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Chapter 5Resource Masters

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ISBN: 0-07-869132-X Advanced Mathematical ConceptsChapter 5 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-x

Lesson 5-1Study Guide . . . . . . . . . . . . . . . . . . . . . . . . . 181Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . . 183








Chapter 5 AssessmentChapter 5 Test, Form 1A . . . . . . . . . . . . 205-206Chapter 5 Test, Form 1B . . . . . . . . . . . . 207-208Chapter 5 Test, Form 1C . . . . . . . . . . . . 209-210Chapter 5 Test, Form 2A . . . . . . . . . . . . 211-212Chapter 5 Test, Form 2B . . . . . . . . . . . . 213-214Chapter 5 Test, Form 2C . . . . . . . . . . . . 215-216Chapter 5 Extended Response

Assessment . . . . . . . . . . . . . . . . . . . . . . . 217Chapter 5 Mid-Chapter Test . . . . . . . . . . . . . 218Chapter 5 Quizzes A & B . . . . . . . . . . . . . . . 219Chapter 5 Quizzes C & D. . . . . . . . . . . . . . . 220Chapter 5 SAT and ACT Practice . . . . . 221-222Chapter 5 Cumulative Review . . . . . . . . . . . 223

SAT and ACT Practice Answer Sheet,10 Questions . . . . . . . . . . . . . . . . . . . . . . . A1

SAT and ACT Practice Answer Sheet,20 Questions . . . . . . . . . . . . . . . . . . . . . . . A2

ANSWERS . . . . . . . . . . . . . . . . . . . . . . A3-A17

Contents

© Glencoe/McGraw-Hill iv Advanced Mathematical Concepts

A Teacher’s Guide to Using theChapter 5 Resource Masters

The Fast File Chapter Resource system allows you to conveniently file theresources you use most often. The Chapter 5 Resource Masters include the corematerials needed for Chapter 5. 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-x 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 5-1. 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 5Resources 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 341. 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 5 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.

© Glencoe/McGraw-Hill vii Advanced Mathematical Concepts

This is an alphabetical list of the key vocabulary terms you will learn in Chapter 5.As you study the chapter, complete each term’s definition or description.Remember to add the page number where you found the term.

Vocabulary Term Foundon Page Definition/Description/Example

ambiguous case

angle of depression

angle of elevation

apothem

arccosine relation

arcsine relation

arctangent relation

circular function

cofunctions

cosecant

(continued on the next page)

Reading to Learn MathematicsVocabulary Builder

NAME _____________________________ DATE _______________ PERIOD ________Chapter

5

© Glencoe/McGraw-Hill viii Advanced Mathematical Concepts


cosine

cotangent

coterminal angles

degree

Hero’s Formula

hypotenuse

initial side

inverse

Law of Cosines

Law of Sines

leg


Reading to Learn MathematicsVocabulary Builder (continued)


5

© Glencoe/McGraw-Hill ix Advanced Mathematical Concepts


NAME _____________________________ DATE _______________ PERIOD ________

3


minute

quadrant angle

reference angle

secant

second

side adjacent

side opposite

sine

solve a triangle

standard position

tangent



5

© Glencoe/McGraw-Hill x Advanced Mathematical Concepts


NAME _____________________________ DATE _______________ PERIOD ________


terminal side

trigonometric function

trigonometric ratio

unit circle

vertex

Chapter

5

© Glencoe/McGraw-Hill 181 Advanced Mathematical Concepts

Study GuideNAME _____________________________ DATE _______________ PERIOD ________

5-1

Angles and Degree MeasureDecimal degree measures can be expressed in degrees(°),minutes(′), and seconds(″).

Example 1 a. Change 12.520° to degrees, minutes, andseconds.12.520° � 12° � (0.520 � 60)′ Multiply the decimal portion of

� 12° � 31.2′ the degrees by 60 to find minutes.� 12° � 31′ � (0.2 � 60)″ Multiply the decimal portion of� 12° � 31′ � 12″ the minutes by 60 to find seconds.

12.520° can be written as 12° 31′ 12″.

b. Write 24° 15′ 33″ as a decimal rounded to thenearest thousandth.

24° 15′ 33″ � 24° � 15′ ��610°′�� 33″ ��36

10°0″��

� 24.259°24° 15′ 33″ can be written as 24.259°.

An angle may be generated by the rotation of one raymultiple times about the origin.

Example 2 Give the angle measure represented by eachrotation.a. 2.3 rotations clockwise

2.3 � �360 � �828 Clockwise rotations have negative measures.The angle measure of 2.3 clockwise rotations is �828°.

b. 4.2 rotations counterclockwise4.2 � 360 � 1512 Counterclockwise rotations have positive

measures.The angle measure of 4.2 counterclockwise rotations is 1512°.

If � is a nonquadrantal angle in standard position, its referenceangle is defined as the acute angle formed by the terminal side ofthe given angle and the x-axis.

Example 3 Find the measure of the reference angle for 220°.Because 220° is between 180° and 270°, theterminal side of the angle is in Quadrant III.220° � 180° � 40°The reference angle is 40°.

Reference

For any angle �, 0° < � < 360°, its reference angle �′ is defined by

Angle Rule

a. �, when the terminal side is in Quadrant I,b. 180° � �, when the terminal side is in Quadrant II,c. � � 180°, when the terminal side is in Quadrant III, andd. 360° � �, when the terminal side is in Quadrant IV.


Angles and Degree Measure

Change each measure to degrees, minutes, and seconds.1. 28.955� 2. �57.327�

Write each measure as a decimal degree to the nearestthousandth.

3. 32� 28′ 10″ 4. �73� 14′ 35″

Give the angle measure represented by each rotation.5. 1.5 rotations clockwise 6. 2.6 rotations counterclockwise

Identify all angles that are coterminal with each angle. Then findone positive angle and one negative angle that are coterminalwith each angle.

7. 43� 8. �30�

If each angle is in standard position, determine a coterminal anglethat is between 0� and 360�, and state the quadrant in which theterminal side lies.

9. 472� 10. �995�

Find the measure of the reference angle for each angle.11. 227� 12. 640�

13. Navigation For an upcoming trip, Jackie plans to sail fromSanta Barbara Island, located at 33� 28′ 32″ N, 119� 2′ 7″ W, toSanta Catalina Island, located at 33.386� N, 118.430� W. Writethe latitude and longitude for Santa Barbara Island as decimalsto the nearest thousandth and the latitude and longitude forSanta Catalina Island as degrees, minutes, and seconds.

PracticeNAME _____________________________ DATE _______________ PERIOD ________

5-1

Reading Mathematics: If and Only If StatementsIf p and q are interchanged in a conditional statement so that pbecomes the conclusion and q becomes the hypothesis, the new state-ment, q → p, is called the converse of p → q.

If p → q is true, q → p may be either true or false.

Example Find the converse of each conditional.a. p → q: All squares are rectangles. (true)

q → p: All rectangles are squares. (false)

b. p → q: If a function ƒ(x) is increasing on aninterval I, then for every a and b contained inI, ƒ(a) � ƒ(b) whenever a � b. (true)q → p: If for every a and b contained in an interval I,ƒ(a) � ƒ(b) whenever a � b then function ƒ(x) isincreasing on I. (true)

In Lesson 3-5, you saw that the two statements in Example 2 can becombined in a single statement using the words “if and only if.”

A function ƒ(x) is increasing on an interval I if and only if forevery a and b contained in I, ƒ(a) � ƒ(b) whenever a � b.

The statement “p if and only if q” means that p implies q andq implies p.

State the converse of each conditional. Then tell if the converse istrue or false. If it is true, combine the statement and its converseinto a single statement using the words “if and only if.”

1. All integers are rational numbers.

2. If for all x in the domain of a function ƒ(x), ƒ(�x) � �ƒ(x), thenthe graph of ƒ(x) is symmetric with respect to the origin.

3. If ƒ(x) and ƒ�1(x) are inverse functions, then [ ƒ ° ƒ�1](x) � [ ƒ�1 ° ƒ](x) � x.


EnrichmentNAME _____________________________ DATE _______________ PERIOD ________

5-1



5-2

Trigonometric Ratios in Right TrianglesThe ratios of the sides of right triangles can be used to definethe trigonometric ratios known as the sine, cosine, andtangent.

Example 1 Find the values of the sine, cosine, and tangentfor �A.

First find the length of B�C�.(AC)2 � (BC)2 � (AB)2 Pythagorean Theorem

102 � (BC)2 � 202 Substitute 10 for AC and 20 for AB.(BC)2 � 300

BC � �3�0�0� or 10�3� Take the square root of each side.Disregard the negative root.

Then write each trigonometric ratio.

sin A � �shidyepo

otpepnoussiete

� cos A � �sihdyepaotdejnaucesne

t� tan A � �s

siiddee

aodpp

jaocseintet�

sin A � �102�0

3�� or ��23�� cos A � �12

00� or �2

1� tan A � �101�0

3�� or �3�

Trigonometric ratios are often simplified but never written asmixed numbers.

Three other trigonometric ratios, called cosecant, secant,and cotangent, are reciprocals of sine, cosine, and tangent,respectively.

Example 2 Find the values of the six trigonometric ratios for �R.

First determine the length of the hypotenuse.(RT)2 � (ST)2 � (RS)2 Pythagorean Theorem

152 � 32 � (RS)2 RT � 15, ST � 3(RS)2 � 234

RS � �2�3�4� or 3�2�6� Disregard the negative root.

sin R � �shidyepo

otpepnoussiete

� cos R � �sihdyepaotdejnaucesne

t� tan R � �s

siiddee

aodpp

jaocseintet�

sin R � �3�

32�6�� or ��26

2�6�� cos R � �3�

152�6�� or �5�

262�6�� tan R � �1

35� or �5

1�

csc R � �shidyepo

otpepnoussiete� sec R � �si

hdyepaotdejnaucesne

t� cot R � �ssiidd

ee

aopdpjaocseitnet

�

csc R � �3�32�6�� or �2�6� sec R � �3�

152�6�� or ��5

2�6�� cot R � �135� or 5



Trigonometric Ratios in Right Triangles

Find the values of the sine, cosine, and tangent for each �B.

1. 2.

3. If tan � � 5, find cot �. 4. If sin � � �38�, f ind csc �.

Find the values of the six trigonometric ratios for each �S.

5. 6.

7. Physics Suppose you are traveling in a car when a beam of lightpasses from the air to the windshield. The measure of the angle of incidence is 55�, and the measure of the angle of refraction is 35� 15′. Use Snell’s Law, �s

siinn

�

�

r

i� � n, to f ind the index of refraction nof the windshield to the nearest thousandth.

5-2



5-2

Using Right Triangles to Find the Area of Another TriangleYou can find the area of a right triangle by using the formula

A � bh. In the formula, one leg of the right triangle can be used as

the base, and the other leg can be used as the height.

The vertices of a triangle can be represented on the coordinate planeby three ordered pairs. In order to find the area of a general triangle,you can encase the triangle in a rectangle as shown in the diagrambelow.

A rectangle is placed around the triangle so that the vertices of thetriangle all touch the sides of the rectangle.Example Find the area of a triangle whose vertices are

A(�1, 3), B(4, 8), and C(8, 5).Plot the points and draw the triangle. Encase the trianglein a rectangle whose sides are parallel to the axes, then find the coordinates of the vertices of the rectangle.

Area �ABC � area ADEF � area �ADB �area �BEC � area �CFA, where �ADB, �BEC,and �CFA are all right triangles.

Area �ABC � 5(9) � (5)(5) � (4)(3) � (2)(9)

� 17.5 square units

Find the area of the triangle having vertices with each set of coordinates.1. A(4, 6), B(–1, 2), C(6, –5) 2. A(–2, –4), B(4, 7), C(6, –1)3. A(4, 2), B(6, 9), C(–1, 4) 4. A(2, –3), B(6, –8), C(3, 5)

1�2

1�2

1�2

1�2



Trigonometric Functions on the Unit Circle

Example 1 Use the unit circle to find cot (�270°).

