Chapter 7 Resource Masters -...

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







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


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


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


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�


7-2



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



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



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�

��

�


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�


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.)


7-3



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


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


7-4



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��


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



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.



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.


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.


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


NAME _____________________________ DATE _______________ PERIOD ________

Enrichment7-5

c�v


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�


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.



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


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


7-6



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


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


7-7



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


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�


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)


7


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


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


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)


7


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


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


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)


7


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


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)


7


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


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. __________________


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)


7


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


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. __________________


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)


7


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


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)


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 ________


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 ________


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.


Chapter 7 SAT and ACT Practice


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)


Chapter 7 SAT and ACT Practice (continued)


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


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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SAT and ACT Practice Answer Sheet(20 Questions)

NAME _____________________________ DATE _______________ PERIOD ________

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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



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lenc

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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

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Mat

hem

atic

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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



© G

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cGra

w-H

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2A

dva

nced

Mat

hem

atic

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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

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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�,�

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�1 12 10 9��4 5� ,

�3 5� ,� 34 �

6.se

c �

��

�5 2� , �� 2�

��

��

7.si

n �

��3 5� ,

0�

��

� 2� �

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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





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

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9A

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Mat

hem

atic

al C

once

pts

NA

ME

____

____

____

____

____

____

____

_ D

ATE

____

____

____

___

PE

RIO

D__

____

__

Enr

ichm

ent

7-5

c � v

© G

lenc

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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�

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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

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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



© G

lenc

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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

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1A

dva

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Mat

hem

atic

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once

pts

Pra

ctic

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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� �

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x�

y�

8�

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x�

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6.p

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, ��

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ach

eq

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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

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�0

�3�13

1�3� �x

��2�

131�3� �

y �

�� 11� 33��

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8.5x

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132�6� �

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2�6� �; 1

1�

9.4x

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10.y

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x�

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11.

2x�

y�

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0

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y �

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0; �� 55� �

; 207

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© G

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hem

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once

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Enr

ichm

ent

NA

ME

____

____

____

____

____

____

____

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ATE

____

____

____

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PE

RIO

D__

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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

Les

son

7-7

for

th

e di

stan

ce f

rom

P(x

1, y

1) t

o �

.

1.U

se t

he

equ

atio

n o

f th

e li

ne

to f

ind

the

coor

din

ates

of

Jan

d K

, th

e x-

an

d y-

inte

rcep

ts

of �

.

J��

, 0�, K

�0, �

�2.

P�Q�

is a

ver

tica

l seg

men

t fr

om P

to �

. F

ind

the

x-co

ordi

nat

e of

Q.

x 1

3.S

ince

Qis

on

�, i

ts c

oord

inat

es m

ust

sat

isfy

th

e eq

uat

ion

of

�.

Use

you

r an

swer

to

Exe

rcis

e 2

to f

ind

the

y-co

ordi

nat

e of

Q.

�x 1

�

4.F

ind

PQ

by f

indi

ng

the

diff

eren

ce b

etw

een

th

e y-

coor

din

ates

of

Pan

d Q

. Wri

te y

our

answ

er a

s a

frac

tion

.

��

5.Tr

ian

gle

KJ

Ois

a r

igh

t tr

ian

gle.

Use

you

r an

swer

s to

Exe

rcis

e 1

and

the

Pyt

hag

orea

n T

heo

rem

to

fin

d K

J. S

impl

ify.

��

6.S

ince

�Q

��

K ,

�J

KO

~ �

PQ

R.

�

Use

you

r an

swer

s to

Exe

rcis

es 1

, 4, a

nd

5 to

fin

d P

R, t

he

dist

ance

from

Pto

�. S

impl

ify.

PR

��(A

x 1�

By 1

�C

) ��

��

A�2 ��

B�2 �

PQ

� KJ

PR

� OJ

C�

A�2 �+�

�B�

2 ��

�A

B

Ax 1

�B

y 1�

C�

� B

C BA B

C BC A

7-7

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dva

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hem

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once

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Dis

tan

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Fro

m a

Po

int

to a

Lin

e

Fin

d t

he

dis

tan

ce b

etw

een

th

e p

oin

t w

ith

th

e g

iven

coo

rdin

ates

and

th

e lin

e w

ith

th

e g

iven

eq

uat

ion

.

1.(�

1, 5

), 3

x�

4y�

1�

02.

(2, 5

), 5

x�

12y

�1

�0

�2 54 ��4 139 �

3.(1

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, 12x

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�3

�0

4.(�

1,�

3), 6

x�

8y�

3�

0�1 11 3�

�3 13 0�

Fin

d t

he

dis

tan

ce b

etw

een

th

e p

aral

lel l

ines

wit

h t

he

giv

eneq

uat

ion

s.

