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PRECALCULUS WITH TRIGONOMETRY COURSE SAMPLER 87

Overview of Solutions Manual

The Precalculus with Trigonometry: Concepts and Applications Solutions Manual containsone possible complete solution, including key steps and commentary where necessary,to each of the problems at the end of each section in the student text.

Solutions are presented in the form your students would be expected to use. Bear in mind, though, that there may be more than one way to solve any given problem using a correct method.

As in the student text, exact answers are displayed using the ellipsis format. When real-world approximations are required in the answer, exact calculations are used until thefinal answer is found, and then the appropriate rounding is indicated.

Where calculator programs are called for, sample programs and commentary areprovided in the Instructor’s Resource Book. The programs can be downloaded to TI-83and TI-84 calculators from the Instructor’s Resource CD or at www.keymath.com/precalc.

Solutions are not provided for journal entries. Student responses are highly individualand will vary from student to student.

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88 PRECALCULUS WITH TRIGONOMETRY COURSE SAMPLER

b.

c. The arc length on the unit circle equals the radian measure.

2. a. 1.3 m

b. 2.6 m for m; 3.9 m for

c.

d.

3. 4.

5. 6.

7. 8.

9. 10.

11. 12.

13. 14.

15. 18− 16. 90−

17. 30− 18. 45−

19. 15− 20. 120−

21. 135− 22. 180−

23. 270− 24. 150−

25.

26.

27.

28.

29.

30.

31.

32.

33.

34.

35. sinD1 0.3 = 0.3046…

sin 1066 = D0.8415…

tan(D2.3) = 1.1192…

cos 2 = D0.4161…

sin 5 = D0.9589…

180

π•3 = 171.8873…−

180

π= 57.2957…−

180

π•1.57 = 89.9543…−

180

π•1.26 = 72.1926…−

180

π•0.62 = 35.5233…−

180

π•0.34 = 19.4805…−

258

180π = 4.5029…

123

180π = 2.1467…

54

180π = 0.9424…

37

180π = 0.6457…

1080

180π = 6π

D225

180π = D

5

4π

450

180π =

5

2π

120

180π =

2

3π

180

180π = π

30

180π =

π

6

45

180π =

π

4

60

180π =

π

3

a = rθ

1.3r m

r = 3 mr = 2

1

2

3

12

3 u

v

r = 1

36.

37.

38.

39.

40.

41.

42.

43. 44.

45.

46.

47.

48.

49.

50.

51.

52.

53.

54.

Problem Set 3-5Q1. Q2. 360−

Q3. Q4.

Q5. Q6.

Q7. 72− Q8. 7

Q9. 5 h Q10.

1. units 2. units

3. units 4. units

5. 60− 6. 30−

7. 45− 8. 90−

9. units 10. units

11. 2 units

12. 1.467 units

13.

14. sin 2 = 0.9092…

tan 1 = 1.5574…

ππ

2

π

4

π

2

π

3

π

6

5% = 0.05

sin 47 = 0.1235…sin 47− = 0.7313…

34

180π = 0.5934…

180−π

= 57.2957…−

π

θ = tanD1 1

2= 26.5650…−

θ = cosD1 3

7= 64.6230…−

x = 100 sec 20− = 106.4177… cm

x = 17 sin 55− = 13.9255… cm

y = 5.5 + 0.5 cos 3617 (θ D 15−)

y = 5 + 7 cos 30(θ D 2−)

tan2

π

3D sec2

π

3= (√3)2 D 22 = D1

cos2 π + sin2

π = (D1)2 + 02 = 1

csc π

6 sin

π

6= 1

sin π

2+ 6 cos

π

3= 1 + 6•

1

2= 4

csc π

4= √2sec 2π = 1

cot π

2= 0

tan π

6=

√3

3

cos π = D1

sin π

3=

√3

2

cscD1 1.001 = 1.5260…

cotD1 3 = 0.3217…

tanD1 5 = 1.3734…

Precalculus with Trigonometry: Solutions Manual Problem Set 3-5 31© 2007 Key Curriculum Press

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15.

16.

17.

18.

19.

20.

21.

22.

23.

24. is undefined.

25. Period H 10Amplitude H 2Phase displacement H C4Sinusoidal axis H C3

26. Period H 3Amplitude H 5Phase displacement H D1Sinusoidal axis H D4

27. Period H 8Amplitude H 6Phase displacement H D1Sinusoidal axis H C2

28. Period H 6Amplitude H 4Phase displacement H C2Sinusoidal axis H C5

y

x

21

y

x

2

–4

y

x

4

–9

y

x

4

5

csc π

tan π

6=

1

√3

cos π

4=

1

√2

sin π

3=

√3

2

secD1 9 = 1.4594…

cscD1 5 = 0.2013…

tanD1 1.4 = 0.9505…

cosD1 0.3 = 1.2661…

cot 4 = 0.8636…

sec 3 = D1.0101… 29. Period H 4Asymptotes at Points of inflection at

30. Period H

Asymptotes at

Points of inflection at

31. Period

Asymptotes at

Critical points at specifically and

32. PeriodAsymptotes at

Critical points at specifically and

33.

