAME _____________________________ DATE _______________ PERIOD ________
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© Glencoe/McGraw-Hill A2 Advanced Mathematical Concepts
SAT and ACT Practice Answer Sheet(20 Questions)
NAME _____________________________ DATE _______________ PERIOD ________
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ress
eac
h v
alu
e as
a t
rig
onom
etri
c fu
nct
ion
of
an a
ng
le in
Qu
and
ran
t I.
5.co
s 89
2�6.
csc
495�
7.si
n �2 33� �
�co
s 8�
csc
45�
�si
n� 3� �
Sim
plif
y ea
ch e
xpre
ssio
n.
8.co
s x
�si
n x
tan
x9.
� tc ao ntA A
�10
.si
n2
�co
s2�
�co
s2�
sec
xco
t2A
�co
s4�
11.K
ite
Fly
ing
Bre
tt a
nd
Tara
are
fly
ing
a ki
te. W
hen
th
e st
rin
gis
tie
d to
th
e gr
oun
d, t
he
hei
ght
of t
he
kite
can
be
dete
rmin
ed b
y th
e fo
rmu
la � HL �
�cs
c �, w
her
e L
is t
he
len
gth
of
the
stri
ng
and
�is
th
e an
gle
betw
een
th
e st
rin
g an
d th
e le
vel g
rou
nd.
Wh
at f
orm
ula
cou
ld B
rett
an
d Ta
ra u
se t
o fi
nd
the
hei
ght
of t
he
kite
if t
hey
know
th
e va
lue
of s
in �
?H
�L
sin
�Pra
ctic
eN
AM
E__
____
____
____
____
____
____
___
DAT
E__
____
____
____
_ P
ER
IOD
____
____
7-1
Answers (Lesson 7-1)
© Glencoe/McGraw-Hill A3 Advanced Mathematical Concepts
© G
lenc
oe/M
cGra
w-H
ill27
7A
dva
nced
Mat
hem
atic
al C
once
pts
Enr
ichm
ent
NA
ME
____
____
____
____
____
____
____
_ D
ATE
____
____
____
___
PE
RIO
D__
____
__
7-1
Th
e P
hy
sic
s o
f S
oc
ce
rR
ecal
l fro
m L
esso
n 7
-1 t
hat
th
e fo
rmu
la f
or t
he
max
imu
m h
eigh
t h
of a
pro
ject
ile
is
,w
her
e �
is t
he
mea
sure
of
the
angl
e of
el
evat
ion
in d
egre
es,v
0is
th
e in
itia
l vel
ocit
y in
fee
t pe
r se
con
d,an
d g
is t
he
acce
lera
tion
du
e to
gra
vity
in f
eet
per
seco
nd
squ
ared
.
Sol
ve. G
ive
answ
ers
to t
he
nea
rest
ten
th.
1.A
soc
cer
play
er k
icks
a b
all a
t an
init
ial v
eloc
ity
of 6
0 ft
/s a
nd
anan
gle
of e
leva
tion
of
40°.
Th
e ac
cele
rati
on d
ue
to g
ravi
ty is
32
ft/s
2 .F
ind
the
max
imu
m h
eigh
t re
ach
ed b
y th
e ba
ll.
23.2
ft
2.W
ith
wh
at in
itia
l vel
ocit
y m
ust
you
kic
k a
ball
at
an a
ngl
e of
35°
in o
rder
for
it t
o re
ach
a m
axim
um
hei
ght
of 2
0 ft
?62
.4 f
t/s
Th
e di
stan
ce d
that
a p
roje
cted
obj
ect
trav
els
is g
iven
by
the
form
ula
3.F
ind
the
dist
ance
tra
vele
d by
th
e ba
ll d
escr
ibed
in E
xerc
ise
1.11
0.8
ft
In o
rder
to
kick
a b
all t
he
grea
test
pos
sibl
e di
stan
ce a
t a
give
n
init
ial v
eloc
ity,
a so
ccer
pla
yer
mu
st m
axim
ize
Sin
ce 2
,v0,
and
gar
e co
nst
ants
,th
is m
ean
s th
e pl
ayer
mu
st
max
imiz
e si
n �
cos
�.
sin
0°co
s 0°
�si
n 90
°cos
90°
�0
sin
10°c
os 1
0°�
sin
80°c
os 8
0°�
0.17
10
sin
20°c
os 2
0°�
sin
70°c
os 7
0°�
0.32
14
4.U
se t
he
patt
ern
s in
th
e ta
ble
to h
ypot
hes
ize
a va
lue
of �
for
wh
ich
sin
�co
s �
wil
l be
max
imal
.Use
a c
alcu
lato
r to
ch
eck
you
r h
ypot
hes
is.A
t w
hat
an
gle
shou
ld t
he
play
er k
ick
the
ball
to
ach
ieve
th
e gr
eate
st d
ista
nce
? 45
°
h�
v 02
sin
2�
�� 2g
d�
.2v
02si
n �
cos
��
� g
d�
.2v
02si
n �
cos
��
� g
Answers (Lesson 7-2)
© Glencoe/McGraw-Hill A4 Advanced Mathematical Concepts
© G
lenc
oe/M
cGra
w-H
ill27
9A
dva
nced
Mat
hem
atic
al C
once
pts
Pra
ctic
eN
AM
E__
____
____
____
____
____
____
___
DAT
E__
____
____
____
_ P
ER
IOD
____
____
Veri
fyin
g T
rig
on
om
etr
ic I
de
ntitie
s
Ver
ify
that
eac
h e
qu
atio
n is
an
iden
tity
.