The terminal side of a �270° angle in standard position is the positive y-axis,which intersects the unit circle at (0, 1).By definition, cot (�270°) � �y

x� or �01�.

Therefore, cot (�270°) � 0.

Trigonometric sin � � �yr� cos � � �

xr� tan � � �

yx�

Functions of an Angle in csc � � �y

r� sec � � �x

r� cot � � �

xy�

Standard Position

Example 2 Find the values of the six trigonometric functionsfor angle � in standard position if a point withcoordinates (�9, 12) lies on its terminal side.

We know that x � �9 and y � 12. We need tofind r.

r � �x�2�� y�2� Pythagorean Theorem

r � �(��9�)2� �� 1�2�2� Substitute �9 for x and 12 for y.

r � �2�2�5� or 15 Disregard the negative root.

sin � � �1125� or �45� cos � � ��15

9� or ��35� tan � � ��12

9� or ��43�

csc � � �1152� or �54� sec � � �

�15

9� or ��53� cot � � ��129� or ��34�

Example 3 Suppose � is an angle in standard position whoseterminal side lies in Quadrant I. If cos � � �35�, find the values of the remaining five trigonometricfunctions of �.

r2 � x2 � y2 Pythagorean Theorem52 � 32 � y2 Substitute 5 for r and 3 for x.16 � y2

4 � y Take the square root of each side.

Since the terminal side of � lies in Quadrant I, y mustbe positive.

sin � � �45� tan � � �43�

csc � � �54� sec � � �35� cot � � �34�

5-3


Trigonometric Functions on the Unit Circle

Use the unit circle to find each value.1. csc 90� 2. tan 270� 3. sin (�90�)

Use the unit circle to find the values of the six trigonometricfunctions for each angle.4. 45�

5. 120�

Find the values of the six trigonometric functions for angle � instandard position if a point with the given coordinates lies on itsterminal side.6. (�1, 5) 7. (7, 0) 8. (�3, �4)


5-3



5-3

Areas of Polygons and CirclesA regular polygon has sides of equal length and angles of equal measure. A regular polygon can be inscribed in or circumscribed about a circle. For n-sided regular polygons, the following area formulas can be used.

Area of circle AC �� r2

Area of inscribed polygon AI � � sin

Area of circumscribed polygon AC � nr2 � tan

Use a calculator to complete the chart below for a unit circle (a circle of radius 1).

1.

2.

3.

4.

5.

6.

7.

8.

9. What number do the areas of the circumscribed and inscribed polygons seem to be approaching as the number of sides of the polygon increases?

180°�

n

360°�

n

nr2

�2

NumberArea of Area of Circle Area of Area of Polygon

of SidesInscribed less Circumscribed lessPolygon Area of Polygon Polygon Area of Circle

3 1.2990381 1.8425545 5.1961524 2.0545598

4

8

12

20

24

28

32

1000



5-4

Applying Trigonometric FunctionsTrigonometric functions can be used to solve problemsinvolving right triangles.

Example 1 If T � 45° and u � 20, find t to the nearest tenth.

From the figure, we know the measures of anangle and the hypotenuse. We want to know themeasure of the side opposite the given angle. Thesine function relates the side opposite the angleand the hypotenuse.

sin T � �ut� sin � �

shidyepo

otpepnoussiete

�

sin 45° � �2t0� Substitute 45° for T and 20 for u.

20 sin 45° � t Multiply each side by 20.14.14213562 � t Use a calculator.

Therefore, t is about 14.1.

Example 2 Geometry The apothem of a regular polygon is themeasure of a line segment from the center of thepolygon to the midpoint of one of its sides. Theapothem of a regular hexagon is 2.6 centimeters.Find the radius of the circle circumscribed aboutthe hexagon to the nearest tenth.

First draw a diagram. Let a be the anglemeasure formed by a radius and its adjacentapothem. The measure of a is 360° 12 or30°. Now we know the measures of an angleand the side adjacent to the angle.

cos 30° � �2r.6� cos � �

sihdyepaotdejnaucesne

t�

r cos 30° � 2.6 Multiply each side by r.

r � �co2s.360°� Divide each side by cos 30°.

r � 3.0022214 Use a calculator.

Therefore, the radius is about 3.0 centimeters.



Applying Trigonometric FunctionsSolve each problem. Round to the nearest tenth.1. If A � 55� 55′ and c � 16, find a.

2. If a � 9 and B � 49�, f ind b.

3. If B � 56� 48′ and c � 63.1, find b.

4. If B � 64� and b � 19.2, find a.

5. If b � 14 and A � 16�, f ind c.

6. Construction A 30-foot ladder leaning against the side of a house makes a 70� 5′ angle with the ground.a. How far up the side of the house does the

ladder reach?

b. What is the horizontal distance between thebottom of the ladder and the house?

7. Geometry A circle is circumscribed about a regular hexagon with an apothem of 4.8 centimeters.a. Find the radius of the circumscribed circle.

b. What is the length of a side of the hexagon?

c. What is the perimeter of the hexagon?

8. Observation A person standing 100 feet from the bottomof a cliff notices a tower on top of the cliff. The angle of elevation to the top of the cliff is 30�. The angle of elevationto the top of the tower is 58�. How tall is the tower?

5-4



5-4

Making and Using a HypsometerA hypsometer is a device that can be used to measure the height ofan object. To construct your own hypsometer, you will need a rectangular piece of heavy cardboard that is at least 7 cm by 10 cm,a straw, transparent tape, a string about 20 cm long, and a smallweight that can be attached to the string.

Mark off 1-cm increments along one short side and one long side ofthe cardboard. Tape the straw to the other short side. Then attachthe weight to one end of the string, and attach the other end of thestring to one corner of the cardboard, as shown in the figure below.The diagram below shows how your hypsometer should look.

To use the hypsometer, you will need to measure the distance fromthe base of the object whose height you are finding to where youstand when you use the hypsometer.

Sight the top of the object through the straw. Note where the free-hanging string crosses the bottom scale. Then use similar trianglesto find the height of the object.

1. Draw a diagram to illustrate how you can use similar trianglesand the hypsometer to find the height of a tall object.

Use your hypsometer to find the height of each of the following.

2. your school’s flagpole3. a tree on your school’s property4. the highest point on the front wall of your school building5. the goal posts on a football field6. the hoop on a basketball court7. the top of the highest window of your school building8. the top of a school bus9. the top of a set of bleachers at your school

10. the top of a utility pole near your school



5-5

Solving Right TrianglesWhen we know a trigonometric value of an angle but not thevalue of the angle, we need to use the inverse of thetrigonometric function.

Example 1 Solve tan x � �3�.

If tan x � �3�, then x is an angle whose tangentis �3�.

x � arctan �3�

From a table of values, you can determine that x equals 60°, 240°, or any angle coterminal withthese angles.

Example 2 If c � 22 and b � 12, find B.

In this problem, we know the side opposite theangle and the hypotenuse. The sine functionrelates the side opposite the angle and thehypotenuse.

sin B � �bc� sin � �shidyepo

otpepnoussiete

�

sin B � �1222� Substitute 12 for b and 22 for c.

B � sin�1��1222�� Definition of inverse

B � 33.05573115 or about 33.1°.

Example 3 Solve the triangle where b � 20 and c � 35, giventhe triangle above.

a2 � b2 � c2

a2 � 202 � 352

a � �8�2�5�a � 28.72281323

55.15009542 � B � 90B � 34.84990458

Therefore, a � 28.7, A � 55.2°, and B � 34.8°.

cos A � �bc�

cos A � �3250�

A � cos-1��3250��

A � 55.15009542

Trigonometric Function Inverse Trigonometric Relationy � sin x x � sin�1 y or x � arcsin yy � cos x x � cos�1 y or x � arccos yy � tan x x � tan�1 y or x � arctan y


Solving Right Triangles

Solve each equation if 0� � x � 360�.

1. cos x � ��22�� 2. tan x � 1 3. sin x � �2

1�

Evaluate each expression. Assume that all angles are in Quadrant I.

4. tan �tan�1 ��33�� 5. tan �cos�1 �23�� 6. cos �arcsin �1

53��

Solve each problem. Round to the nearest tenth.7. If q � 10 and s � 3, find S.

8. If r � 12 and s � 4, find R.

9. If q � 20 and r � 15, find S.

Solve each triangle described, given the triangle at the right.Round to the nearest tenth, if necessary.10. a � 9, B � 49�

11. A � 16�, c � 14

12. a � 2, b � 7

13. Recreation The swimming pool at Perris Hill Plunge is 50 feetlong and 25 feet wide. The bottom of the pool is slanted so thatthe water depth is 3 feet at the shallow end and 15 feet at thedeep end. What is the angle of elevation at the bottom of thepool?


5-5


NAME _____________________________ DATE _______________ PERIOD ________

Enrichment5-5

Disproving Angle TrisectionMost geometry texts state that it is impossible to trisect an arbitraryangle using only a compass and straightedge. This fact has beenknown since ancient times, but since it is usually stated withoutproof, some geometry students do not believe it. If the students setout to find a method for trisecting angles, they will probably try thefollowing method. It is based on the familiar construction whichallows a segment to be divided into any desired number of congruentsegments. You can use inverse trigonometric functions to show thatapplication of the method to the trisection of angles is not valid.

Given: � AClaim: � A can be trisected using the following method.

Method: Choose point C on one ray of � A .Through C construct a perpendicular to the other ray, intersecting it at B.Construct M and N, the points that divide C�B� into three congruent segments. Draw A�M� and A�N�, whichtrisect �CAB into the congruent angles�1, �2, and �3.

The proposed method has been used to construct the figure below.CM � MN � NB � 1. AB � 5. Follow the instructions to show thatthe three angles �1, �2, and �3, are not congruent. Find anglemeasures to the nearest tenth of a degree.

1. Express m�3 as the value of an inverse function.

2. Find the measure of �3.

3. Write m�MAB as the value of an inverse function.

4. Find the measure of �MAB.5. Find the measure of �2.6. Find m�CAB and use it to find m �1.7. Explain why the proposed method for trisecting an angle fails.


The Law of Sines

Given the measures of two angles and one side of a triangle,we can use the Law of Sines to find one unique solution forthe triangle.

Example 1 Solve � ABC if A � 30°, B � 100°, and a � 15.

First find the measure of �C.C � 180° � (30° � 100°) or 50°

Use the Law of Sines to find b and c.

�sina

A� � �sinb

B� �sinc

C� � �sina

A�

�sin15

30°� � �sinb100°� �sin

c50°� � �sin

1530°�

�15ssinin

3100°0°� � b c � �15

sisnin30

5°0°�

29.54423259 � b c � 22.98133329

Therefore, C � 50°, b � 29.5, and c � 23.0.

The area of any triangle can be expressed in terms of twosides of a triangle and the measure of the included angle.

Example 2 Find the area of �ABC if a � 6.8, b � 9.3, and C � 57°.

K � �12�ab sin C

K � �12�(6.8)(9.3) sin 57°

K � 26.51876336

The area of �ABC is about 26.5 square units.


5-6

Law of Sines �sin

aA

� � �sin

bB

� � �sin

cC

�

Area (K) of a Triangle K � �12

�bc sin A K � �12

�ac sin B K � �12

�ab sin C



The Law of Sines

Solve each triangle. Round to the nearest tenth.1. A � 38�, B � 63�, c � 15 2. A � 33�, B � 29�, b � 41

3. A � 150�, C � 20�, a � 200 4. A � 30�, B � 45�, a � 10

Find the area of each triangle. Round to the nearest tenth.5. c � 4, A � 37�, B � 69� 6. C � 85�, a � 2, B � 19�

7. A � 50�, b � 12, c � 14 8. b � 14, C � 110�, B � 25�

9. b � 15, c � 20, A � 115� 10. a � 68, c � 110, B � 42.5�

11. Street Lighting A lamppost tilts toward the sunat a 2� angle from the vertical and casts a 25-footshadow. The angle from the tip of the shadow to thetop of the lamppost is 45�. Find the length of thelamppost.