34.

35.

36.

37.

38. y = cot π4x

y = csc π6x

y = 0.25 + 0.05 cos π4 (x + 1)

y = D2 + 5 cos π15 (x + 5)

y = 4 + 9 cos 10πx

y = 5 + 2 cos π3 (x D 1)

x

y

2π

3

(π

2± (2n + 1)π, D3

)(π

2± 2nπ, 3

)π

2± nπ,

±nπ= 2π

x

y

2π

2

(±(2n + 1)π, 1)(±2nπ, 3)±nπ,

π

2± nπ

= 2π

x

y

4

1

±1

2n

1

4±

1

2n

1

2

x

y

4

4

2 ± 4n±4n

32 Problem Set 3-5 Precalculus with Trigonometry: Solutions Manual© 2007 Key Curriculum Press

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39.

40.

41.

42.

43.

z(50) is 0.9079… below the sinusoidal axis.

44.

E(10,000) is 2.7553… above the sinusoidal axis.

45. a. Horizontal translation of ;

b. Horizontal translation of the graph would coincidewith itself and appear unchanged.

c. or or any multiple of

d. A horizontal translation by a multiple of results in agraph that coincides with itself. The period of the sinefunction is

46. a. Since the length of the hypotenuse H the radius of the

circle H 1, ,

and .

b. Answers will vary. The second angle measure is double the first, but the moving points always have the same x-values.

c. As k increases, the period decreases, and vice versa. Theperiod is always .

47. a. This lets u, v, x, and y all be represented on the samediagram—x is now an arc, and y is now either u or v,depending on whether we are talking about or

b. A radian measure corresponds to an angle measure, using , but because a radian measure is a

pure number, it can represent something other than anangle in an application problem.

48. a.

b. The description says that the circle is a unit circle.

Hence , and

.

c.

x sin x tan x

0.1 0.0998… 0.1003…

0.01 0.0099… 0.0100…

0.001 0.0009… 0.0010…

d. , but approaches 1 as x approaches 0; ,

but also approaches 1 as x approaches 0.

49. Journal entries will vary.

tan x

x > 1

sin x

x < 1

= tan xAD =AD

1=

AD

OA=

opp

adj

BC =BC

1=

BC

OB=

opp

hyp= sin x

=x

1= xmR

(∠AOB) =arc

radius

mR(θ) = m−(θ)• π180−

y = sin x.y = cos x

2πk

y = sin 2x =opp

hyp=

v2

radius of circle=

v2

1= v2

y = sin x =opp

hyp=

v1

radius of circle=

v1

1= v1

2π.

2π

±2πD2π,+2π

+2π;

sin x = cos

(x D

π

2

)+

π

2

E(10,000) = D2.4 + 7.2 cos π800 (10,000 D 100) = 0.3553…

E(1234) = D2.4 + 7.2 cos π800 (1234 D 100) = D4.2452…

= D8.9079…z(50) = D8 + 2 sin 5π (50 D 0.17)= D8.9079…z(0.4) = D8 + 2 sin 5π (0.4 D 0.17)

E = D2.4 + 7.2 cos π

800 (r D 100)

z = D8 + 2 sin 5π (t D 0.17)

y = D2 + sec x

y = 3 tan x Problem Set 3-6

Q1. Q2. 90−

Q3. 30− Q4.

Q5. Q6.

Q7.

Q8. Circle of radius 3 and center (0, 0)

Q9. Q10. Periodic

1. 7.4424…, 11.4070…,13.7256…

2. 6.7342…, 12.1153…,13.0173…

3. 8.0553…,10.7942…, 14.3385…

4. 8.3775…,10.4719…, 14.6607…

5. a.

b.

c. 5.0483…, 20.9516…, 25.0483…

d.

5.0483…, 20.9516…, 25.0483…

e.

6. a. 4.4, 11.6, 16.4, 23.6, 28.4

b.

c. 4.3509…, 11.6490…, 16.3509…, 23.6490…,28.3509…

d.

, 4.3509…, 11.6490…, 16.3509…, 23.6490…,28.3509…

e.

7. a. 1.1, 3.5, 5.1

b.

c. 3.4608…, 5.1391…

d.

D0.5391…, 1.1391…, 3.4608…, 5.1391…

e. x = 0.3 +2

π

(cosD1

1

4+ 50π

)= 101.1391…

x = D2.8608…,

(cosD1

D1 + 2

4 + 2πn

)x = 0.3 ±

2

π

x = D2.8608…, D0.5391…, 1.1391…,

y = D2 + 4 cos π2 (x D 0.3)