1.� co
txcs �c
tx anx
��
cos
x
� cotc xs �c
tx anx
��
��s si in n
x xc co o
s sx x
��� co
s2c xo �sx si
n2x
��
�co1s
x�
�co
s x
2.� si
ny1
�1
��
� sin
y1�
1�
��
2 se
c2y
� sin
y1
�1
��
� sin
y1
�1
��
�� �
co
2 s2y
��
�2
sec
2y
3.si
n3
x�
cos3
x�
(1�
sin
xco
s x)
(sin
x�
cos
x)
sin
3x
�co
s3x
�(s
in x
�co
s x)
(sin
2x
�si
n x
cos
x�
cos2
x)
�(s
in x
�co
s x)
(1�
sin
xco
s x)
�(1
�si
n x
cos
x)(s
in x
�co
s x)
4.ta
n �
�� 1
�cos si
� n�
��
sec
�
tan
��
� 1c �
o ss in��
��
� cs oin s� �
��
� 1c �
o ss in��
��
��
sec
�
Fin
d a
nu
mer
ical
val
ue
of o
ne
trig
onom
etri
c fu
nct
ion
of
x.
5.si
n x
cot
x�
16.
sin
x�
3 co
s x
7.co
s x
�co
t x
cos
x�
1ta
n x
�3
csc
x�
1 o
r si
n x
�1
8.P
hys
ics
Th
e w
ork
don
e in
mov
ing
an o
bjec
t is
giv
en b
y th
e fo
rmu
la
W�
Fd
cos
�, w
her
ed
is t
he
disp
lace
men
t, F
is t
he
forc
e ex
erte
d, a
nd
�is
th
e an
gle
betw
een
th
e di
spla
cem
ent
and
the
forc
e. V
erif
y th
at
W�
Fd
� cc so ct� �
�is
an
equ
ival
ent
form
ula
.
W�
Fd� cc so ct
� ��
�Fd
�Fd
�c so ins��
��
sin
��
Fdco
s �
�c so ins��
�� � si
n1�
�
1�
sin
��
��
(co
s �)
(1�
sin
�)
sin
��
sin2
��
cos2
��
��
(co
s �)
(1�
sin
�)
sin
y�
1�
sin
y�
1�
��
sin2
y�
1
� sin1
x�
��
�c so insxx
��
� cs oin sx x
�
7-2
© G
lenc
oe/M
cGra
w-H
ill28
0A
dva
nced
Mat
hem
atic
al C
once
pts
Enr
ichm
ent
NA
ME
____
____
____
____
____
____
____
_ D
ATE
____
____
____
___
PE
RIO
D__
____
__
7-2
Bu
ildin
g f
rom
1�
1B
y st
arti
ng
wit
h t
he
mos
t fu
nda
men
tal i
den
tity
of
all,
1 �
1, y
ou c
ancr
eate
new
iden
titi
es a
s co
mpl
ex a
s yo
u w
ould
like
th
em t
o be
.
Fir
st, t
hin
k of
way
s to
wri
te 1
usi
ng
trig
onom
etri
c id
enti
ties
. Som
eex
ampl
es a
re t
he
foll
owin
g.
1 �
cos
Ase
c A
1 �
csc2
A�
cot2
A
1 �
Ch
oose
tw
o su
ch e
xpre
ssio
ns
and
wri
te a
new
iden
tity
.
cos
Ase
c A
�cs
c2A
�co
t2A
Now
mu
ltip
ly t
he
term
s of
th
e id
enti
ty b
y th
e te
rms
of a
not
her
iden
tity
of y
our
choo
sin
g, p
refe
rabl
y on
e th
at w
ill a
llow
som
e si
mpl
ific
atio
nu
pon
mu
ltip
lica
tion
.
cos
Ase
c A
�cs
c2 A�
cot2 A
�
�ta
n A
____
____
____
____
____
____
____
____
_si
n A
sec
A�
tan
Acs
c2A
�co
t A
Beg
inn
ing
wit
h 1
�1,
cre
ate
two
trig
onom
etri
c id
enti
ties
. A
nsw
ers
will
var
y.
1.__
____
____
____
____
____
____
____
____
____
____
___
2.__
____
____
____
____
____
____
____
____
____
____
___
Ver
ify
that
eac
h o
f th
e id
enti
ties
you
cre
ated
is a
n id
enti
ty.