5-6



5-6

Triangle ChallengeA surveyor took the following measurements from two irregularly-shaped pieces of land. Some of the lengths and angle measurementsare missing. Find all missing lengths and angle measurements. Roundlengths to the nearest tenth and angle measurements to the nearestminute.

1.

2.

e

G

H

a

f

e

J

b

c



The Ambiguous Case for the Law of SinesIf we know the measures of two sides and a nonincluded angle of a triangle, three situations are possible: no triangle exists, exactly one triangle exists, or two triangles exist. A triangle with two solutions is called the ambiguous case.

Example Find all solutions for the triangleif a � 20, b � 30, and A � 40°. If no solutions exist, write none.

Since 40° � 90°, consider Case 1.b sin A � 30 sin 40°b sin A � 19.28362829Since 19.3 � 20 � 30, there are two solutions forthe triangle.Use the Law of Sines to find B.

�sin20

40°� � �si3n0B� �sin

aA� � �sin

bB�

sin B � �30 s2in0

40°�

B � sin�1��30 s2in0

40°��B � 74.61856831

So, B � 74.6°. Since we know there are two solutions,there must be another possible measurement for B.In the second case, B must be less than 180° andhave the same sine value. Since we know that if � � 90, sin � � sin (180 � �), 180° � 74.6° or 105.4°is another possible measure for B. Now solve thetriangle for each possible measure of B.

Solution I

C � 180° � (40° � 74.6°) or 65.4°

�sina

A� � �sinc

C�

�sin20

40°� � �sin 6c5.4°�

c � �20ssiinn

4605°.4°�

c � 28.29040558

One solution is B � 74.6°,C � 65.4°, and c � 28.3.

5-7

Solution II

C � 180° � (40° � 105.4°) or 34.6°

�sina

A� � �sinc

C�

�sin20

40°� � �sin 3c4.6°�

c � �20ssiinn

4304°.6°�

c � 17.66816088

Another solution is B � 105.4°,C � 34.6°, and c � 17.7.

Case 1: A � 90° for a, b, and Aa � b sin A no solutiona � b sin A one solutiona � b one solutionb sin A � a � b two solutions

Case 2: A � 90°a b no solutiona � b one solution


The Ambiguous Case for the Law of Sines

Determine the number of possible solutions for each triangle.

1. A � 42�, a � 22, b � 12 2. a � 15, b � 25, A � 85�

3. A � 58�, a � 4.5, b � 5 4. A � 110�, a � 4, c � 4

Find all solutions for each triangle. If no solutions exist, writenone. Round to the nearest tenth.

5. b � 50, a � 33, A � 132� 6. a � 125, A � 25�, b � 150

7. a � 32, c � 20, A � 112� 8. a � 12, b � 15, A � 55�

9. A � 42�, a � 22, b � 12 10. b � 15, c � 13, C � 50�

11. Property Maintenance The McDougalls plan to fence a triangular parcel of theirland. One side of the property is 75 feet in length. It forms a 38� angle with anotherside of the property, which has not yet been measured. The remaining side of theproperty is 95 feet in length. Approximate to the nearest tenth the length of fenceneeded to enclose this parcel of the McDougalls’ lot.


5-7



5-7

Spherical TrianglesSpherical trigonometry is an extension of plane trigonometry.Figures are drawn on the surface of a sphere. Arcs of greatcircles correspond to line segments in the plane. The arcs ofthree great circles intersecting on a sphere form a sphericaltriangle. Angles have the same measure as the tangent linesdrawn to each great circle at the vertex. Since the sides arearcs, they too can be measured in degrees.

Example Solve the spherical triangle given a � 72°, b � 105°, and c � 61°.

Use the Law of Cosines.

0.3090 � (–0.2588)(0.4848) � (0.9659)(0.8746) cos Acos A � 0.5143

A � 59°

–0.2588 � (0.3090)(0.4848) � (0.9511)(0.8746) cos Bcos B � –0.4912

B � 119°

0.4848 � (0.3090)(–0.2588) � (0.9511)(0.9659) cos Ccos C � 0.6148

C � 52°

Check by using the Law of Sines.

� � � 1.1

Solve each spherical triangle.

1. a � 56°, b � 53°, c � 94° 2. a � 110°, b � 33°, c � 97°

3. a � 76°, b � 110°, C � 49° 4. b � 94°, c � 55°, A � 48°

sin 61°��sin 52°

sin 105°��sin 119°

sin 72°��sin 59°

The sum of the sides of a spherical triangle is less than 360°.The sum of the angles is greater than 180° and less than 540°.The Law of Sines for spherical triangles is as follows.

� �

There is also a Law of Cosines for spherical triangles.cos a � cos b cos c � sin b sin c cos Acos b � cos a cos c � sin a sin c cos Bcos c � cos a cos b � sin a sin b cos C

sin c�sin C

sin b�sin B

sin a�sin A


The Law of CosinesWhen we know the measures of two sides of a triangle andthe included angle, we can use the Law of Cosines to findthe measure of the third side. Often times we will use boththe Law of Cosines and the Law of Sines to solve a triangle.

Example 1 Solve �ABC if B � 40°, a � 12, and c � 6.

b2 � a2 � c2 � 2ac cos B Law of Cosinesb2 � 122 � 62 � 2(12)(6) cos 40°b2 � 69.68960019b � 8.348029719

So, b � 8.3.

�sinb

B� � �sinc

C� Law of Sines

�sin8.

430°� � �sin

6C�

sin C � �6 si8n.3

40°�

C � sin�1��6 si8n.3

40°��C � 27.68859159

So, C � 27.7°.A � 180° � (40° � 27.7°) � 112.3°

The solution of this triangle is b � 8.3, A � 112.3°, and C � 27.7°.

Example 2 Find the area of �ABC if a � 5, b � 8, and c � 10.

First, find the semiperimeter of �ABC.s � �12�(a � b � c)

s � �12�(5 � 8 � 10)

s � 11.5

Now, apply Hero’s Formulak � �s(�s�� a�)(�s�� b�)(�s�� c�)�k � �1�1�.5�(1�1�.5� �� 5�)(�1�1�.5� �� 8�)(�1�1�.5� �� 1�0�)�k � �3�9�2�.4�3�7�5�k � 19.81003534

The area of the triangle is about 19.8 square units.


5-8

a2 � b2 � c2 � 2bc cos ALaw of Cosines b2 � a2 � c2 � 2ac cos B

c2 � a2 � b2 � 2ab cos C



The Law of Cosines

Solve each triangle. Round to the nearest tenth.

1. a � 20, b � 12, c � 28 2. a � 10, c � 8, B � 100�

3. c � 49, b � 40, A � 53� 4. a � 5, b � 7, c � 10

Find the area of each triangle. Round to the nearest tenth.

5. a � 5, b � 12, c � 13 6. a � 11, b � 13, c � 16

7. a � 14, b � 9, c � 8 8. a � 8, b � 7, c � 3

9. The sides of a triangle measure 13.4 centimeters,18.7 centimeters, and 26.5 centimeters. Find the measureof the angle with the least measure.

10. Orienteering During an orienteering hike, two hikers start at point A and head in a direction 30� west of south to point B. They hike 6 miles from point A to point B. From point B, they hike to point C and then from point C back to point A, which is 8 miles directly north of point C. How many miles did they hike from point B to point C?

5-8



5-8

The Law of Cosines and the Pythagorean TheoremThe law of cosines bears strong similarities to thePythagorean Theorem. According to the Law ofCosines, if two sides of a triangle have lengths a andb and if the angle between them has a measure of x,then the length, y, of the third side of the trianglecan be found by using the equation

y2 � a2 � b2 � 2ab cos (x°).

Answer the following questions to clarify the relationship between the Law of Cosines and the Pythagorean Theorem.

1. If the value of x becomes less and less, what number is cos (x°)close to?

2. If the value of x is very close to zero but then increases, whathappens to cos (x°) as x approaches 90?

3. If x equals 90, what is the value of cos (x°)? What does the equa-tion of y2 � a2 � b2 � 2ab cos (x°) simplify to if x equals 90?

4. What happens to the value of cos (x°) as x increases beyond 90and approaches 180?

5. Consider some particular values of a and b, say 7 for a and19 for b. Use a graphing calculator to graph the equation youget by solving y2 � 72 � 192 � 2(7)(19) cos (x°) for y.

a. In view of the geometry of the situation, what range of valuesshould you use for X on a graphing calculator?

b. Display the graph and use the TRACE function. What do themaximum and minimum values appear to be for the function?

c. How do the answers for part b relate to the lengths 7 and 19?Are the maximum and minimum values from part b ever actually attained in the geometric situation?



5 Chapter 5 Test, Form 1A

Write the letter for the correct answer in the blank at the right ofeach problem.

1. Change 128.433° to degrees, minutes, and seconds. 1. ________A. 128° 25′ 58″ B. 128° 25′ 59″ C. 128° 25′ 92″ D. 128° 26′ 00″

2. Write 43° 18′ 35″ as a decimal to the nearest thousandth of a degree. 2. ________A. 43.306° B. 43.308° C. 43.309° D. 43.310°

3. Give the angle measure represented by 3.25 rotations clockwise. 3. ________A. �1170° B. �90° C. 90° D. 1170°

4. Identify all coterminal angles between �360° and 360° 4. ________for the angle �420°.A. �60° and 300° B. �30° and 330°C. 30° and �330° D. 60° and �300°

5. Find the measure of the reference angle for 1046°. 5. ________A. �56° B. 56° C. 34° D. �34°

6. Find the value of the tangent for �A. 6. ________A. �2�

25�� B. ��2

5��

C. �23� D. ��35��

7. Find the value of the secant for �R. 7. ________A. ��5

7�0�� B. �3�14

1�4��

C. ��35�� D. ��3

1�4��

8. Which of the following is equal to csc θ ? 8. ________A. �sin

1θ� B. �co

1s θ� C. �ta

1n θ� D. �se

1c θ�

9. If cot θ � 0.85, find tan θ . 9. ________A. 0.588 B. 0.85 C. 1.176 D. 1.7

10. Find cos (�270°). 10. ________A. undefined B. �1 C. 1 D. 0

11. Find the exact value of sec 300°. 11. ________A. �2 B. ��2�

33�� C. 2 D. �2�

33��

12. Find the value of csc θ for angle θ in standard position if 12. ________the point at (5, �2) lies on its terminal side.A. ��2

2�9�� B. ��2�29

2�9�� C. ��52�9�� D. �5�

292�9��

13. Suppose θ is an angle in standard position whose terminal side 13. ________lies in Quadrant II. If sin θ � �11

23� , find the value of sec θ .

A. ��153� B. ��15

3� C. ��152� D. �1

123�


For Exercises 14 and 15, refer to the figure. The angle of elevation from the end of the shadow to the top of the building is 63° and the distance is 220 feet.

14. Find the height of the building to the nearest foot. 14. ________A. 100 ft B. 196 ftC. 432 ft D. 112 ft

15. Find the length of the shadow to the nearest foot. 15. ________A. 100 ft B. 196 ftC. 432 ft D. 112 ft

16. If 0° ≤ x ≤ 360°, solve the equation sec x � �2. 16. ________A. 150° and 210° B. 210° and 330°C. 120° and 240° D. 240° and 300°

17. Assuming an angle in Quadrant I, evaluate csc �cot�1 �43��. 17. ________

A. �35� B. �53� C. �45� D. �54�

18. Given the triangle at the right, find B to the 18. ________nearest tenth of a degree if b � 10 and c � 14.A. 44.4° B. 35.5°C. 54.5° D. 45.6°

For Exercises 19 and 20, round answers to the nearest tenth.19. In �ABC, A � 27° 35′, B � 78° 23′, and c � 19. Find a. 19. ________

A. 8.6 B. 9.2 C. 12.8 D. 19.4

20. If A � 42.2°, B � 13.6°, and a � 41.3, find the area of �ABC. 20. ________A. 138.8 units2 B. 493.8 units2 C. 327.4 units2 D. 246.9 units2

21. Determine the number of possible solutions if A � 62°, a � 4, 21. ________and b � 6.A. none B. one C. two D. three

22. Determine the greatest possible value for B if A � 30°, a � 5, 22. ________and b � 8.A. 23.1° B. 53.1° C. 126.9° D. 96.9°

For Exercises 23-25, round answers to the nearest tenth.23. In �ABC, A � 47°, b � 12, and c � 8. Find a. 23. ________

A. 6.3 B. 8.7 C. 8.8 D. 18.4

24. In �ABC, a � 7.8, b � 4.2, and c � 3.9. Find B. 24. ________A. 15.1° B. 148.7° C. 78.9° D. 16.2°

25. If a � 22, b � 14, and c � 30, find the area of �ABC. 25. ________A. 33 units2 B. 121.0 units2 C. 130.2 units2 D. 143.8 units2

Bonus The terminal side of an angle θ in standard position Bonus: ________coincides with the line 4x � y � 0 in Quadrant II. Find sec θto the nearest thousandth.