D0.5,x M D2.9,

x = 2 +6

π

(cosD1

1

3+ 16π

)= 100.3509…

x = D0.3509…

(cosD1

5 D 4

3 + 2πn

)x = 2 ±

6

π

x = D0.3509…,

y = 4 + 3 cos π6 (x D 2)

x M D0.4,

x = 3 D10

π

(cosD1

4

5D 10π

)= 100.9516…

x = 0.9516...,

(cosD1

6 D 2

5+ 2πn

)x = 3 ±

10

π

x = 0.9516…,

y = 2 + 5 cos π10 (x D 3)

x M 1, 5, 21, 25

± cosD1(D0.5) + 2πn = 2.0943…, 4.1887…,

± cosD1(D0.2) + 2πn = 1.7721…, 4.5110…,

± cosD1 0.4 + 2πn = 0.4510…, 5.8321…,

± cosD1 0.9 + 2πn = 1.1592…, 5.1239…,

y = abx

tanD1 3

7= 23.1985…−

y

1

90°θ

y

1

90°θ

π

4

π

2

Precalculus with Trigonometry: Solutions Manual Problem Set 3-6 33© 2007 Key Curriculum Press

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Overview of Instructor’sResource Book

Precalculus with Trigonometry: Concepts and Applications is designed to be used byinstructors with a wide spectrum of teaching styles. It is possible for you to use the textin a passive lecture–and–note-taking mode, but the text is most effective in a cooperativelearning environment in which you and the students interact during class, and in whichstudents are expected to arrive at conclusions on their own. Your role in this mode is toprovide guidance and to follow up and reinforce what the students discover. TheInstructor’s Resource Book contains the materials to help you do the job. All of thematerial in this text may be reproduced for direct classroom use with your students.

• The blackline masters are enlarged, reproducible copies of graphs that are needed tocomplete examples and problems in the student text. Also included are reproduciblecopies of special types of graph paper.

• The Exploration masters enable you to help students learn mathematical concepts byexploring them before reading the material in the text. Often the Explorations areintended for cooperative groups. Complete solutions to each Exploration are alsoprovided.

• The technology activities use The Geometer’s Sketchpad, Fathom Dynamic Data, orCBL 2 to help students visualize and experience concepts in dynamic or real-worldenvironments. These are provided as an optional enhancement to selected lessonsthat you can choose to assign given your technology resources.

• The programs for graphing calculators include the programs called for in specificplaces in the student text.

Note on the Precalculus with TrigonometryElectronic Instructor’s ResourcesInstructor’s resources in electronic format are available on the Instructor’s Resource CDthat accompanies the Instructor’s Guide and also at www.keymath.com/keyonline, whereyou can become a registered user of Precalculus with Trigonometry: Concepts andApplications. Certain resources, such as dynamic sketches and data sets, are onlyavailable electronically. All of the resources listed below are available in electronicformat.

• PDF files of all blackline masters, Exploration masters and solutions, and technologyactivities

• Dynamic Precalculus Explorations using any Java-enabled Web browser

• Presentation sketches using The Geometer’s Sketchpad

• Sketchpad and Fathom files required for some of the technology activities

• Program files for graphing calculators using TI-Connect™

• Data sets in TI List, Excel, and Fathom formats for problems in the Student Editionand other supplemental problems

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Precalculus with Trigonometry: Instructor’s Resource Book Blackline Masters 15© 2007 Key Curriculum Press

Name: Group Members:

Trigonometric Ratios Table Date:

Fill in this table as you work through Chapters 2 and 3. Wait until you have encountered eachparticular concept before filling in the column below it. For example, you will learn about sineand cosine in Section 2-3, but you won’t learn about radians until Section 3-4.

Degrees Radians Sine Cosine Tangent Cotangent Secant Cosecant

0º

30º

45º

60º

90º

120º

135º

150º

180º

210º

225º

240º

270º

300º

315º

330º

360º

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16 Blackline Masters Precalculus with Trigonometry: Instructor’s Resource Book© 2007 Key Curriculum Press


Problem Set 3-2/Pages 103–105 Date:

5.

6.

7.

8.

�2° 8°

y

�40

�20

�30

θ

10° 70°

y

2

�3

θ

54°44°

34°24°14°4°

�6°

�2

�16°

y

18

θ

20°�25°�70° 65° 110° 155° 200°

y

15

3θ

9.

10.

11.

12.

7°3°

y

�5000

5000

θ

7°3°

y

�5000

5000

θ

5.3°0.3°

y

50

10 θ

�16° 2°

y

0.34

2.56

θ

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68 Exploration Masters Precalculus with Trigonometry: Instructor’s Resource Book© 2007 Key Curriculum Press


Exploration 3-1a: Periodic Daily Temperatures Date:

Objective: Transform the cosine function so that it fits, approximately, data on the averagedaily temperatures for a city.

Month Temperature (−F)

Jan. 61.7

Feb. 66.3

Mar. 73.7

Apr. 80.3

May 85.6

June 91.8

1. On the graph paper, plot the average daily hightemperatures for two years. Assume that January ismonth 1 and so forth. Determine a time-efficient wayfor your group members to do the plotting. Whatshould you plot for month zero? Connect the pointswith a smooth curve.

2. The graph of completes a cycle each 360−(angle, not temperature). What horizontal dilationfactor would make it complete a cycle each 12−, asshown? Write an equation for this transformedsinusoid and plot it on your grapher.