3.__
____
____
____
____
____
____
4.__
____
____
____
____
____
___
____
____
____
____
____
____
____
____
____
____
____
____
___
____
____
____
____
____
____
____
____
____
____
____
____
___
____
____
____
____
____
____
____
____
____
____
____
____
___
____
____
____
____
____
____
____
____
____
____
____
____
___
____
____
____
____
____
____
____
____
____
____
____
____
___
cos
(A�
360°
)�
�co
s (3
60°
�A
)
sin
A� co
s A
Answers (Lesson 7-3)
© Glencoe/McGraw-Hill A5 Advanced Mathematical Concepts
© G
lenc
oe/M
cGra
w-H
ill28
2A
dva
nced
Mat
hem
atic
al C
once
pts
Su
m a
nd
Dif
fere
nc
e I
de
ntitie
s
Use
su
m o
r d
iffe
ren
ce id
enti
ties
to
fin
d t
he
exac
t va
lue
of e
ach
tri
gon
omet
ric
fun
ctio
n.
1.co
s �5 1� 2�
2.si
n (
�16
5�)
3.ta
n 3
45�
��6�
� 4�
2��
��2�
� 4�
6��
�3�
�2
4.cs
c 91
5�5.
tan
���7 1� 2�
�6.
sec
� 1� 2�
��
6��
�2�
2�
�3�
�6�
��
2�
Fin
d e
ach
exa
ct v
alu
e if
0�
x�
�� 2�an
d 0
�y
��� 2�
.
7.co
s (x
�y)
if s
in x
�� 15 3�
and
sin
y�
� 54 �
�1 66 5�
8.si
n (
x�
y) if
cos
x�
� 18 7�an
d co
s y
�� 53 �
� 81 53 �
9.ta
n (
x�
y) if
csc
x�
�1 53 �an
d co
t y
�� 34 �
�� 61 36 �
Ver
ify
that
eac
h e
qu
atio
n is
an
iden
tity
.
10.c
os (
180�
��)�
�co
s �
11.
sin
(36
0��
�)�
sin
�co
s (1
80�
��)
sin
(360
��
�)�
cos
180�
cos
��
sin
180�
sin
��
sin
360�
cos
��
cos
360�
sin
��
(�1)
co
s �
�0
�si
n �
�0
�co
s �
�1
�si
n �
��
cos
��
sin
�
12.P
hys
ics
Sou
nd
wav
es c
an b
e m
odel
ed b
y eq
uat
ion
s of
th
e fo
rm y
�20
sin
(3t
��).
Det
erm
ine
wh
at t
ype
of in
terf
eren
cere
sult
s w
hen
sou
nd
wav
es m
odel
ed b
y th
e eq
uat
ion
s y
�20
sin
(3t
�90
�) a
nd
y�
20 s
in (
3t�
270�
) ar
e co
mbi
ned
.(H
int:
Ref
er t
o th
e ap
plic
atio
n in
Les
son
7-3
.)T
he in
terf
eren
ce is
des
truc
tive
. The
wav
es c
ance
lea
ch o
ther
co
mp
lete
ly.
Pra
ctic
eN
AM
E__
____
____
____
____
____
____
___
DAT
E__
____
____
____
_ P
ER
IOD
____
____
7-3
© G
lenc
oe/M
cGra
w-H
ill28
3A
dva
nced
Mat
hem
atic
al C
once
pts
Enr
ichm
ent
NA
ME
____
____
____
____
____
____
____
_ D
ATE
____
____
____
___
PE
RIO
D__
____
__
7-3
Ide
ntitie
s fo
r th
e P
rod
uc
ts o
f S
ine
s a
nd
Co
sin
es
By
addi
ng
the
iden
titi
es f
or t
he
sin
es o
f th
e su
m a
nd
diff
eren
ce o
f th
em
easu
res
of t
wo
angl
es, a
new
iden
tity
is o
btai
ned
.
sin
(�
��
)�
sin
� c
os �
�co
s �
sin
�si
n (
��
�)
�si
n �
cos
��
cos
�si
n �
(i)
sin
(�
��
)�si
n (
��
�)
�2
sin
� c
os �
Th
is n
ew id
enti
ty is
use
ful f
or e
xpre
ssin
g ce
rtai
n p
rodu
cts
as s
um
s.
Exa
mp
leW
rite
sin
3�
cos
�as
a s
um
.
In t
he
righ
t si
de o
f id
enti
ty (
i) le
t �
�3�
and
��
�so
that
2 s
in 3
�co
s �
�si
n (
3��
�)
�si
n (
3��
�).
Th
us,
sin
3�
cos�
�si
n 4
��
sin
2�.
By
subt
ract
ing
the
iden
titi
es f
or s
in (
��
�)
and
sin
(�
��
), y
ou o
btai
n a
sim
ilar
iden
tity
for
exp
ress
ing
apr
odu
ct a
s a
diff
eren
ce.
(ii)
sin
(�
��
)�si
n (
��
�)
�2
cos
�si
n �
Exa
mp
leV
erif
y th
e id
enti
ty�c sio ns
22 xxcs oin s
x x�
�.