A. �0.243 B. �4.123 C. 0.243 D. 4.123

Chapter 5 Test, Form 1A (continued)


5


Chapter 5 Test, Form 1B

NAME _____________________________ DATE _______________ PERIOD ________


1. Change 110.23° to degrees, minutes, and seconds. 1. ________A. 110° 13′ 00″ B. 110° 13′ 8″ C. 110° 13′ 28″ D. 110° 13′ 48″

2. Write 24° 38′ 42″ as a decimal to the nearest thousandth of a degree. 2. ________A. 24.645° B. 24.646° C. 24.647° D. 24.648°

3. Give the angle measure represented by 1.75 rotations counterclockwise. 3. ________A. �630° B. �90° C. 90° D. 630°

4. Identify the coterminal angle between �360° and 360° for the angle �120°. 4. ________A. �240° B. 60° C. 240° D. 300°

5. Find the measure of the reference angle for 295°. 5. ________A. 25° B. �65° C. �25° D. 65°

6. Find the value of the cosine for �A. 6. ________

A. �12� B. ��23��

C. ��33�� D. 2

7. Find the value of the cosecant for �R. 7. ________

A. ��26�� B. ��3

6��

C. ��31�5�� D. ��2

1�0��

8. Which of the following is equal to sec �? 8. ________A. �sin

1�

� B. �co1s �� C. �ta

1n �� D. �se

1c ��

9. If tan � � 0.25, find cot �. 9. ________A. 0.25 B. 4 C. 0.5 D. 14

10. Find cot (�180°). 10. ________A. undefined B. �1 C. 1 D. 0

11. Find the exact value of tan 240°. 11. ________A. ��3� B. ��

33�� C. �3� D. ��

33��

12. Find the value of sec � for angle � in standard position if the 12. ________point at (�2, �4) lies on its terminal side.A. ��2

5�� B. �5� C. ��25�� D. ��5�

13. Suppose � is an angle in standard position whose terminal side 13. ________lies in Quadrant III. If sin � � ��11

23�, find the value of cot �.

A. ��153� B. ��15

3� C. �152� D. �1

123�

Chapter

5


For Exercises 14 and 15, refer to the figure. The angleof elevation from the end of the shadow to the top ofthe building is 56° and the distance is 120 feet.14. Find the height of the building to the nearest foot. 14. ________

A. 99 ft B. 67 ftC. 178 ft D. 81 ft


16. If 0° x 360°, solve the equation tan x � �1. 16. ________A. 135° and 315° B. 45° and 225° C. 45° and 315° D. 225° and 315°

17. Assuming an angle in Quadrant I, evaluate tan �cos�1 �45��. 17. ________

A. �34� B. �53� C. �45� D. �54�

18. Given the triangle at the right, find A to the nearest 18. ________tenth of a degree if b � 10 and c � 14.A. 44.4° B. 35.5°C. 54.5° D. 45.6°

For Exercises 19 and 20, round answers to the nearest tenth.19. In �ABC, A � 41° 15′, B � 107° 39′, and c � 19. Find b. 19. ________

A. 10.0 B. 24.3 C. 35.1 D. 54.6

20. If A � 52.6°, B � 49.8°, and a � 33.8, find the area of �ABC. 20. ________A. 117.9 units2 B. 338.2 units2 C. 536.4 units2 D. 1072.8 units2

21. Determine the number of possible solutions if A � 62°, a � 7, and b � 6. 21. ________A. none B. one C. two D. three

For Exercises 22–25, round answers to the nearest tenth.22. Determine the least possible value for B if A � 30°, a � 5, 22. ________

and b � 8.A. 23.1° B. 53.1° C. 126.9° D. 96.9°

23. In �ABC, B � 52°, a � 14, and c � 9. Find b. 23. ________A. 8.2 B. 11.0 C. 11.1 D. 18.4

24. In �ABC, a � 7.8, b � 4.2, and c � 3.9. Find A. 24. ________A. 15.1° B. 78.9° C. 148.7° D. 16.2°

25. If a � 32, b � 26, and c � 40, find the area of �ABC. 25. ________A. 49 units2 B. 121.0 units2 C. 298.6 units2 D. 415.2 units2

Bonus The terminal side of an angle � in standard position Bonus: ________coincides with the line 2x � y � 0 in Quadrant III.Find cos � to the nearest ten-thousandth.

A. �0.4472 B. 0.4472 C. �0.8944 D. 0.8944

Chapter 5 Test, Form 1B (continued)


5


Chapter 5 Test, Form 1C

NAME _____________________________ DATE _______________ PERIOD ________


1. Change 36.3� to degrees, minutes, and seconds. 1. ________A. 36� 18′ 00″ B. 36� 18′ 16″ C. 36� 18′ 24″ D. 36� 18′ 28″

2. Write 21� 44′ 3″ as a decimal to the nearest thousandth of a degree. 2. ________A. 21.731� B. 21.732� C. 21.733� D. 21.734�

3. Give the angle measure represented by 0.5 rotation clockwise. 3. ________A. �180� B. �90� C. 90� D. 180�

4. Identify the coterminal angle between 0� and 360� for the angle 480�. 4. ________A. 30� B. 60� C. 120� D. 240�

5. Find the measure of the reference angle for 235�. 5. ________A. �125� B. 55� C. 25� D. �55�

6. Find the value of the sine for �A. 6. ________A. ��1�

51�9�� B. ��1

1�21�9��

C. �152� D. �15

2�

7. Find the value of the cotangent for �R. 7. ________A. �43� B. �4

3�

C. �45� D. �45�

8. Which of the following is equal to cot �? 8. ________A. �sin

1�

� B. �co1s �� C. �se

1c �� D. �tan

1�

�

9. If cos � � 0.5, find sec �. 9. ________A. 0.25 B. 0.5 C. 1 D. 2

10. Find tan 180�. 10. ________A. undefined B. �1 C. 1 D. 0

11. Find the exact value of cos 135�. 11. ________A. �1 B. ��2

2�� C. 1 D. ��22��

12. Find the value of csc � for angle � in standard position if the point at 12. ________(3, �1) lies on its terminal side.A. ��1�0� B. ��1

1�00�� C. ��3

1�0�� D. �3

13. Suppose � is an angle in standard position whose terminal side lies 13. ________in Quadrant II. If cos � � ��11

23�, find the value of tan �.

A. ��152� B. ��15

3� C. ��152� D. �1

123�

Chapter

5


For Exercises 14 and 15, refer to the figure. The angle of elevationfrom the end of the shadow to the top of the building is 70� andthe distance is 180 feet.14. Find the height of the building to the nearest foot. 14. ________

A. 62 ft B. 66 ftC. 169 ft D. 495 ft


16. If 0� x 360�, solve the equation sin x � ��23��. 16. ________

A. 120� and 240� B. 240� and 300�C. 210� and 330� D. 150� and 210�

17. Assuming an angle in Quadrant I, evaluate cos �tan�1 �43��. 17. ________

A. �35� B. �53� C. �45� D. �45�

18. Given the triangle at the right, find B to the 18. ________nearest tenth of a degree if b � 8 and c � 12.A. 33.7� B. 41.8�C. 48.2� D. 56.3�

For Exercises 19 and 20, round answers to the nearest tenth.19. In �ABC, A � 102� 12�, B � 23� 21�, and c � 19.8. Find a. 19. ________

A. 8.0 B. 23.8 C. 48.8 D. 64.4

20. If A � 32.2�, b � 21.5, and c � 11.3, find the area of �ABC. 20. ________A. 129.5 units2 B. 102.8 units2 C. 64.7 units2 D. 32.6 units2

21. Determine the number of possible solutions if A � 48�, a � 5, and b � 6. 21. ________A. none B. one C. two D. three

22. Determine the least possible value for B if A � 20�, 22. ________a � 7, and b � 11.A. 12.6� B. 32.5� C. 147.5� D. 96.9�

For Exercises 23–25, round answers to the nearest tenth.23. In �ABC, A � 52�, b � 9, and c � 14. Find a. 23. ________

A. 6.3 B. 8.7 C. 8.8 D. 11.0

24. In �ABC, a � 2.4, b � 8.2, and c � 10.1. Find B. 24. ________A. 15.1� B. 21.7� C. 28.9� D. 33.3�

25. If a � 12, b � 30, and c � 22, find the area of �ABC. 25. ________A. 33.7 units2 B. 93.4 units2 C. 113.1 units2 D. 143.8 units2

Bonus The terminal side of an angle � in standard position Bonus: ________coincides with the line y � �12�x in Quadrant I. Find sin � to the nearest thousandth.

A. 0.233 B. 0.447 C. 0.508 D. 0.693

Chapter 5 Test, Form 1C (continued)


5


Chapter 5 Test, Form 2A

NAME _____________________________ DATE _______________ PERIOD ________

1. Change 225.639� to degrees, minutes, and seconds. 1. __________________

2. Write 23� 16′ 25″ as a decimal to the nearest thousandth of a degree. 2. ____________________

3. State the angle measure represented by 2.4 rotations clockwise. 3. ____________________

4. Identify all coterminal angles between �360� and 360� for the 4. ____________________angle �540�.

5. Find the measure of the reference angle for 562�. 5. ____________________

6. Find the value of the sine for �A. 6. ____________________

7. Find the value of the cotangent for �A. 7. ____________________

8. Find the value of the secant for �A. 8. ____________________

9. If csc � � �2, find sin �. 9. ____________________

10. Find sin (�270�). 10. ____________________

11. Find the exact value of cot 330�. 11. ____________________

12. Find the exact value of sec � for angle � in standard position if 12. ____________________the point at (�3, 2) lies on its terminal side.

13. Suppose � is an angle in standard position whose terminal side 13. ____________________lies in Quadrant IV. If cos � � �11

23�, find the value of csc �.

Chapter

5

For Exercises 6–8, refer to the figure.

Exercises 6–8


For Exercises 14 and 15, refer to the figure. The angle of elevationfrom the far side of the pool to the top of the waterfall is 75�, andthe distance is 185 feet.14. Find the height of the waterfall 14. __________________

to the nearest foot.

15. Find the width across the pool 15.to the nearest foot.

16. If 0� x 360�, solve cot x � ��3�. 16. __________________

17. Assuming an angle in Quadrant I, evaluate sec �tan�1 �34��. 17. __________________

18. Given triangle at the right, 18. __________________find B to the nearest tenth of a degree if a � 8 and b � 20.

For Exercises 19 and 20, round answers to the nearest tenth.19. In �ABC, A � 47� 15′, B � 58� 33′, and c � 23. Find a. 19. __________________

20. If A � 37.2�, B � 17.9�, and a � 22.3, find the area of �ABC. 20. __________________

21. Determine the number of possible solutions if A � 47�, 21. __________________a � 4, and b � 5.

22. Determine the least possible value for c if A � 30�, 22. __________________a � 5, and b � 8.

For Exercises 23-25, round answers to the nearest tenth.23. In �ABC, A � 118�, b � 8, and c � 6. Find a. 23. __________________

24. In �ABC, a � 9, b � 5, and c � 12. Find B. 24. __________________

25. If a � 12, b � 24, and c � 30, find the area of �ABC. 25. __________________

Bonus The terminal side of an angle � in standard Bonus: __________________position coincides with the line 3x � y � 0 in Quadrant II. Find csc � to the nearest thousandth.