3. Earth rotates 360− around the Sun in 12 months.How do these numbers relate to the dilation factoryou used in Problem 2?

y

θ

12° 24°

1

�1

y = cos θ

y

x6 12 18 24

100

90

80

70

60

50

40

30

20

10

Months

Tem

per

atu

re (

°F)

Month Temperature (−F)

July 94.9

Aug. 94.6

Sept. 89.3

Oct. 81.5

Nov. 70.7

Dec. 64.6

4. The temperature graph in Problem 1 has a high pointat months. What transformation would youapply to the sinusoid in Problem 2 (dashed in thenext figure) to make it have a high point at (solid) instead of at Write the equation andconfirm it by plotting it on your grapher.

5. The average of the highest and lowest temperaturesin the table is Write an equation forthe transformation that would translate the graph inProblem 4 upward by 78.3 units.

6. The 94.9 high point in Problem 1 is 16.6 units above78.3, and the 61.7 low point is 16.6 units below 78.3.Write an equation for the transformation that woulddilate the sinusoid in Problem 5 by a factor of 16.6so that it looks like this graph. Confirm your answerby grapher.

7. On your grapher, plot the points you plotted inProblem 1. How well does the sinusoidal equation inProblem 6 fit the points?

8. What did you learn as a result of doing thisExploration that you did not know before?

y

θ

7°

61.778.3

94.9

94.9 + 61.72 = 78.3.

y

θ

12°

7°

24°

1

�1

θ = 0−?θ = 7−

x = 7

Here are average daily high temperatures for San Antonio, by month, based on data collectedover the past 100 years and published by NOAA, the National Oceanic and AtmosphericAdministration. Such data are used, for example, in the design of heating and airconditioning systems.

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Precalculus with Trigonometry: Instructor’s Resource Book Exploration Masters 69© 2007 Key Curriculum Press


Exploration 3-1b: Sine and Cosine Graphs, Manually Date:

Objective: Find the shape of sine and cosine graphs by plotting them on graph paper.

3. Find sin 45− and cos 65−. Show that thecorresponding points are on the graphs inProblems 1 and 2, respectively.

4. Find the inverse trigonometric functions. Show that the

corresponding points are on the graphs inProblems 1 and 2, respectively.

θ = sinD1 0.4 and θ = cosD1 0.8

5. What are the ranges of the sine and cosinefunctions?

6. Name a real-world situation where variables arerelated by a periodic graph like sine or cosine.


1. On your grapher, make a table of values of for each 10− from 0− to 90−. Set themode to round to 2 decimal places. Plot the values on this graph paper. Also plot for each 90− through 720−. Connect the points with a smooth curve, observing the shapeyou plotted for 0− to 90−.

2. Plot the graph of pointwise, the way you did for sine in Problem 1.

y

90° 180° 270° 360° 450° 540° 630° 720°

�1

1

θ

y = cos θ

y

90° 180° 270° 360° 450° 540° 630° 720°

�1

1

θ

y = sin θy = sin θ

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FPC2_CS_B.qxd 1/10/06 3:22 AM 5


1. Write the horizontal dilation factor, period,amplitude, phase displacement, and verticaldisplacement, and sketch the graph.

Horizontal dilation factor:

Period:

Amplitude:

Phase displacement:

Vertical displacement:

2. Write the horizontal dilation factor, period,amplitude, phase displacement, and verticaldisplacement, and sketch the graph.


Period:

Amplitude:

Phase displacement:


y

θ

y = D2 + 4 sin 30(θ + 1−)

y

θ

y = 4 + 3 cos 2(θ D 70−)

3. Once you know the connection between the equationof a sinusoid and its graph, you can go backwardsand write the equation from a given graph. For thefollowing sinusoid, write the period, horizontaldilation factor, amplitude, phase displacement (forthe cosine function), and vertical displacement. Thenwrite the particular equation.

Period:


Amplitude:

Phase displacement:


Equation:

4. Confirm that your answer to Problem 3 is correct byentering the equation in the grapher and plotting thegraph. Does your graph agree with the given figure?

5. By the most time-efficient method possible, find yfor your equation in Problem 3 if Write theanswer to as many decimal places as your grapherwill give. Draw something on the given graph toshow that your answer is reasonable.


θ = 35−.

y

10°

�8

2θ



Exploration 3-2a: Transformed Sinusoid Graphs Date:

Objective: Given the equation of a transformed sinusoid, sketch the graph, and vice versa.

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1. Sketch two cycles of this sinusoid:

2. Write a particular equation (cos) for this sinusoid:

3. Write a particular equation for the sinusoid inProblem 2 using sine.

4. Plot the equation in Problem 2 as y1 on your grapher.Plot the equation in Problem 3 as y2. Use a differentstyle for each graph. Do both graphs agree with thegiven graph?

y

θ

12°�3°

10

100

y

θ

y = D3 + 5 sin 4(θ D 20−)

5. This is a half-cycle of a sinusoid. Write a particularequation.

6. This is a quarter-cycle of a sinusoid. Write aparticular equation.

7. On the sinusoid in Problem 2, mark a point ofinflection. Mark another point at which the graph isincreasing but concave down.


y

�60

�100

θ

36°24°

y

600°400°

�3

4

θ


Exploration 3-2b: Sinusoidal Equations from Graphs Date:

Objective: Given the equation, sketch the sinusoid, and vice versa.