In t
he
righ
t si
des
of id
enti
ties
(i)
and
(ii)
let
� �
2x
and
��
x.
Th
en w
rite
th
e fo
llow
ing
quot
ien
t.�
By
sim
plif
yin
g an
d m
ult
iply
ing
by t
he
con
juga
te, t
he
iden
tity
is v
erif
ied.
�
.
�
Com
ple
te.
1.U
se t
he
iden
titi
es f
or c
os (
��
�)
and
cos
(��
�)
to f
ind
iden
titi
es f
or e
xpre
ssin
g th
epr
odu
cts
2 co
s �
cos
�an
d 2
sin
�si
n �
as a
su
m o
r di
ffer
ence
.co
s (�
��
) �co
s �
cos
��
sin
�si
n �
cos
(��
�) �
cos
�co
s �
�si
n �
sin
�2
cos
�co
s �
�co
s (�
��
)�co
s (�
��
)2
sin
�si
n �
�co
s (�
��
)�co
s (�
��
)2.
Fin
d th
e va
lue
of s
in 1
05°
cos
75°
by u
sin
g th
e id
enti
ty a
bove
.
sin
105°
cos
75°
�(s
in 1
80°
� s
in 3
0°)
1 � 2
(sin
3x
� s
in x
)2�
�si
n2
3x�
sin
2 x
sin
3x
� s
in x
��
sin
3x
�si
n x
sin
3x
� s
in x
��
sin
3x
�si
n x
cos
2xsi
n x
��
sin
2x
cos
x
sin
(2x
� x
) �
sin
(2x
� x
)�
��
�si
n (
2x �
x)�
sin
(2x
� x
)2
cos
2x s
in x
��
2 si
n 2
x co
s x
(sin
3x�
sin
x)2
��
�si
n2
3x�
sin
2x
1 � 21 � 2
�(0
�
) �
or
0.25
1 � 41 � 2
1 � 2
© G
lenc
oe/M
cGra
w-H
ill28
6A
dva
nced
Mat
hem
atic
al C
once
pts
Enr
ichm
ent
NA
ME
____
____
____
____
____
____
____
_ D
ATE
____
____
____
___
PE
RIO
D__
____
__
7-4
Re
ad
ing
Ma
the
ma
tic
s: U
sin
g E
xam
ple
sM
ost
mat
hem
atic
s bo
oks,
incl
udi
ng
this
on
e, u
se e
xam
ples
to
illu
stra
teth
e m
ater
ial o
f ea
ch le
sson
. E
xam
ples
are
ch
osen
by
the
auth
ors
tosh
ow h
ow t
o ap
ply
the
met
hod
s of
th
e le
sson
an
d to
poi
nt
out
plac
esw
her
e po
ssib
le e
rror
s ca
n a
rise
.
1.E
xpla
in t
he
purp
ose
of E
xam
ple
1c in
Les
son
7-4
.to
illu
stra
te h
ow
to
use
the
do
uble
-ang
le id
enti
ty f
or
the
tan-
gen
t; t
o s
how
ho
w t
o f
ind
tan
�fr
om
info
rmat
ion
alre
ady
kno
wn
2.E
xpla
in t
he
purp
ose
of E
xam
ple
3 in
Les
son
7-4
.to
illu
stra
te h
ow
a d
oub
le-a
ngle
iden
tity
can
be
app
lied
to
are
al-w
orl
d s
itua
tio
n3.
Exp
lain
th
e pu
rpos
e of
Exa
mpl
e 4
in L
esso
n 7
-4.
to il
lust
rate
the
ver
ifica
tio
n o
f a
trig
ono
met
ric
iden
tity
invo
lv-
ing
a d
oub
le-a
ngle
iden
tity
To m
ake
the
best
use
of
the
exam
ples
in a
less
on, t
ry f
ollo
win
g th
ispr
oced
ure
:
a.W
hen
you
com
e to
an
exa
mpl
e, s
top.
Th
ink
abou
t w
hat
you
hav
eju
st r
ead.
If
you
don
’t u
nde
rsta
nd
it, r
erea
d th
e pr
evio
us
sect
ion
.
b.
Rea
d th
e ex
ampl
e pr
oble
m. T
hen
inst
ead
of r
eadi
ng
the
solu
tion
,tr
y so
lvin
g th
e pr
oble
m y
ours
elf.
c.A
fter
you
hav
e so
lved
th
e pr
oble
m o
r go
ne
as f
ar a
s yo
u c
an g
o,st
udy
th
e so
luti
on g
iven
in t
he
text
. Com
pare
you
r m
eth
od a
nd
solu
tion
wit
h t
hos
e of
th
e au
thor
s. I
f n
eces
sary
, fin
d ou
t w
her
eyo
u w
ent
wro
ng.
If
you
don
’t u
nde
rsta
nd
the
solu
tion
, rer
ead
the
text
or
ask
you
r te
ach
er f
or h
elp.
4.E
xpla
in t
he
adva
nta
ge o
f w
orki
ng
an e
xam
ple
you
rsel
f ov
er
sim
ply
read
ing
the
solu
tion
giv
en in
th
e te
xt.