Chapter 5 Test, Form 2A (continued)


5


Chapter 5 Test, Form 2B

NAME _____________________________ DATE _______________ PERIOD ________

1. Change 124.63° to degrees, minutes, and seconds. 1. __________________

2. Write 48° 32′ 15″ as a decimal to the nearest thousandth of 2. __________________a degree.

3. State the angle measure represented by 1.25 rotations 3. __________________clockwise.

4. Identify all coterminal angles between �360° and 360° for 4. __________________the angle 630°.

5. Find the measure of the reference angle for 310°. 5. __________________

6. Find the value of the cosine for �A. 6. __________________

7. Find the value of the cosecant for �A. 7. __________________

8. Find the value of the cotangent for �A. 8. __________________

9. If sec � � �4, find cos �. 9. __________________

10. Find tan (�180°). 10. __________________

11. Find the exact value of sec 240°. 11. __________________

12. Find the exact value of sec � for angle � in standard 12. __________________position if the point at (�4, 5) lies on its terminal side.

13. Suppose � is an angle in standard position whose terminal 13. __________________side lies in Quadrant IV. If cos � � �11

23�, find the value of cot �.

Chapter

5

Exercises 6–8

For Exercises 6–8, refer to the figure.


For Exercises 14 and 15, refer to the figure. The angle of elevationfrom the far side of the pool to the top of the waterfall is 54°, andthe distance is 230 feet.14. Find the height of the waterfall 14. __________________


15. Find the width across the pool 15. __________________to the nearest foot.

16. If 0° x 360°, solve the equation csc x � �2. 16. __________________

17. Assuming an angle in Quadrant I, evaluate cos �cot�1 �152��. 17. __________________

18. Given the triangle at the right, 18. __________________find B to the nearest tenth of a degree if a � 12 and c � 22.

For Exercises 19 and 20, round answers to the nearest tenth.19. In �ABC, A � 42°, B � 68°, and c � 15. Find a. 19. __________________

20. If A � 27.2°, B � 67.4°, and a � 12.8, find the area of �ABC. 20. __________________

21. Determine the number of possible solutions if A � 110°, 21. __________________a � 5, and b � 4.

For Exercises 22–25, round answers to the nearest tenth.22. Determine the greatest possible value for c if A � 30°, 22. __________________

a � 5, and b � 8.

23. In �ABC, A � 59°, b � 12, and c � 4. Find a. 23. __________________

24. In �ABC, a � 4, b � 11, and c � 8. Find B. 24. __________________

25. If a � 21, b � 15, and c � 28, find the area of � ABC. 25. __________________

Bonus The terminal side of an angle � in standard Bonus: _________________position coincides with the line 3x � y � 0 in Quadrant III. Find sin � to the nearest ten-thousandth.

Chapter 5 Test, Form 2B (continued)


5


Chapter 5 Test, Form 2C

NAME _____________________________ DATE _______________ PERIOD ________


2. Write 75� 30′ as a decimal to the nearest thousandth of a 2. __________________degree.

3. State the angle measure represented by 1.5 rotations 3. __________________counterclockwise.

4. Identify a coterminal angle between 0� and 360� for the 4. __________________angle �225�.

5. Find the measure of the reference angle for 235�. 5. __________________

6. Find the value of the sine for �A. 6. __________________


8. Find the value of the secant for �A. 8. __________________

9. If tan � � �3, find cot �. 9. __________________

10. Find tan 180�. 10. __________________

11. Find the exact value of cos 330�. 11. __________________

12. Find the exact value of sin � for angle � in standard position 12. __________________if the point at (�1, 4) lies on its terminal side.

13. Suppose � is an angle in standard position whose 13. __________________terminal side lies in Quadrant II. If sin � � �11

23�, find

the value of sec �.

Chapter

5

Exercises 6–8

For Excercises 6–8, refer to the figure.


For Exercises 14 and 15, refer to the figure. The angle of elevationfrom the far side of the pool to the top of the waterfall is 68� andthe distance is 200 feet.14. Find the height of the waterfall 14. __________________


15. Find the width across the pool 15.to the nearest foot.

16. If 0� x 360�, solve sin x � ��23��. 16. __________________

17. Assuming an angle in Quadrant I, evaluate cos �tan�1 �152��. 17. __________________

18. Given the triangle at the right, 18. __________________find B to the nearest tenth of a degree if b � 12 and c � 18.

For Exercises 19 and 20, round answers to the nearest tenth.19. In �ABC, A � 47�, B � 58�, and b � 23. Find a. 19. __________________

20. If C � 37.2�, a � 17.9, and b � 22.3, find the area of �ABC. 20. __________________

21. Determine the number of possible solutions if A � 47�, 21. __________________a � 2, and b � 4.

22. Determine the greatest possible value for c if A � 15�, 22. __________________a � 8, and b � 13.

For Exercises 23–25, round answers to the nearest tenth.23. In �ABC, A � 67�, b � 10, and c � 5. Find a. 23. __________________

24. In �ABC, a � 8, b � 6, and c � 12. Find C. 24. __________________

25. If a � 18, b � 22, and c � 30, find the area of �ABC. 25. __________________

Bonus The terminal side of an angle � in standard Bonus: __________________position coincides with the line y � x in Quadrant I. Find tan � to the nearest thousandth.

Chapter 5 Test, Form 2C (continued)


5


Chapter 5 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 answers. You may show your solution in more than one way or investigate beyond therequirements of the problem.

1. The point at (�3, ��3�) lies on the terminal side of an angle in standard position.

a. Give the degree measure of three angles that fit the description.

b. Tell how to find the cosine of such angles. Give the cosine of these angles.

c. Name angles in the first, second, and fourth quadrants that have the same reference angle as those above.

d. Write the coordinates of a point in Quadrant II. Find the values of the six trigonometric functions of an angle in standard position with this point on its terminal side.

2. A children’s play area is being built next to a circular fountain in the park. A fence will be erected around the play area for safety. A diagram of the area is shown below.

a. How long will the fence need to be in order to enclose the area?

b. The park commission is planning to enlarge the play area. Do you think it should be enlarged to the east or to the west? Why?

Chapter

5


Chapter 5 Mid-Chapter Test (Lessons 5-1 through 5-4)



2. If a �470� angle is in standard position, determine a 2. __________________coterminal angle that is between 0� and 360�. State the quadrant in which the terminal side lies.

For Exercises 3 and 4, use right triangle ABC to find each value.

3. Find the value of the cosine for �A. 3. __________________


5. If csc � � �3, find sin �. 5. __________________

6. Use the unit circle to find tan 180�. 6. __________________

7. Find the exact value of sin 300�. 7. __________________

8. Find the value of csc � for angle � in standard position if 8. __________________the point at (2, �1) lies on its terminal side.

For Exercises 9 and 10, use right triangle ABC to find each value to the nearest tenth.

9. Find b. 9. __________________

10. Find c. 10. __________________

1. Change 47.283� to degrees, minutes, and seconds. 1. __________________2. Write 122� 43′ 12″ as a decimal to the nearest thousandth 2. __________________

of a degree.3. Give the angle measure represented by 2.25 rotations 3. __________________

counterclockwise.4. Identify all coterminal angles between �360� and 360� 4. __________________

for the angle 480�.5. Find the measure of the reference angle for 323�. 5. __________________6. Find the value of the sine for �A. 6. __________________7. Find the value of the tangent for �A. 7. __________________8. Find the value of the secant for �A. 8. __________________

9. If cot � � ��23�, find tan �. 9. __________________

10. If sin � � 0.5, find csc �. 10. __________________

Use the unit circle to find each value. 1. __________________

1. sin (�90�) 2. csc 180� 2. __________________3. Find the exact value of cos 210�. 3. __________________4. Find the exact value of tan 135�. 4. __________________5. Find the value of sec � for angle � in standard position if 5. __________________

the point at (4, �5) lies on its terminal side.6. Suppose � is an angle in standard position whose terminal 6. __________________

side lies in Quadrant III. If tan � � �152�, find the value of sin �.

Refer to the figure. Find each value to the nearest tenth.

7. Find a. 7. __________________8. Find c. 8. __________________

A 100-foot cable is stretched from a stake in the ground to the topof a pole. The angle of elevation is 57°.

9. Find the height of the pole to the nearest tenth. 9. __________________10. Find the distance from the base of the pole to the 10. __________________

stake to the nearest tenth.

Chapter 5, Quiz B (Lessons 5-3 and 5-4)

NAME _____________________________ DATE _______________ PERIOD ________

Chapter 5, Quiz A (Lessons 5-1 and 5-2)

NAME _____________________________ DATE _______________ PERIOD ________


Chapter

5

Chapter

5

Exercises 6–8

1. If 0� x 360�, solve: csc x � �2. 1. __________________

2. Assuming an angle in Quadrant I, evaluate tan �sec�1 �153��. 2. __________________

3. Given right triangle ABC, find B to the 3. __________________nearest tenth of a degree if b � 7 and c � 12.

Find each value. Round to the nearest tenth.4. In �ABC, A � 58� 21�, C � 97� 07�, and b � 23.8. Find a. 4. __________________

5. If B � 29.5�, C � 64.5�, and a � 18.8, find the area of �ABC. 5. __________________

1. Determine the number of possible solutions for �ABC 1. __________________if A � 28�, a � 4, and b � 11.

Find each value. Round to the nearest tenth.

2. For �ABC, determine the least possible value for B 2. __________________if A � 49�, a � 12, and b � 15.

3. In �ABC, B � 32�, a � 11, and c � 2.4. Find b. 3. __________________

4. In �ABC, a � 3.1, b � 5.4, and c � 4.7. Find C. 4. __________________

5. If a � 28.2, b � 36.5, and c � 40.1, find the area of �ABC. 5. __________________

Chapter 5, Quiz D (Lessons 5-7 and 5-8)

NAME _____________________________ DATE _______________ PERIOD ________

Chapter 5, Quiz C (Lessons 5-5 and 5-6)

NAME _____________________________ DATE _______________ PERIOD ________


Chapter

5

Chapter

5


Chapter 5 SAT and ACT Practice

NAME _____________________________ DATE _______________ PERIOD ________

After working each problem, record thecorrect answer on the answer sheetprovided or use your own paper.

Multiple Choice1. The length of a diagonal of a square

is 6 units. Find the length of a side.A 9�2� unitsB �2� unitsC 9 unitsD 3�2� unitsE None of these

2. If the area of �ABC is 30 square meters,what is the length of segment CD?A 2 mB 3 m C 5 mD 8 mE 10 m

3. The midpoint of a diameter of a circle is(3, 4). If the coordinates of one endpointof the diameter are (�3, 6), what arethe coordinates of the other endpoint?A (9, 2)B (9, 1) C (8, 2)D (3, 2)E (9, 14)

4. The endpoints of a diameter of a circleare (3, 2) and (11, 8). Find the area ofthe circle.A 5 units2

B 25 units2

C 25� units2

D 10� units2

E 5� units2

5. What is the arithmetic mean of �13� and �14�?A �16� B �7

1�

C �112� D �1

72�

E �274�

6. Which of the following products has thegreatest value less than 100?

A 2 � 4 � 6B 2 � 4 � 9C 4 � 4 � 9D 3 � 3 � 9E 4 � 4 � 6

7. A diagonal of a rectangle is �1�5�inches. The length of the rectangle is�1�2� inches. Find the area of the rectangle.A 3�2� in2

B 6 in2

C 9 in2

D 6�5� in2

E None of these

8. The diagonals of a rhombus are perpendicular and bisect each other.If the length of one side of a rhombus is 25 meters and the length of onediagonal is 14 meters, find the lengthof the other diagonal.A 7 mB 12 mC 24 mD 48 mE 144 m

9. If c � �a1b�, what is the value of a when

c � 12�1 and b � 3?A �36B ��3

16�

C �4D 4E None of these

10. How many times do the graphs of y � x2 and xy � 27 intersect?A 0B 1C 2D 3E 4

Chapter

5


Chapter 5 SAT and ACT Practice (continued)


5

7 and 12 8 and 11

11. If �143x� is an integer, then the value of x

CANNOT be which of the following?A �112B �32C 0D 6E 8

12. Which of the following statementsmakes the expression a � b representa negative number?A b � aB a � bC b � 0D a � 0E b � a

13. A hiker travels 5 miles due north, then3 miles due west, and then 2 miles duenorth. How far is the hiker from hisbeginning point?A About 8.2 miB About 7.1 miC About 9.4 miD About 6 miE About 7.6 mi

14. A rectangular swimming pool is 100 meters long and 25 meters wide.Lucia swims from one corner of thepool to the opposite corner and back10 times. How far did she swim?A About 193.6 mB About 1030.8 mC About 2061.6 mD About 10,308.8 mE None of these

15. 4 � 3�2� is a root of which equation?A x2 � 8x � 2 � 0B x2 � 8x � 18 � 0C x2 � 16x � 2 � 0D x2 � 16x � 18 � 0E None of these

16. If ƒ(x) � x4 � 4x3 � 16x � 16 has azero of �2, with a multiplicity of 3,what is another zero of ƒ?A 3B 2C �1D 1E 8

17–18. Quantitative ComparisonA if the quantity of 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 informationgiven

Column A Column B17. The value of x in each figure

18. The length of the hypotenuse of aright triangle with the given leglengths

19. Grid-In A tent with a rectangularfloor has a diagonal length of 7 feetand a width of 5 feet. What is the areaof the floor to the nearest square foot?