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No graphers allowed for Problems 1–7.

1. The reciprocal property states that

Without your grapher, use this property to sketchthe graph of on the same axes as the graphof the parent function In particular, showwhat happens to the secant graph wherever

2. Write the quotient property expressing tan θ as aquotient of two other trigonometric functions.

3. The next figure shows the parent functionsBased on the answer to

Problem 2, determine where the asymptotes are forthe graph of and mark them on the figure.

y

540°450°360°270°180°90°�90°

1

θ

y = tan θ,

y = sin θ and y = cos θ.

y

540°450°360°270°180°90°�90°

1

θ

cos θ = 0.

y = cos θ.y = sec θ

sec θ =1

cos θ

4. Based on the quotient property, find out where theθ-intercepts are for the graph of Markthese intercepts on the figure in Problem 3.

5. At are equal. Based on thisfact, what does tan 45− equal? Mark this point on thegraph in Problem 3. Mark all other points where

tan 45− H

6. Use the points and asymptotes you have marked tosketch the graph of on the figure inProblem 3. (No graphers allowed!)

7. Check your graphs with your instructor.

Graphers allowed for the remaining problems.

8. On your grapher, plot the graph of Sketchthe result here.

9. On your grapher, plot the graph of Sketchthe result here.

10. At what values of θ are the points of inflection forExplain why the tangent function has no

critical points.

11. Explain why the graph of has no points ofinflection, even though the graph goes from concaveup to concave down at various places.


y = sec θ

y = tan θ?

y = cot θ.

y = csc θ.

y = tan θ

|sin θ | = |cos θ |.

θ = 45−, sin θ and cos θ

y = tan θ.



Exploration 3-3a: Tangent and Secant Graphs Date:

Objective: Discover what the tangent and secant function graphs look like and how they relateto sine and cosine.

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1. For state

The horizontal dilation:

The period:

The horizontal translation:

The vertical dilation:

The vertical translation:

2. Sketch the graph of showingvertical asymptotes, horizontal axis, points ofinflection, and other significant points.

3. For the next graph, state


The period:

The horizontal translation (for cotangent):



4. Write a particular equation for the graph inProblem 3. Check your answer by plotting onyour grapher.

y

36° θ21°�9° 6° 51°

�1

y

θ

y = 3 + 12 tan 5(θ D 7−),

y = 3 + 12 tan 5(θ D 7−), 5. For give


The period:

The horizontal translation:



6. Sketch the graph of showingvertical asymptotes, horizontal axis, and critical points.

7. For the next graph, give


The period:

The horizontal translation (for secant):



8. Write a particular equation for the graph in Problem 7.Check your answer by plotting on your grapher.


y

θ

70°�20° 160°

1

4

7

y

θ

y = 1 + 3 csc 4(θ + 10−),

y = 1 + 3 csc 4(θ + 10−),


Exploration 3-3b: Transformed Tangent Date:

and Secant GraphsObjective: Sketch transformed tangent, cotangent, secant, and cosecant graphs, and findequations from given graphs.

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Exploration 3-4a: Radian Measure of Angles Date:

Objective: Discover how angles are measured in radians by measuring around a circle with aflexible ruler.

1. The figure shows a circle centered at the origin of auv-coordinate system. On a flexible ruler (an indexcard will do), mark off a length equal to the radiusof the circle. Start at the point where the circleintersects the positive u-axis and bend the ruler tomark off arcs of lengths 1, 2, and 3 unitscounterclockwise around the circle.

2. Draw a ray from the origin through the pointcorresponding to 1 radius length. Your drawingshould look like this:

3. The resulting angle in standard position has measure1 radian. Measure the number of degrees in thisangle.

u

1 radian

12

3

1 radius length

v

4. From geometry, recall that the circumference ofa circle is where r is the radius. So there are

radians in a complete revolution. The fact thatthere are 360− in a complete revolution gives youa way to transform between degrees and radians.Calculate exactly the number of degrees in 1 radian. How does the measured value in Problem 3compare with this exact answer?

5. Calculate the exact number of degrees in 3 radians.Draw a 3-radian angle on the previous figure. Howclose is the degree measure of your drawn angle tothe exact value?


2π2πr,

u

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1. At the board, plot horizontal and vertical u- and v-axes. Obtain a roll of masking tape and place itwith its center at the origin. Draw a circle on theboard by tracing around the outside of the roll.

2. Remove the roll of tape from the board. Mark a“ruler” on a piece of string, with units equal to theradius of the circle you drew. Then attach the stringto the roll of tape.

3. Put the roll back on the board in such a way that thestarting point on the string is on the positive side ofthe u-axis. Wrap the string counterclockwise aroundthe tape roll. Make marks on the board at the points1, 2, 3, 4, 5, and 6 on the string. Then remove thetape roll again.