Sam
ple
ans
wer
: Thi
s m
etho
d c
heck
s yo
ur u
nder
stan
din
g o
fth
e m
ater
ial r
athe
r th
an y
our
ab
ility
to
fo
llow
the
aut
hors
’lo
gic
. By
allo
win
g e
rro
rs t
o a
rise
in y
our
so
luti
on,
it h
elp
s yo
ufin
d a
reas
of
mis
und
erst
and
ing
. The
n it
giv
es y
ou
a m
etho
d f
or
corr
ecti
ng y
our
err
ors
and
che
ckin
g y
our
so
luti
on.
© G
lenc
oe/M
cGra
w-H
ill28
5A
dva
nced
Mat
hem
atic
al C
once
pts
Pra
ctic
eN
AM
E__
____
____
____
____
____
____
___
DAT
E__
____
____
____
_ P
ER
IOD
____
____
Do
ub
le-A
ng
le a
nd
Ha
lf-A
ng
le I
de
ntitie
s
Use
a h
alf-
ang
le id
enti
ty t
o fi
nd
th
e ex
act
valu
e of
eac
h f
un
ctio
n.
1.si
n 1
05�
2.ta
n �� 8�
3.co
s �5 8� �
�2�
�1
Use
th
e g
iven
info
rmat
ion
to
fin
d s
in 2
�, c
os 2
�, a
nd
tan
2�.
4.si
n �
��1 12 3�
, 0�
��
�90
�5.
tan
��
�1 2� , �
��
��3 2� �
�1 12 60 9�,�
�1 11 69 9�,�
�1 12 10 9��4 5� ,
�3 5� ,� 34 �
6.se
c �
��
�5 2� , �� 2�
��
��
7.si
n �
��3 5� ,
0�
��
� 2� �
��4�
252�1� �
,��1 27 5�
, �4�
172�1� �
�2 24 5�, �
27 5�, �
2 74 �
Ver
ify
that
eac
h e
qu
atio
n is
an
iden
tity
.
8.1
�si
n 2
x�
(sin
x�
cos
x)2
1�
sin
2x
� (s
in x
�co
s x)
2
1�
sin
2x
�si
n2x
�2
sin
xco
s x
�co
s2x
1�
sin
2x
�1
�2
sin
xco
s x
1�
sin
2x
�1
�si
n 2
x
9.co
s x
sin
x�
�sin 22x �
cos
xsi
n x
��si
n 22x
�
cos
x si
nx
��2
sin
x 2co
sx
�
cos
xsi
n x
�co
s x
sin
x
10.B
ase
ball
Aba
tter
hit
s a
ball
wit
h a
n in
itia
l vel
ocit
y v 0
of
100
feet
per
sec
ond
at a
n a
ngl
e �
to t
he
hor
izon
tal.
An
ou
tfie
lder
catc
hes
th
e ba
ll 2
00 f
eet
from
hom
e pl
ate.
Fin
d �
if t
he
ran
ge
of a
pro
ject
ile
is g
iven
by
the
form
ula
R�
� 31 2�v 02
sin
2�.
abo
ut 2
0�
�2�
����
2���
� 2�
2���
��3��
�� 2
7-4
Answers (Lesson 7-4)
© Glencoe/McGraw-Hill A6 Advanced Mathematical Concepts
Answers (Lesson 7-5)
© Glencoe/McGraw-Hill A7 Advanced Mathematical Concepts
Th
e S
pe
ctr
um
In s
ome
way
s, li
ght
beh
aves
as
thou
gh it
wer
e co
mpo
sed
of w
aves
. Th
ew
avel
engt
h o
f vi
sibl
e li
ght
ran
ges
from
abo
ut
4 �
10�
5cm
for
vio
let
ligh
t to
abo
ut
7 �
10�
5cm
for
red
ligh
t.
As
ligh
t pa
sses
th
rou
gh a
med
ium
, its
vel
ocit
y de
pen
ds u
pon
th
e w
avel
engt
h o
f th
e li
ght.
Th
e gr
eate
r th
e w
avel
engt
h, t
he
grea
ter
the
velo
city
. Sin
ce w
hit
e li
ght,
incl
udi
ng
sun
ligh
t, is
com
pose
d of
ligh
t of
vary
ing
wav
elen
gth
s, w
aves
wil
l pas
s th
rou
gh t
he
med
ium
at
an
infi
nit
e n
um
ber
of d
iffe
ren
t sp
eeds
. Th
e in
dex
of r
efra
ctio
n n
of
the
med
ium
is d
efin
ed b
y n
�, w
her
e c
is t
he
velo
city
of
ligh
t in
ava
cuu
m (
3 �
1010
cm/s
), a
nd
v is
th
e ve
loci
ty o
f li
ght
in t
he
med
ium
. A
s yo
u c
an s
ee, t
he
inde
x of
ref
ract
ion
of
a m
ediu
m is
not
a c
onst
ant.