20. Grid-In Julio wants to bury a waterpipe from one corner of his field to theopposite corner. How many feet ofpipe, to the nearest foot, does he needif the rectangular field is 200 feet by300 feet?


Chapter 5, Cumulative Review (Chapters 1-5)

NAME _____________________________ DATE _______________ PERIOD ________

1. If ƒ(x) � x � 2 and g(x) � �21x� , find g( ƒ(x)). 1. __________________

2. Determine whether the graphs of 2x � y � 5 � 0 2. __________________and x � 2y � 6 are parallel, coinciding, perpendicular, or none of these.

3. Graph the inequality 2x � 4y 8. 3.

4. Solve the following system of equations algebraically. 4. __________________5x � y � 162x � 3y � 3

5. Find BC if B � � � and C � � �. 5. __________________

6. Find the inverse of � �, if it exists. 6. __________________

7. Determine whether the graph of y � 2x4 � x2 � 3 is 7. __________________symmetric to the x-axis, the y-axis, both, or neither.

8. Describe the transformations relating the graph of 8. __________________y � ��x � 2� � 5 to its parent function, y � �x �.

9. Find the inverse of y � 2x � 6. State whether the inverse 9. __________________is a function.

10. Solve x2 � 6x � 6 � 0. 10. __________________

11. Solve �3

x� �� 2� � 7 � 4. 11. __________________

12. Find csc (�90°). 12. __________________

13. Given the right triangle ABC, 13.find side c to the nearest tenth.

14. In �ABC, a � 8, b � 6, and c � 10. Find B to the 14. __________________nearest tenth.

�13

24

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61

Chapter

5

© 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)

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th

e co

ncl

usi

on a

nd

q be

com

es t

he

hyp

oth

esis

, th

e n

ewst

atem

ent,

q →

p, is

cal

led

the

con

vers

eof

p →

q.

If p

→q

is t

rue,

q →

pm

ay b

e ei

ther

tru

e or

fal

se.

Exa

mp

leF

ind

th

e co

nve

rse

of e

ach

con

dit

ion

al.

a.p

→q

: All

sq

uar

es a

re r

ecta

ngl

es.

(tru

e)q

→p:

All

rec

tan

gles

are

squ

ares

.(f

alse

)

b.

p→

q: I

f a

fun

ctio

n ƒ

(x)

is i

ncr

easi

ng

on a

nin

terv

al I

, th

en f

or e

very

aan

d b

con

tain

ed i

nI,

ƒ(a

)�

ƒ(b)

wh

enev

er a

�b.

(tru

e)q

→p:

If

for

ever

y a

and

bco

nta

ined

in a

n in

terv

al I

,ƒ(

a)�

ƒ(b)

wh

enev

era

�b

then

fu

nct

ion

ƒ(x

) is

incr

easi

ng

on I

.(t

rue)

In L

esso

n 3

-5, y

ou s

aw t

hat

th

e tw

o st

atem

ents

in E

xam

ple

2 ca

n b

eco

mbi

ned

in a

sin

gle

stat

emen

t u

sin

g th

e w

ords

“if

an

d on

ly if

.”

Afu

nct

ion

ƒ(x

) is

incr

easi

ng

on a

n in

terv

al I

if

and

on

ly i

ffo

rev

ery

aan

d b

con

tain

ed in

I, ƒ

(a)�

ƒ(b)

wh

enev

era

�b.

Th

e st

atem

ent

“p if

an

d on

ly if

q”

mea

ns

that

p im

plie

s q

and

q im

plie

s p.

Sta

te t

he

con

vers

e of

eac

h c

ond

itio

nal

. T

hen

tel

l if

the

con

vers

e is

tru

e or

fal

se.

If it

is t

rue,

com

bin

e th

e st

atem

ent

and

its

con

vers

ein

to a

sin

gle

sta

tem

ent

usi

ng

th

e w

ord

s “i

f an

d o

nly

if.”

1.A

ll in

tege

rs a

re r

atio

nal

nu

mbe

rs.

All

rati

ona

l num

ber

s ar

e in

teg

ers;

fal

se2.

If f

or a

ll x

in t

he

dom

ain

of

a fu

nct

ion

ƒ(x

), ƒ

(�x)

��

ƒ(x)

, th

enth

e gr

aph

of

ƒ(x)

is s

ymm

etri

c w

ith

res

pect

to

the

orig

in.

If a

fun

ctio

n ha

s a

gra

ph

that

is s

ymm

etri

c w

ith

resp

ect

toth

e o

rig

in, t

hen

ƒ(�

x)�

�ƒ

(x) f

or

all x

in t

he d

om

ain

of

ƒ(x

);tr

ue. A

fun

ctio

n ha

s a

gra

ph

that

is s

ymm

etri

c w

ith

resp

ect

to t

he o

rig

in if

and

onl

y if

ƒ(�

x)�

�ƒ

(x) f

or

all x

in t

hed

om

ain

of

ƒ(x

).3.

If ƒ

(x)

and

ƒ�1 (

x) a

re in

vers

e fu

nct

ion

s, t

hen

[ƒ

°ƒ�

1 ](x

)�[ƒ

�1

°ƒ

](x)

�x.

If [

ƒ °

ƒ�

1 ](x

)�[ƒ

�1

°ƒ

](x)

�x,

the

n ƒ

(x) a

nd ƒ

�1 (

x) a

re in

vers

e fu

ncti

ons

; tru

e. T

wo

fun

ctio

ns, ƒ

(x) a

nd ƒ

�1 (

x), a

re in

vers

e fu

ncti

ons

if a

nd o

nly

if [ƒ

°ƒ

�1 ]

(x)�

[ƒ�

1°

ƒ](

x)�

x.

© G

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cGra

w-H

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3A

dva

nced

Mat

hem

atic

al C

once

pts

Enr

ichm

ent

NA

ME

____

____

____

____

____

____

____

_ D

ATE

____

____

____

___

PE

RIO

D__

____

__

5-1



© G

lenc

oe/M

cGra

w-H

ill18

5A

dva

nced

Mat

hem

atic

al C

once

pts

Pra

ctic

eN

AM

E__

____

____

____

____

____

____

___

DAT

E__

____

____

____

_ P

ER

IOD

____

____

Trig

on

om

etr

ic R

atio

s in

Rig

ht

Tria

ng

les

Fin

d t

he

valu

es o

f th

e si

ne,

cos

ine,

an

d t

ang

ent

for

each

�B

.

1.2.

sin

B�

�3 8� ; co

s B

��

85�5� �;

sin

B�

�2�5

5��

; co

s B

�� 55� �

;

tan

B�

�3�55

5�5� �ta

n B

�2

3.If

tan

��

5, f

ind

cot

�.

4.If

sin

��

�3 8� , f

ind

csc

�.

�1 5��8 3�

Fin

d t

he

valu

es o

f th

e si

x tr

igon

omet

ric

rati

os f

or e

ach

�S

.

5.6.

sin

S�

�3�10

1�0� �; c

os

S�

�� 11� 00 ��

;si

n S

�� 17 9�

; co

s S

��2�

197�8� �

;

tan

S�

3; c

sc S

��

31�0� �;

tan

S�

�7 1� 57� 68 ��

; csc

S�

�1 79 �;

sec

S�

�1 �0�

; co

t S

��1 3�

sec

S�

�191� 56

7�8��

; co

t S

��2�

77�8� �

7.P

hys

ics

Su

ppos

e yo

u a

re t

rave

lin

g in

a c

ar w

hen

a b

eam

of

ligh

tpa

sses

fro

m t

he

air

to t

he

win

dsh

ield

. Th

e m

easu

re o

f th

e an

gle

of in

cide

nce

is 5

5�, a

nd

the

mea

sure

of

the

angl

e of

ref

ract

ion

is

35�

15′.

Use

Sn

ell’s

Law

, �ss ii nn

�� ri�

�n

,to

fin

d th

e in

dex

of r

efra

ctio

n n

of t

he

win

dsh

ield

to

the

nea

rest

th

ousa

ndt

h.

abo

ut 1

.419

5-2

© G

lenc

oe/M

cGra

w-H

ill18

6A

dva

nced

Mat

hem

atic

al C

once

pts

Enr

ichm

ent

NA

ME

____

____

____

____

____

____

____

_ D

ATE

____

____

____

___

PE

RIO

D__

____

__

5-2

Usi

ng

Rig

ht

Tria

ng

les

to F

ind

th

e A

rea

of

An

oth

er

Tria

ng

leYo

u c

an f

ind

the

area

of

a ri

ght

tria

ngl

e by

usi

ng

the

form

ula

A�

bh. I

n t

he

form

ula

, on

e le

g of

th

e ri

ght

tria

ngl

e ca

n b

e u

sed

as

the

base

, an

d th

e ot

her

leg

can

be

use

d as

th

e h

eigh

t.

Th

e ve

rtic

es o

f a

tria

ngl

e ca

n b

e re

pres

ente

d on

th

e co

ordi

nat

e pl

ane

by t

hree

ord

ered

pai

rs. I

n or

der

to fi

nd t

he a

rea

of a

gen

eral

tri

angl

e,yo

u c

an e

nca

seth

e tr

ian

gle

in a

rec

tan

gle

as s

how

n in

th

e di

agra

mbe

low

.

Are

ctan

gle

is p

lace

d ar

oun

d th

e tr

ian

gle

so t

hat

th

e ve

rtic

es o

f th

etr

ian

gle

all t

ouch

th

e si

des

of t

he

rect

angl

e.E

xam

ple

Fin

d t

he

area

of

a tr

ian

gle

wh

ose

vert

ices

are

A

(�1,

3),

B(4

, 8),

an

d C

(8, 5

).P

lot

the

poin

ts a

nd d

raw

the

tri

angl

e. E

ncas

e th

e tr

iang

lein

a r

ecta

ngl

e w

hos

e si

des

are

para

llel

to

the

axes

, th

en f

ind

the

coor

din

ates

of

the

vert

ices

of

the

rect

angl

e.

Are

a�

AB

C�

area

AD

EF

�ar

ea�

AD

B�

area

�B

EC

�ar

ea �

CFA

, wh

ere

�A

DB

,�

BE

C,

and

�C

FAar

e al

l rig

ht

tria

ngl

es.

Are

a�

AB

C�

5(9)

�(5

)(5)

�(4

)(3)

�(2

)(9)

�17

.5 s

quar

e u

nit

s

Fin

d t

he

area

of

the

tria

ng

le h

avin

g v

erti

ces

wit

h e

ach

set

of

coor

din

ates

.1.