4. Draw rays through the points you marked on theboard, like this:

5. The central angles formed by the rays you drew havemeasures of 1, 2, 3, . . . radians. By measuring with aprotractor, find out approximately how manydegrees are in 1 radian.

6. An angle of 6 radians is not quite a completerevolution. How many radians would it take to makea complete revolution? Provide the exact value.

u

v

1 radian

12

3

4

5

6

1 radius length

u

Draw this circle.

Roll of tape

v

7. You should have answered “ radians” forProblem 6. The fact that there are 360− in a completerevolution gives you a way to transform degrees toradians, and the other way around. Calculate exactlythe number of degrees in 1 radian. How does themeasured value in Problem 5 compare with thisexact answer?

8. Calculate the exact number of degrees in 3 radians.Show a 3-radian angle on your board drawing. Howclose is the degree measure of your drawn angle tothe exact value?

9. Explain why the size of a radian would be the sameno matter what size circular object you use in placeof the roll of tape in Problem 1.


2π


Exploration 3-4b: Radian Measure of Angles Date:

Objective: Discover how angles are measured in radians by wrapping a string arounda circle.

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1. Sketch the parent trigonometric function



4. Set your grapher to radian mode. Set the windowwith an x-range of and the y-range as shownon the given graphs. Then plot the graph of thecircular function Sketch the result.

5. With your grapher still in radian mode, plot thegraph of the circular function Sketch theresult.

y

2π 3π 4ππ

�1

1

x

y = cos x.

y

2π 3π 4ππ

�1

1

x

y = sin x.

[0, 4π]

720°360°

y

1

�1

θ

y = tan θ.

y

720°360°

�1

1

θ

y = cos θ.

y

720°360°

�1

1

θ

y = sin θ. 6. With your grapher still in radian mode, plot thegraph of the circular function Sketch theresult.

7. The only difference between the parent graphs forthe circular function sinusoid and the ordinarytrigonometric function sinusoid is the period.Explain how the periods of the two types of sinusoidrelate to degrees and radians.

8. The graph here is a transformed circular functionsinusoid. Using what you have learned abouttransformations, find a particular equation of thissinusoid. Confirm by grapher that your equation iscorrect.


y

x10

1

4π3ππ 2π

y

1

�1

x

y = tan x.



Exploration 3-5a: Circular Function Parent Graphs Date:

Objective: Plot circular function sinusoids and tangent graphs.

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Exploration 3-6a: Sinusoids, Given y, Date:

Find x NumericallyObjective: Find a particular equation for a given sinusoid and use it to graphically andnumerically find x-values for a given y-value.

1. For the sinusoid shown, draw the line Readfrom the graph the six values of x for which the linecrosses the part of the graph shown. Write youranswers to one decimal place.

x H , , ,

, , .

2. Write an equation for this sinusoid.

3. Plot the equation from Problem 2 on your grapher.Does it look like the given graph?

4. Trace your graph in Problem 3 to Does yourgraph have a high point there?

5. Circle the leftmost point on the given graph atwhich Plot the line , and use the intersectfeature to find the value of x at this point.

x H

6. Other values of x for which can be found byadding multiples of the period to the value of x inProblem 5. Let n be the number of periods you add.Find two more values of x for which Circle thethree x-values in Problem 1 that are also answers toProblem 5 and this problem.

Multiple, n H 1: x H

Multiple, n H 2: x H

y = 5.

y = 5

y = 5y = 5.

x = 17.

y = 5. 7. Put a box on the figure at a point whose x-value isnot an answer to Problem 5 or 6. Use the intersectfeature to find one of these x-values.

x H

8. Add multiples of the period to the x-values inProblems 5 or 7 to find the other two x-values thatare also on the graph. Tell what multiple of theperiod you added.

Multiple, n H : x H

Multiple, n H : x H

9. By adding an appropriate multiple of the period tothe answer to Problem 5 or 7, find the first value of xgreater than 1000 for which At this value of x,will y be increasing or decreasing? How can you tell?

Multiple, n H : x H


y = 5.

y

x5�5 10 15 20 25

16

2

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Exploration 3-6b: Given y, Find x Algebraically Date:

Objective: Given the particular equation for a sinusoid and a value of y, calculate thecorresponding x-values algebraically.

1. The sinusoid has the equation

Confirm that this equation gives the correct value ofy when

2. Your objective is to find algebraically the valuesof x given Substitute 5 for y. Then do thealgebra necessary to get x using an arccosine.Write the general solution in the form

x H (number) C (period)n or (number) C (period)n

y = 5.

x = 15.

y = 9 + 7 cos 2π

13(x D 4)

3. Write the two values of x from the general solutionin the row of this table. By adding andsubtracting multiples of the period, fill in the otherrows in the table with more possible values of x.

n x1 x2

D1

0

1

2

4. Circle the points on the given graph where the linecuts the graph. For each point, tell the value of

n at that point.