It
depe
nds
on
th
e w
avel
engt
h a
nd
the
velo
city
of
ligh
t pa
ssin
g th
rou
gh it
.(T
he
inde
x of
ref
ract
ion
of
diam
ond
give
n in
th
e le
sson
is a
n a
vera
ge.)
1.F
or a
ll m
edia
, n>
1. I
s th
e sp
eed
of li
ght
in a
med
ium
gre
ater
th
anor
less
th
an c
? E
xpla
in.
less
tha
n; v
�. S
ince
n >
1, v
< c
.
2.A
beam
of
viol
et li
ght
trav
els
thro
ugh
wat
er a
t a
spee
d of
2.
234
�10
10cm
/s. F
ind
the
inde
x of
ref
ract
ion
of
wat
er f
or v
iole
tli
ght.
1.
343
Th
e di
agra
m s
how
s w
hy
a pr
ism
spl
its
wh
ite
ligh
tin
to a
spe
ctru
m. B
ecau
se t
hey
tra
vel a
t di
ffer
ent
velo
citi
es in
th
e pr
ism
, wav
es o
f li
ght
of d
iffe
ren
t co
lors
are
ref
ract
ed d
iffe
ren
t am
oun
ts.
3.B
eam
s of
red
an
d vi
olet
ligh
t st
rike
cro
wn
gla
ss a
t an
an
gle
of
20°.
Use
Sn
ell’s
Law
to
fin
d th
e di
ffer
ence
bet
wee
n t
he
angl
es o
fre
frac
tion
of
the
two
beam
s.
viol
et li
ght:
n�
1.53
1
red
ligh
t: n
�1.
513
abo
ut 0
.16°
c � n
© G
lenc
oe/M
cGra
w-H
ill28
9A
dva
nced
Mat
hem
atic
al C
once
pts
NA
ME
____
____
____
____
____
____
____
_ D
ATE
____
____
____
___
PE
RIO
D__
____
__
Enr
ichm
ent
7-5
c � v
© G
lenc
oe/M
cGra
w-H
ill28
8A
dva
nced
Mat
hem
atic
al C
once
pts
So
lvin
g T
rig
on
om
etr
ic E
qu
atio
ns
Sol
ve e
ach
eq
uat
ion
for
pri
nci
pal
val
ues
of
x. E
xpre
ssso
luti
ons
in d
egre
es.
1.co
s x
�3
cos
x�
22.
2 si
n2
x�
1�
00�
�45
�
Sol
ve e
ach
eq
uat
ion
for
0�
�x
�36
0�.
3.se
c2x
�ta
n x
�1
�0
4.co
s 2x
�3
cos
x�
1�
00�
, 135
�, 1
80�,
315
�60
�, 3
00�
Sol
ve e
ach
eq
uat
ion
for
0 �
x�
2�.
5.4
sin
2x
�4
sin
x�
1�
06.
cos
2x�
sin
x�
1�� 6� ,
�5 6� �0,
�� 6� , �5 6� �
, �
Sol
ve e
ach
eq
uat
ion
for
all
real
val
ues
of
x.7.
3 co
s 2x
�5
cos
x�
18.
2 si
n2
x�
5 si
n x
�2
�0
�2 3� ��
2�k,
�4 3� ��
2�k
�� 6��
2�k,
�5 6� ��
2�k
9.3
sec2
x�
4�
010
.ta
n x
(tan
x�
1)�
0�� 6�
��
k, �5 6� �
��
k�
k, �� 4�
��
k
11.A
via
tion
An
air
plan
e ta
kes
off
from
th
e gr
oun
d an
d re
ach
es
a h
eigh
t of
500
fee
t af
ter
flyi
ng
2 m
iles
. Giv
en t
he
form
ula
H
�d
tan
�, w
her
e H
is t
he
hei
ght
of t
he
plan
e an
d d
is t
he
dist
ance
(al
ong
the
grou
nd)
th
e pl
ane
has
flo
wn
, fin
d th
e an
gle
of a
scen
t �
at w
hic
h t
he
plan
e to
ok o
ff.
abo
ut 2
.7�
Pra
ctic
eN
AM
E__
____
____
____
____
____
____
___
DAT
E__
____
____
____
_ P
ER
IOD
____
____
7-5
Answers (Lesson 7-6)
© Glencoe/McGraw-Hill A8 Advanced Mathematical Concepts
© G
lenc
oe/M
cGra
w-H
ill29
2A
dva
nced
Mat
hem
atic
al C
once
pts
Slo
pe
s o
f P
erp
en
dic
ula
r L
ine
sT
he
deri
vati
on o
f th
e n
orm
al f
orm
of
a li
nea
r eq
uat
ion
use
s th
is
fam
ilia
r th
eore
m, f
irst
sta
ted
in L
esso
n 1
-6: T
wo
non
vert
ical
lin
es a
repe
rpen
dicu
lar
if a
nd
only
if t
he
slop
e of
on
e is
th
e n
egat
ive
reci
proc
alof
th
e sl
ope
of t
he
oth
er.