A(4

, 6),

B(–

1, 2

), C

(6, –

5)31

.52.

A(–

2, –

4), B

(4, 7

), C

(6, –

1)35

3.A

(4, 2

), B

(6, 9

), C

(–1,

4)

19.5

4.A

(2, –

3), B

(6, –

8), C

(3, 5

)18

.5

1 � 21 � 2

1 � 2

1 � 2



© G

lenc

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w-H

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9A

dva

nced

Mat

hem

atic

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once

pts

Enr

ichm

ent

NA

ME

____

____

____

____

____

____

____

_ D

ATE

____

____

____

___

PE

RIO

D__

____

__

5-3

Are

as

of

Po

lyg

on

s a

nd

Cir

cle

sA

regu

lar

poly

gon

has

sid

es o

f eq

ual

len

gth

an

d an

gles

of

equ

al

mea

sure

. A

regu

lar

poly

gon

can

be

insc

ribe

d in

or

circ

um

scri

bed

abou

t a

circ

le.

For

n-s

ided

reg

ula

r po

lygo

ns,

th

e fo

llow

ing

area

fo

rmu

las

can

be

use

d.

Are

a of

cir

cle

AC

��

r2

Are

a of

insc

ribe

d po

lygo

nA

I�

�si

n

Are

a of

cir

cum

scri

bed

poly

gon

AC

�n

r2�

tan

Use

a c

alcu

lato

r to

com

ple

te t

he

char

t b

elow

for

a u

nit

cir

cle

(a c

ircl

e of

rad

ius

1).

1. 2. 3. 4. 5. 6. 7. 8. 9.W

hat

nu

mbe

r do

th

e ar

eas

of t

he

circ

um

scri

bed

and

insc

ribe

d po

lygo

ns

seem

to

be a

ppro

ach

ing

as t

he

nu

mbe

r of

sid

es o

f th

e po

lygo

n in

crea

ses?

�

180°

�n

360°

�n

nr2

�2

Num

ber

Are

a o

fA

rea

of

Cir

cle

Are

a o

fA

rea

of

Po

lyg

on

of

Sid

esIn

scri

bed

less

Cir

cum

scri

bed

less

Po

lyg

on

Are

a o

f P

oly

go

nP

oly

go

nA

rea

of

Cir

cle

31.

2990

381

1.84

2554

55.

1961

524

2.05

4559

8

42

1.14

1592

74

0.85

8407

3

82.

8284

271

0.31

3165

53.

3137

085

0.17

2115

8

123

0.14

1592

73.

2153

903

0.07

3797

7

203.

0901

699

0.05

1422

73.

1676

888

0.02

6096

2

243.

1058

285

0.03

5764

13.

1596

599

0.01

8067

3

283.

1152

931

0.02

6299

63.

1548

423

0.01

3249

7

323.

1214

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0.02

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0.01

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1000

3.14

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00.

0000

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0000

103

© G

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Mat

hem

atic

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once

pts

Trig

on

om

etr

ic F

un

ctio

ns

on

th

e U

nit C

irc

le

Use

th

e u

nit

cir

cle

to f

ind

eac

h v

alu

e.1.

csc

90�

2.ta

n 2

70�

3.si

n (

�90

�)1

und

efin

ed�

1

Use

th

e u

nit

cir

cle

to f

ind

th

e va

lues

of

the

six

trig

onom

etri

cfu

nct

ion

s fo

r ea

ch a

ng

le.

4.45

�

sin

45�

�� 22� �

csc

45�

��

2�

cos

45�

�� 22� �

sec

45�

��

2�

tan

45�

�1

cot

45�

�1

5.12

0�si

n 12

0��

�� 23� �cs

c 12

0��

�2�3

3��

cos

120�

��

�1 2�se

c 12

0��

�2

tan

120�

��

�3�

cot

120�

��

�� 33� �

Fin

d t

he

valu

es o

f th

e si

x tr

igon

omet

ric

fun

ctio

ns

for

ang

le �

inst

and

ard

pos

itio

n if

a p

oin

t w

ith

th

e g

iven

coo

rdin

ates

lies

on

its

term

inal

sid

e.6.

(�1,

5)

7.(7

, 0)

8.(�

3,�

4)

sin

��

�5�26

2�6� �si

n �

�0

sin

��

��4 5�

cos

��

�� 22� 66 �

�co

s �

�1

cos

��

��3 5�

tan

��

�5

tan

��

0ta

n �

�� 34 �

csc

��

��52�6� �

csc

��

und

efin

ed

csc

��

��5 4�

sec

��

��

2�6�se

c �

�1

sec

��

��5 3�

cot

��

��1 5�

cot

��

und

efin

edco

t �

�� 43 �

Pra

ctic

eN

AM

E__

____

____

____

____

____

____

___

DAT

E__

____

____

____

_ P

ER

IOD

____

____

5-3



© G

lenc

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cGra

w-H

ill19

1A

dva

nced

Mat

hem

atic

al C

once

pts

Pra

ctic

eN

AM

E__

____

____

____

____

____

____

___

DAT

E__

____

____

____

_ P

ER

IOD

____

____

Ap

ply

ing

Tri

go

no

me

tric

Fu

nc

tio

ns

Sol

ve e

ach

pro

ble

m. R

oun

d t

o th

e n

eare

st t

enth

.1.

If A

�55

�55

′an

d c

�16

, fin

d a.

13.3

2.If

a�

9 an

d B

�49

�, f

ind

b.10

.4

3.If

B�

56�

48′a

nd

c�

63.1

, fin

d b.

52.8

4.If

B�

64�

and

b�

19.2

, fin

d a.

9.4

5.If

b�

14 a

nd

A�

16�,

fin

d c.

14.6

6.C

onst

ruct

ion

A30

-foo

t la

dder

lean

ing

agai

nst

th

e si

de o

f a

hou

se m

akes

a 7

0�5′

angl

e w

ith

th

e gr

oun

d.a.

How

far

up

the

side

of

the

hou

se d

oes

the

ladd

er r

each

?ab

out

28.

2 ft

b.

Wh

at is

th

e h

oriz

onta

l dis

tan

ce b

etw

een

th

ebo

ttom

of

the

ladd

er a

nd

the

hou

se?

abo

ut 1

0.2

ft

7.G

eom

etry

Aci

rcle

is c

ircu

msc

ribe

d ab

out

a re

gula

r h

exag

on w

ith

an

ap

oth

em o

f 4.

8 ce

nti

met

ers.

a.F

ind

the

radi

us

of t

he

circ

um

scri

bed

circ

le.

abo

ut 5

.5 c

m

b.

Wh

at is

th

e le

ngt

h o

f a

side

of

the

hex

agon

?ab

out

5.5

cm

c.W

hat

is t

he

peri

met

er o

f th

e h

exag

on?

abo

ut 3

3 cm

8.O

bser

vati

onA

pers

on s

tan

din

g 10

0 fe

et f

rom

th

e bo

ttom

of a

cli

ff n

otic

es a

tow

er o

n t

op o

f th

e cl

iff.

Th

e an

gle

of

elev

atio

n t

o th

e to

p of

th

e cl

iff

is 3

0�. T

he

angl

e of

ele

vati

onto

th

e to

p of

th

e to

wer

is 5

8�. H

ow t

all i

s th

e to

wer

?ab

out

102

.3 f

t

5-4

© G

lenc

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2A

dva

nced

Mat

hem

atic

al C

once

pts

Enr

ichm

ent

NA

ME

____

____

____

____

____

____

____

_ D

ATE

____

____

____

___

PE

RIO

D__

____

__

5-4

Ma

kin

g a

nd

Usi

ng

a H

yp

som

ete

rA

hyp

som

eter

is a

dev

ice

that

can

be

use

d to

mea

sure

th

e h

eigh

t of

an o

bjec

t. T

o co

nst

ruct

you

r ow

n h

ypso

met

er, y

ou w

ill n

eed

a re

ctan

gula

r pi

ece

of h

eavy

car

dboa

rd t

hat

is a

t le

ast

7 cm

by

10 c

m,

a st

raw

, tra

nsp

aren

t ta

pe, a

str

ing

abou

t 20

cm

lon

g, a

nd

a sm

all

wei

ght

that

can

be

atta

ched

to

the

stri

ng.

Mar

k of

f 1-

cm in

crem

ents

alo

ng

one

shor

t si

de a

nd

one

lon

g si

de o

fth

e ca

rdbo

ard.

Tap

e th

e st

raw

to

the

oth

er s

hor

t si

de. T

hen

att

ach

the

wei

ght

to o

ne

end

of t

he

stri

ng,

an

d at

tach

th

e ot

her

en

d of

th

est

rin

g to

on

e co

rner

of

the

card

boar

d, a

s sh

own

in t

he

figu

re b

elow

.T

he

diag

ram

bel

ow s

how

s h

ow y

our

hyp

som

eter

sh

ould

look

.

To u

se t

he

hyp

som

eter

, you

wil

l nee

d to

mea

sure

th

e di

stan

ce f

rom

the

base

of

the

obje

ct w

hos

e h

eigh

t yo

u a

re f

indi

ng

to w

her

e yo

ust

and

wh

en y

ou u

se t

he

hyp

som

eter

.

Sig

ht

the

top

of t

he

obje

ct t

hro

ugh

th

e st

raw

. Not

e w

her

e th

e fr

ee-

han

gin

g st

rin

g cr

osse

s th

e bo

ttom

sca

le. T

hen

use

sim

ilar

tri

angl

esto

fin

d th

e h

eigh

t of

th

e ob

ject

.

1.D

raw

a d

iagr

am t

o il

lust

rate

how

you

can

use

sim

ilar

tri

angl

esan

d th

e h

ypso

met

er t

o fi

nd

the

hei

ght

of a

tal

l obj

ect.

See

stu

den

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iag

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s.U

se y

our

hyp

som

eter

to

fin

d t

he

hei

gh

t of

eac

h o

f th

e fo

llow

ing

.

2.yo

ur

sch

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fla

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a tr

ee o

n y

our

sch

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pro

pert

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the

hig

hes

t po

int

on t

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fron

t w

all o

f yo

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sch

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uil

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g5.

the

goal

pos

ts o

n a

foo

tbal

l fie

ld6.

the

hoo

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the

top

of t

he

hig

hes

t w

indo

w o

f yo

ur

sch

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uil

din

g8.

the

top

of a

sch

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us

9.th

e to

p of

a s

et o

f bl

each

ers

at y

our

sch

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10.t

he

top

of a

uti

lity

pol

e n

ear

you

r sc

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Dis

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sin

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eth

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roba

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foll

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bas

ed o

n t

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nst

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allo

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be

divi

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into

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y de

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d n

um

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ofco

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uen

tse

gmen

ts.

You

can

use

inve

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trig

onom

etri

c fu

nct

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s to

sh

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hat

appl

icat

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of

the

met

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f an

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ot v

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can

be

tris

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eth

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oose

poi

nt

Con

on

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�A

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th

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ray

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it a

t B

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. D

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A �M�

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into

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3.

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d �

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func

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no

t lin

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rat

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two

ang

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pool

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th

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Tria

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Sp

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ters

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an

ceT

he

McD

ouga

lls

plan

to

fen

ce a

tri

angu

lar

parc

el o

f th

eir

lan

d. O

ne

side

of

the

prop

erty

is 7

5 fe

et in

len

gth

. It

form

s a

38�

angl

e w

ith

an

oth

ersi

de o

f th

e pr

oper

ty, w

hic

h h

as n

ot y

et b

een

mea

sure

d. T

he

rem

ain

ing

side

of

the

prop

erty

is 9

5 fe

et in

len

gth

. App

roxi

mat

e to

th

e n

eare

st t

enth

th

e le

ngt

h o

f fe

nce

nee

ded

to e

ncl

ose

this

par

cel o

f th

e M

cDou

gall

s’lo

t.

abo

ut 3

12.1

ft

Pra

ctic

eN

AM

E__

____

____

____

____

____

____

___

DAT

E__

____

____

____

_ P

ER

IOD

____

____

5-7



© G

lenc

oe/M

cGra

w-H

ill20

4A

dva

nced

Mat

hem

atic

al C

once

pts

Enr

ichm

ent

NA

ME

____

____

____

____

____

____

____

_ D

ATE

____

____

____

___

PE

RIO

D__

____

__

5-8

The

La

w o

f C

osi

ne

s a

nd

th

e P

yth

ag

ore

an

Th

eo

rem

Th

e la

w o

f co

sin

es b

ears

str

ong

sim

ilar

itie

s to

th

eP

yth

agor

ean

Th

eore

m.