5. Find the two values of x if

6. Find the first value of x greater than 1000 for whichWhat does n equal there?


y = 5.

n = 100.

y = 5

n = 0

y

x5�5 10 15 20 25

16

2

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Exploration 3-7a: Chemotherapy Problem Date:

Objective: Use sinusoids to predict events in the real world.

1. Draw the graph of the sinusoid on the given axes.Show enough cycles to fill the graph paper.

2. Write a particular equation for the (circular) sinusoidin Problem 1. It is recommended that you use thecosine function.

3. Enter your equation in your grapher. Plot the graphusing the window shown. Explain how the graphverifies that your equation is correct.

4. The woman feels “good” if the red blood cell count is700 or more, “bad” if the count is 300 or less, and“so-so” if the count is between 300 and 700. Howwill she be feeling on her birthday, March 19? Explainhow you arrived at your answer.

5. Show on your graph the interval of dates betweenwhich the woman will feel “good” as she comes backfrom the low point after the January 13 treatment.

6. Find precisely the values of x at the beginning andend of the interval in Problem 5 by setting and using appropriate numeric or graphicalmethods. Describe what you did.

x H and x H


y = 700

Chemotherapy Problem: A woman has cancer and must have a chemotherapy treatment onceevery three weeks. One side effect is that her red blood cell count goes down and then comesback up between treatments. On January 13 (day 13 of the year), she gets a treatment. At thattime, her red blood cell count is at a high of 800. Halfway between treatments, the count dropsto a low of 200. Assume that the red blood cell count varies sinusoidally with the day of theyear, x.

y (red cell count)

10 20 30 40 50

x (days)

1000

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Exploration 3-7b: Oil Well Problem Date:

Objective: Use sinusoids to predict events in the real world.

1. Find a particular equation for y as a function of x.

2. Plot the graph on your grapher. Use a window withan x-range of Describe how the graphconfirms that your equation is correct.

3. Find graphically the first interval of x-values in theavailable land for which the top surface of theformation is no more than 1600 feet deep. Draw asketch showing what you did.

[D100, 900].

4. Find algebraically the values of x at the ends of theinterval in Problem 3.

5. Suppose that the original measurements wereslightly inaccurate and that the value of x shown atD65 feet was at instead. Would this factmake much difference in the answer to Problem 3?Use a time-efficient method to reach your answer.Explain what you did.


x = D64

The figure shows a vertical cross section through a piece of land. The y-axis is drawn comingout of the ground at the fence bordering land owned by your boss, Earl Wells. Earl owns theland to the left of the fence and is interested in buying land on the other side to drill a new oilwell. Geologists have found an oil-bearing formation, which they believe to be sinusoidal inshape, beneath Earl’s land. At feet, the top surface of the formation is, at its deepest,

feet. A quarter-cycle closer to the fence, at feet, the top surface is only2000 feet deep. The first 700 feet of land beyond the fence is inaccessible. Earl wants to drill at the first convenient site beyond ft.

y

y = �2500 ft

�100 �65 �30

Top surface

Fence

y = �2000 ft

x = 700 ft

xInaccessible land Available land

x = 700

x = D65y = D2500x = D100

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Exploration 3-8a: Introduction to Angular Date:

and Linear VelocityObjective: Distinguish between linear and angular velocity of a point on a rotating object.

1. Suppose that you have conducted the rotationillustrated in the figure. With a flexible ruler you findthat the curved length of the outer arc is 60 cm andthe curved length of the inner one is 24 cm. At whataverage speed (in cm/s) did each pen move in the5 seconds while they were making the arcs? Whichpen moved faster?

2. Suppose that the angle from the initial position tothe terminal position of the rotating ruler is about115−. At what number of degrees per second did theouter pen move? At what number of degrees persecond did the inner pen move? Which pen movedfaster in deg/s?

3. Suppose that the angle from the initial position tothe terminal position is exactly 2 radians, that theinner pen is 12 cm from the center of rotation, andthat the outer pen is 30 cm from the center ofrotation. Explain why you can find the arc lengths inProblem 1 by multiplying the radius of the arc bythe number of radians. Show that the data in thisproblem are consistent with the data in Problem 1.

4. The numbers of cm/s in Problem 1 are called linearvelocities because they tell how fast a distance (arclength) changes. The numbers of deg/s in Problem 2are called angular velocities because they tell howfast the angle changes. What is the angular velocityin radians per second of the rotating ruler inProblem 3?

5. Show that you can find the linear velocities inProblem 1 by multiplying the radius of the arc bythe angular velocity in radians per second.

6. Write some conclusions about the angular velocitiesand the linear velocities of points on the samerotating object. Include definitions for linear velocityand angular velocity.


The figure illustrates a rotating ruler attached by suction cup to a dry-erase board.Marking pens are put into two of the holes in the ruler, which is then rotated slowly fromits initial position (dotted), coming to rest at its final position after 5 seconds. Each penleaves a curved line on the board. Question: “Which pen moved faster, the inner one or theouter one?” In this Exploration you will learn that there are two answers to that question,depending on whether you want to know how fast the pen moved across the board or howfast the angle changed.

Rotate

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1. Adam Ant sits on the second hand of a clock, 12 cmfrom the center. What is his angular velocity inrevolutions per minute?