You
can
use
tri
gon
omet
ric
iden
titi
es t
o pr
ove
that
ifli
nes
are
per
pen
dicu
lar,
th
en t
hei
r sl
opes
are
n
egat
ive
reci
proc
als
of e
ach
oth
er.
� 1an
d � 2
are
perp
endi
cula
r li
nes
. �
1an
d �
2ar
e th
e an
gles
th
at �
1an
d � 2,
re
spec
tive
ly, m
ake
wit
h t
he
hor
izon
tal.
Let
m1
�sl
ope
of �
1
m2
�sl
ope
of �
2
Com
ple
te t
he
follo
win
g e
xerc
ises
to
pro
ve t
hat
m1
��
.
1.E
xpla
in w
hy
m1
�ta
n �
1an
d m
2�
tan
�2.
tan
��
�m
2.A
ccor
din
g to
th
e di
ffer
ence
iden
tity
for
th
e co
sin
e fu
nct
ion
,co
s (�
2�
�1)
�co
s �
2co
s �
1�
sin
�2 si
n �
1. E
xpla
in w
hy
the
left
sid
e of
th
e eq
uat
ion
is e
qual
to
zero
.�
2�
�1
�90
°an
d c
os
90°
�0.
3.co
s �
2co
s �
1�
sin
�2
sin
�1
�0
sin
�2
sin
�1
��
cos
�2
cos
�1
��
Com
plet
e u
sin
g th
e ta
nge
nt
fun
ctio
n.
____
____
____
�__
____
____
__
Com
plet
e, u
sin
g m
1an
d m
2.__
____
____
__ �
____
____
____
cos
�2
� sin
�2
sin
�1
� cos
�1
chan
ge
in y
��
chan
ge
in x
1� m
2
Enr
ichm
ent
NA
ME
____
____
____
____
____
____
____
_ D
ATE
____
____
____
___
PE
RIO
D__
____
__
7-6
m1
�1
� m2
tan
�1
�1
� tan
�2
© G
lenc
oe/M
cGra
w-H
ill29
1A
dva
nced
Mat
hem
atic
al C
once
pts
Pra
ctic
eN
AM
E__
____
____
____
____
____
____
___
DAT
E__
____
____
____
_ P
ER
IOD
____
____
No
rma
l F
orm
of
a L
ine
ar
Eq
ua
tio
n
Wri
te t
he
stan
dar
d f
orm
of
the
equ
atio
n o
f ea
ch li
ne,
giv
enp
, th
e le
ng
th o
f th
e n
orm
al s
egm
ent,
an
d �
, th
e an
gle
th
en
orm
al s
egm
ent
mak
es w
ith
th
e p
osit
ive
x-ax
is.
1.p
�4,
��
30�
2.p
�2�
2�, �
�� 4� �
�3�
x�
y�
8�
0x
�y
�4
�0
3.p
�3,
��
60�
4.p
�8,
��
�5 6� �
x�
�3�
y�
6�
0�
3� x
�y
�16
�0
5.p
�2�
3�, �
��7 4� �
6.p
�15
, ��
225�
�2�
x�
�2�
y�
4�3�
�0
�2�
x�
�2�
y�
30�
0
Wri
te e
ach
eq
uat
ion
in n
orm
al f
orm
. Th
en f
ind
th
e le
ng
th o
fth
e n
orm
al a
nd
th
e an
gle
it m
akes
wit
h t
he
pos
itiv
e x-
axis
.
7.3x
�2y
�1
�0
�3�13
1�3� �x
��2�
131�3� �
y �
�� 11� 33��
�0;
�� 131�3� �
; 326
�
8.5x
�y
�12
�0
�5�26
2�6� �x
��� 22� 66�
�y
��6�
132�6� �
�0;
�6�13
2�6� �; 1
1�
9.4x
�3y
�4
�0
�4 5� x�
�3 5� y�
�4 5��
0;�4 5� ;
37�
10.y
�x
�5
��� 22� �
x�
�� 22� �y
��5�
22�
��
0;�5�
22�
�; 1
35�
11.