Acc

ordi

ng

to t

he

Law

of

Cos

ines

, if t

wo

side

s of

a t

rian

gle

have

leng

ths

a an

db

and

if t

he a

ngle

bet

wee

n th

em h

as a

mea

sure

of x

,th

en t

he

len

gth

, y, o

f th

e th

ird

side

of

the

tria

ngl

eca

n b

e fo

un

d by

usi

ng

the

equ

atio

n

y2�

a2�

b2�

2ab

cos

(x°)

.

An

swer

th

e fo

llow

ing

qu

esti

ons

to c

lari

fy t

he

rela

tion

ship

bet

wee

n

the

Law

of

Cos

ines

an

d t

he

Pyt

hag

orea

n T

heo

rem

.

1.If

th

e va

lue

of x

beco

mes

less

an

d le

ss, w

hat

nu

mbe

r is

cos

(x°

)cl

ose

to?

12.

If t

he

valu

e of

xis

ver

y cl

ose

to z

ero

but

then

incr

ease

s, w

hat

hap

pen

s to

cos

(x°

) as

xap

proa

ches

90?

dec

reas

es, a

pp

roac

hes

03.

If x

equ

als

90, w

hat

is t

he

valu

e of

cos

(x°

)? W

hat

doe

s th

eeq

uat

ion

of

y2�

a2�

b2�

2ab

cos

(x°)

sim

plif

y to

if x

equ

als

90?

0, y

2�

a2

�b

2

4.W

hat

hap

pen

s to

th

e va

lue

of c

os (

x°)

as x

incr

ease

s be

yon

d 90

and

appr

oach

es 1

80?

dec

reas

es t

o –

15.

Con

side

r so

me

part

icu

lar

valu

es o

f a

and

b, s

ay 7

for

aan

d19

for

b.

Use

a g

raph

ing

calc

ula

tor

to g

raph

th

e eq

uat

ion

you

get

by s

olvi

ng

y2

�72

�19

2�

2(7)

(19)

cos

(x°

) fo

r y.

See

stu

den

ts’ g

rap

hs.

a.In

vie

w o

f the

geo

met

ry o

f the

sit

uati

on, w

hat

rang

e of

val

ues

shou

ld y

ou u

se f

or X

on

a g

raph

ing

calc

ula

tor?

Xm

in �

0; X

max

�18

0b

.D

ispl

ay t

he g

raph

and

use

the

TR

AC

E fu

ncti

on.

Wha

t do

the

max

imum

and

min

imum

val

ues

appe

ar t

o be

for

the

func

tion

?S

ee s

tud

ents

’ gra

phs

; Y

min

�12

, Ym

ax�

26c.

How

do

the

answ

ers

for

part

bre

late

to

the

len

gth

s 7

and

19?

Are

th

e m

axim

um

an

d m

inim

um

val

ues

fro

m p

art

bev

er

actu

ally

att

ain

ed in

th

e ge

omet

ric

situ

atio

n?

min

�(1

9�

7); m

ax �

(19

�7)

; no

© G

lenc

oe/M

cGra

w-H

ill20

3A

dva

nced

Mat

hem

atic

al C

once

pts

Pra

ctic

eN

AM

E__

____

____

____

____

____

____

___

DAT

E__

____

____

____

_ P

ER

IOD

____

____

Th

e L

aw

of

Co

sin

es

Sol

ve e

ach

tri

ang

le. R

oun

d t

o th

e n

eare

st t

enth

.

1.a

�20

, b�

12, c

�28

2.a

�10

, c�

8, B

�10

0�A

�38

.2�,

B�

21.8

�,

b�

13.8

, A�

45.5

�, C

�34

.5�

C�

120.

0�

3.c

�49

, b�

40, A

�53

�4.

a�

5, b

�7,

c�

10a

�40

.5, B

�52

.0�,

A

�27

.7�,

B�

40.5

�, C

�11

1.8�

C�

75.0

�

Fin

d t

he

area

of

each

tri

ang

le. R

oun

d t

o th

e n

eare

st t

enth

.

5.a

�5,

b�

12, c

�13

6.a

�11

, b�

13, c

�16

30.0

uni

ts2

71.0

uni

ts2

7.a

�14

,b�

9, c

�8

8.a

�8,

b�

7, c

�3

33.7

uni

ts2

10.4

uni

ts2

9.T

he

side

s of

a t

rian

gle

mea

sure

13.

4 ce

nti

met

ers,

18

.7 c

enti

met

ers,

an

d 26

.5 c

enti

met

ers.

Fin

d th

e m

easu

reof

th

e an

gle

wit

h t

he

leas

t m

easu

re.

abo

ut 2

8.3�

10.O

rien

teer

ing

Du

rin

g an

ori

ente

erin

g h

ike,

tw

o h

iker

s st

art

at p

oin

t A

and

hea

d in

a d

irec

tion

30�

wes

t of

sou

th

to p

oin

t B

.Th

ey h

ike

6 m

iles

fro

m p

oin

t A

to p

oin

t B

. Fro

m

poin

t B

,th

ey h

ike

to p

oin

t C

and

then

fro

m p

oin

t C

back

to

poin

t A

,wh

ich

is 8

mil

es d

irec

tly

nor

th o

f po

int

C.H

ow

man

y m

iles

did

th

ey h

ike

from

poi

nt

Bto

poi

nt

C?

4.1

mi

5-8


Page 205

1. B

2. D

3. A

4. A

5. C

6. B

7. D

8. A

9. C

10. D

11. C

12. A

13. B

Page 206

14. B

15. A

16. C

17. B

18. D

19. B

20. D

21. A

22. C

23. C

24. D

25. D

Bonus: B

Page 207

1. D

2. A

3. D

4. C

5. D

6. B

7. D

8. B

9. B

10. A

11. C

12. D

13. C

Page 208

14. A

15. B

16. A

17. A

18. A

19. C

20. C

21. B

22. B

23. B

24. C

25. D

Bonus: A

Chapter 5 Answer KeyForm 1A Form 1B


Chapter 5 Answer Key

Page 209

1. A

2. B

3. A

4. C

5. B

6. B

7. A

8. D

9. D

10. D

11. B

12. A

13. A

Page 210

14. C

15. A

16. B

17. A

18. B

19. B

20. C

21. C

22. B

23. D

24. D

25. C

Bonus: B

Page 211

1. 225° 38� 20.4

2. 23.274°

3. �864°

4.�180° and 180°

5. 22°

6. �41�16��

7. �52�

46��

8. �151�

9. � �12

�

10. 1

11. ��3�

12. ��31�3��

13. ��153�

Page 212

14. 179 feet

15. 48 feet

16. 150° and 330°

17. �45�

18. 68.2°

19. 17.6

20. 103.7 units2

21. two

22. 3.9

23. 12.0

24. 22.2°

25. 136.8 units2

Bonus: 1.054

Form 1C Form 2A


Chapter 5 Answer KeyForm 2B Form 2C

Page 213

1. 124° 37� 48

2. 48.538°

3. �450°

4. �90° and 270°

5. 50°

6. ��73�3��

7. �47�

8. ��43�3��

9. � �41�

10. 0

11. �2

12. ��44�1��

13. � �152�

Page 214

14. 186 feet

15. 135 feet

16. 210° and 330°

17. �1132�

18. 56.9°

19. 10.7

20. 164.9 units2

21. one

22. 9.9

23. 10.5

24. 129.8°

25. 154.7 units2

Bonus: �0.9487

Page 215

1. 25° 36� 00

2. 75.500°

3. 540°

4. 135°

5. 55°

6. ��96�5��

7. �4�65

6�5��

8. �49�

9. � �31�

10. 0

11. ��23��

12. �4�17

1�7��

13. � �153�

Page 216

14. 185 feet

15. 75 feet

16. 240° and 300°

17. �153�

18. 41.8°

19. 19.8

20. 120.7 units2

21. none

22. 19.8

23. 9.3

24. 117.3

25. 196.7 units2

Bonus: 1.000


Chapter 5 Answer KeyCHAPTER 5 SCORING RUBRIC

Level Specific Criteria

3 Superior • Shows thorough understanding of the concepts standard position, degree measure, quadrant, reference angle, and the six trigonometric functions of an angle.

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

2 Satisfactory, • Shows understanding of the concepts standard position,with Minor degree measure, quadrant, reference angle, and the sixFlaws trigonometric functions of an angle.

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

1 Nearly • Shows understanding of most of the concepts standard Satisfactory, position, degree measure, quadrant, reference angle, with Serious and the six trigonometric functions of an angle.Flaws • May not use appropriate strategies to solve problems.

• Computations are mostly correct.• Written explanations are satisfactory.• Satisfies most requirements of problems.

0 Unsatisfactory • Shows little or no understanding of the concepts standard position, degree measure, quadrant, reference angle, and the six trigonometric functions of an angle.

• May not use appropriate strategies to solve problems.• Computations are incorrect.• Written explanations are not satisfactory.• Does not satisfy requirements of problems.


Page 217

1a. 210°, 570°, 930°

1b. The cosine is �xr

� � �2�

�33�

�, or

��23��.

1c. first quadrant: 30°, second quadrant: 150°, fourth quadrant: 330°

1d. Sample answers:(�3, 4); sin A � �

54�,

cos A � � �35

�, tan A � � �43

�,

csc A � �54

�, sec A � � �53

�,

cot A � � �43�

2a. The length of the missing sideto the east is given by

�s1in5

4ft0.°

� � �sin

x60°� , or 20.2 feet.

The length of the missing sidearound the fountain is given by �1

6� 20�, or 10.5 feet.

The total length isapproximately 140.7 feet.

2b. Answers will vary but mightinclude issues such as thelength of fence required versusthe increase in park size, access and/orproximity to roads,maintenance and/orenhancement of the fountain, and so on.

Chapter 5 Answer KeyOpen-Ended Assessment


Mid-Chapter TestPage 218

1. 65° 46� 55.2

2. 250°; III

3. �2�11

1�0��

4. �2�91�0��

5. � �31�

6. 0

7. ��23��

8. ��5�

9. 9.2

10. 11.0

Quiz APage 219

1. 47° 16� 59

2. 122.720°

3. 810°

4. 120° and �240°

5. 37°6. �3�

343�4��

7. �53�

8. ��53�4��

9. � �23�

10. 2

Quiz BPage 219

1. �1

2. undefined3. ��

23��

4. �15. ��

44�1��

6. ��153�

7. 6.6

8. 9.8

9. 83.9 feet

10. 54.5 feet

Quiz CPage 220

1. 210° and 330°

2. �152�

3. 35.7°

4. 48.8

5. 78.7 units2

Quiz DPage 220

1. none

2. 70.6°

3. 9.1

4. 60.1°

5. 498.0 units2

Chapter 5 Answer Key


Page 221

1. D

2. D

3. A

4. C

5. E

6. E

7. B

8. D

9. D

10. B

Page 222

11. D

12. A

13. E

14. C

15. A

16. B

17. B

18. A

19. 24

20. 361

Page 223

1. �2x

1� 4�

2. perpendicular

3.

4. (3, �1)

5. � �

6.

7. y-axisreflected over x-axis; translated left 2 units

8. and down 5 units

9. y � �x �2

6� ; yes

10. 3 � �3�

11. �29

12. �1

13. 15.4

14. 36.9°

38�15

�3437

Chapter 5 Answer KeySAT/ACT Practice Cumulative Review

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�

�110�

�15

�

�130�

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