5.2x

�3y

�4

�0

6.4x

�y

�1

�0

y�

�2 3� x�

54x

�y

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�0

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�9�17

1�7� �

Fin

d e

qu

atio

ns

of t

he

lines

th

at b

isec

t th

e ac

ute

an

d o

btu

sean

gle

s fo

rmed

by

the

lines

wit

h t

he

giv

en e

qu

atio

ns.

7.x

�2y

�3

�0

x�

y�

4�

0

(�2�

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E__

____

____

____

____

____

____

___

DAT

E__

____

____

____

_ P

ER

IOD

____

____

7-7


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

1. B

2. C

3. A

4. D

5. D

6. C

7. B

8. A

9. C

10. B

11. C

12. B

Page 300

13. D

14. A

15. D

16. B

17. A

18. D

19. A

20. B

Bonus: A

Chapter 7 Answer KeyForm 1A


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

15. A

16. D

17. C

18. B

19. D

20. A

Bonus: D

Page 303

1. cos �

2. ��45�

3. tan �

4. 2 sin �

5. 1

6. ��6�4� �2��

7. �48275

�

8. sin �

9. � �78

�

10. � �2245� or �0.96

11. � �274�

Page 304

12. �21��2� �� 2��

13. � � 2�k

14. 30� and 90�

15. �76�� x � �11

6��

16.

17. x � �3�y � 10 � 0

18. �4�5

5�� units

19. �3�10

1�0�� units

(2�1�3� � 3�5�)x �

(�1�3� � 2�5�)y �

20. 5�1�3� � 6�5� � 0

�632356� or

Bonus: 0.5376

�3�13

1�3��x � �2�13

1�3��y �

�4�13

1�3�� 0


Form 2CForm 2B


Page 305

1. sec �

2. �54�

3. sin �

4. cot2 �

5. �31�

6. ��2� �4

�6��

7. ��6156�

8. cos �

9. �97�

10. �225

4� or 0.96

11. � �274�

Page 306

12. �

13. ��3

�, �43��

14. �30�, 30�

15. �k

16.

17. �3�x � y � 14 � 0

18. �91�05��

19. ��55��

(3�5� � 2�1�0�)x �

(�5� � �1�0�)y �

20. 6�5� � �1�0� � 0

Bonus: sin2 �

Page 307

1. cos �

2. ��34�

3. 1

4. cos2 �

5. �21�

6. ��2� �4

�6��

7. ��1663�

8. sin �

9. ��21�

10. �225

4� or 0.96

11. � �274�

Page 308

12.

13. ��6

�, �56��

14. �60°

15. ��4

� � �k

16.

17. x � �3�y � 18 � 0

18. �7�29

2�9�� units

19. �3�4

2�� units

20. 7x � 56y � 25 � 0

They areperpendicularto each

Bonus: other.

�2� �� 2��2�2� �� 3��2

�2�29

2�9��x � �5�29

2�9��y �

�3�29

2�9�� 0

��31�3

1�3��x � �2�13

1�3��y �

�6�13

1�3�� 0


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.


Page 309

1a.�1

c�ossin�

�� 1 c

�ossin�

�� 0

� 0

�(c1o�s2

s�in

��c) c

oos2

s��

� � 0

�(1 � sin

0�) cos �� 0

0 � 0

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

�csoins

AA

� � tan Atan A � tan A

2a. ��

31�3��x � �

�21�3��y � �

�61�3�� 0 or

��3�13

1�3��x � �2�1 3

1�3��y � �6�13

1�3�� 0

Length of normal � ��

61�3��

or �6�13

1�3�� , tan � �

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)

��3�6�

4(��3)

1��6�16� , or � �2�

131�3�� .

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��

��3�13

1�3��

� ��23�.


Mid-Chapter TestPage 310

1. �21�

2. ��54�

3. sin x

4. cos �

5. �31�

6. ��6�4� �2��

7. �3536�

8. �2245�

9. �87�

10. �12

� �2� �� 2��

Quiz APage 311

1. �31�

2. � �35�

3. csc2 x

4. sin4 �

5. �21�

Quiz BPage 311

1. ��2� �4

�6��

2. � �3536�

3. ��112609

�

4. ��274�

5. �21��2� �� 3��

Quiz CPage 312

1. �32��

2. 30�, 150°

3. ��4�� k

4.

5. x � �3� y � 6 � 0

Quiz DPage 312

1. �2�5

5�� units

2. 0 units

3. �8�5

5�� units

4. 6 units(�5� � �1�0�)x �

(3�5� � 2�1�0�)y �

5. 3�5� � 2�1�0� � 0


��22�66��x � �5�

262�6��y �

�4�13

2�6�� 0


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) �

3. 2 � �x� �� 5�; no

4. 2

5. x3 � 3x2 � 4x � 12 � 0

6. �2; no

7. 41.4�

8. 43.8 units2

9. 13; � ; 2�

10. 0

11. �60�, 60�

12. �5�

532

85

Chapter 7 Answer KeySAT/ACT Practice Cumulative Review

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