2x�
y�
1�
0
��2�
55�
�x
��� 55� �
y �
�� 55� ��
0; �� 55� �
; 207
�
12.x
�y
�5
�0
�� 22� �x
��� 22� �
y �
�5�2
2��
�0;
�5�2
2��
; 45�
7-6
Answers (Lesson 7-7)
© Glencoe/McGraw-Hill A9 Advanced Mathematical Concepts
© G
lenc
oe/M
cGra
w-H
ill29
5A
dva
nced
Mat
hem
atic
al C
once
pts
Enr
ichm
ent
NA
ME
____
____
____
____
____
____
____
_ D
ATE
____
____
____
___
PE
RIO
D__
____
__
De
rivi
ng
th
e P
oin
t-L
ine
Dis
tan
ce
Lin
e �
has
th
e eq
uat
ion
Ax
�B
y�
C�
0.A
nsw
er t
hes
e qu
esti
ons
to d
eriv
e th
e fo
rmu
la g
iven
in
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7-7
© Glencoe/McGraw-Hill A10 Advanced Mathematical Concepts
Form 1BPage 297
1. A
2. C
3. D
4. D
5. B
6. A
7. C
8. A
9. D
10. A
11. C
Page 298
12. B
13. B
14. C
15. A
16. D
17. A
18. D
19. C
20. C
Bonus: D
Page 299
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2. C
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4. D
5. D
6. C
7. B
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11. C
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Page 300
13. D
14. A
15. D
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17. A
18. D
19. A
20. B
Bonus: A
Chapter 7 Answer KeyForm 1A
© Glencoe/McGraw-Hill A11 Advanced Mathematical Concepts
Form 1C Form 2A
Chapter 7 Answer Key
Page 301
1. B
2. D
3. B
4. C
5. A
6. B
7. D
8. C
9. A
10. C
11. D
Page 302
12. B
13. B
14. C
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Page 303
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Page 304
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© Glencoe/McGraw-Hill A12 Advanced Mathematical Concepts
Form 2CForm 2B
Chapter 7 Answer Key
Page 305
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Page 306
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© Glencoe/McGraw-Hill A13 Advanced Mathematical Concepts
Chapter 7 Answer KeyCHAPTER 7 SCORING RUBRIC
Level Specific Criteria
3 Superior • Shows thorough understanding of the concepts proof, identity, normal to a line, and distance from a point to a line.
• Uses appropriate strategies to prove identities and write equations in normal form.
• Computations are correct.• Written explanations are exemplary.• Graphs are accurate and appropriate.• Goes beyond requirements of some or all problems.
2 Satisfactory, • Shows understanding of the concepts proof, identity, with Minor normal to a line, and distance from a point to a line.Flaws • Uses appropriate strategies to prove identities and
write equations in normal form.• Computations are mostly correct.• Written explanations are effective.• Graphs are mostly accurate and appropriate.• Satisfies all requirements of problems.
1 Nearly • Shows understanding of most of the concepts proof, Satisfactory, identity, normal to a line, and distance from a point to a line.with Serious • May not use appropriate strategies to prove identities and Flaws write equations in normal form.
• Computations are mostly correct.• Written explanations are satisfactory.• Graphs are mostly accurate and appropriate.• Satisfies most requirements of problems.
0 Unsatisfactory • Shows little or no understanding of the concepts proof, identity, normal to a line, and distance from a point to a line.
• May not use appropriate strategies to prove identities and write equations in normal form.
• Computations are incorrect.• Written explanations are not satisfactory.• Graphs are not accurate and appropriate.• Does not satisfy requirements of problems.
© Glencoe/McGraw-Hill A14 Advanced Mathematical Concepts
Page 309
1a.�1
c�ossin�
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1b. There are many ways to makea simple expressioncomplicated but few ways to simplify a complicatedexpression. Thus, it is easier to find the propersimplification.
1c. No, because in the third line ofthe attempted proof, theexpression has been treated asan equality by multiplying bothsides by cos A. A correctverifications follows.sec A sin A � tan A�co
1s A� sin A � tan A
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Since cos � � 0 and sin � 0, � is in Quadrant II. The angle ofthe normal with the positive x-axis � 146�.
2b. The point at (0, 3) is on theline with equation 2y � 3x � 6.The distance from the point at(0, 3) to the line with equation6x � 4y � 16 � 0 is
d ��6(�0)
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1��6�16� , or � �2�
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The negative sign indicatesthat the point and the originare on the same side of the line.
2c. No, because the origin and anypoint on the line with equation6x � 4y � 16 � 0 are onopposite sides of the line withequation 2y � 3x � 6.
2d. For parallel lines, d will havethe same sign only when theorigin is between the lines.
cos2 � � (1 � sin2 �)���(1 � sin �)cos �
Chapter 7 Answer KeyOpen-Ended Assessment
�2�1 3
1�3��
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1�3��
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© Glencoe/McGraw-Hill A15 Advanced Mathematical Concepts
Mid-Chapter TestPage 310
1. �21�
2. ��54�
3. sin x
4. cos �
5. �31�
6. ���6�4� �2��
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Quiz DPage 312
1. �2�5
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4. 6 units(�5� � �1�0�)x �
(3�5� � 2�1�0�)y �
5. 3�5� � 2�1�0� � 0
Chapter 7 Answer Key
���22�66��x � �5�
262�6��y �
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2�6�� � 0
© Glencoe/McGraw-Hill A16 Advanced Mathematical Concepts
Page 313
1. C
2. C
3. A
4. E
5. A
6. E
7. D
8. C
9. B
Page 314
10. D
11. E
12. C
13. E
14. B
15. A
16. B
17. D
18. D
19. 70
20. 120
Page 315
1. 3x � y � 5 � 0
2. � �ƒ�1(x) �
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4. 2
5. x3 � 3x2 � 4x � 12 � 0
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7. 41.4�
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11. �60�, 60�
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85
Chapter 7 Answer KeySAT/ACT Practice Cumulative Review
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