2. What is Adam’s angular velocity in radians perminute?

3. What is Adam’s linear velocity in cm/min?

4. What is Adam’s linear velocity in cm/s?

5. Adam crawls to a point on the second hand that is5 cm from the center. What are his angular andlinear velocities now?

6. Adam crawls to the very center of the second hand.What are his angular and linear velocities now?

7. Adam crawls off the clock and onto a rotatable rulerheld to the dry-erase board by suction cup. Phoebegives the ruler a spin. Adam finds that he is moving48 in./s. Is this an angular velocity or a linearvelocity?

8. If Adam is 8 in. from the center of the ruler inProblem 7, how many radians per second is hegoing?

9. In Problems 7 and 8, what is Adam’s angular velocityin revolutions per minute?

10. For the formula v = rω, suppose that r is in cm andω is in radians per second. What do you need toassume about the units of r to make v have theproper units, cm/s? Explain why you can think of ras having these units. A picture might help.




Exploration 3-8b: Adam Ant Problem Date:

Objective: Calculate linear and angular velocities of points on rotating objects.

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Exploration 3-8c: Motorcycle Problem Date:

Objective: Find angular and linear velocities of connected rotating objects.

1. If the small sprocket turns at 1200 rev/min, what isits angular velocity in radians per second?

2. The small sprocket has a radius of 3 cm. If it turnsat 100 rad/s, what is the linear velocity of its rim?

3. If the linear velocity of the rim of the small sprocketis 500 cm/s, how fast is the chain going?

4. If the chain is going 400 cm/s, what is the linearvelocity of the rim of the rear sprocket?

5. The rear sprocket has a diameter of 20 cm. If itsrim has a linear velocity of 600 cm/s, what is itsangular velocity?

6. The rear wheel’s tire has an outside radius of 25 cm. If the rear sprocket has an angular velocityof 70 rad/s, what is the angular velocity of a pointon the outside of the tire?

7. If the rear wheel’s tire has an angular velocity of35 rad/s, what is the linear velocity of a point on theoutside of the tire?

8. If the rear wheel has an angular velocity of 47 rad/s,what are the angular and linear velocity of the centerof the wheel?

9. If the rear wheel’s tire has a linear velocity of900 cm/s, how fast is the motorcycle going?

10. The front wheel’s tire has an outside radius of15 cm. If the rear wheel has a linear velocity of900 cm/s, as in Problem 9, what are the angular andlinear velocities at the outside of the front wheel?

11. If the linear velocity of the rear wheel is 900 cm/s,as in Problems 9 and 10, how fast is the motorcyclegoing in km/h?

12. Suppose that points on the rim of the rear wheelhave a linear velocity of 50 mi/h. To a person on theground, the bottom of the rear tire, touching theground, appears to be stopped! How fast does thetop of the tire appear to be going?


Rhoda rides her motorcycle. The motor turns a small sprocket, which in turndrives a larger sprocket connected to the rear wheel. The front wheel has asmaller diameter than the rear one. In this Exploration you will find linearand angular velocities at various points. Note that the motorcycle is notnecessarily going the same speed in all parts of the problem.

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Degrees and Radians Problems: The figure shows a unitcircle in a uv-coordinate system. The x-axis from an xy-coordinate system is placed tangent to the circle,with its origin, at the point

1. Suppose the x-axis is wrapped around the unit circle.Sketch the points at which and D1 willmap onto the unit circle.

2. Show an angle of 3 radians in standard position onthe figure.

3. Show an angle of D1 radian in standard position onthe figure.

4. Calculate the number of degrees in 3 radians. Writethe answer in ellipsis decimal form.

5. Calculate the number of radians in 240−. Simplify theanswer but leave it exact (no decimals).

6. If you draw a circle of radius 10 cm and thenmeasure an arc of length 7 cm on it with a flexibleruler, what would be the radian measure of thecentral angle that subtends the arc?

x = 3, 2, 1,

u

1

1

v x

(u, v) = (1, 0).x = 0,

Archaeology Problem: The figure shows part of an ancientwall. Archaeologists presume that the height of the wallwas a sinusoidal function of the distance from the leftend of the wall, with a low point of m at m andthe next high point of m at m. The rest of thesinusoidal wall has crumbled away. However, the end ofthe wall at m is clearly visible. AnswerProblems 7–10.

7. Mark all of the high points and all of the low pointsthe wall had between and inclusive.Then sketch the graph.

8. Write a particular equation for y as a function of x.

9. Based on your mathematical model, what did y equalat the right end of the wall?

10. Find numerically the first value of x beyond 10 m atwhich the height of the wall was 3.6 m.

x = 27,x = 0

x

y

15 20 25105

5

3

x = 27

x = 4y = 7x = 0y = 3



Exploration 3-9a: Rehearsal for Sinusoids Test Date:

Objective: Use circular function sinusoids as mathematical models.

(Over)

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Tide Problem: The (average) depth of the water at aparticular point on the beach varies sinusoidally with timedue to the motion of the tides. The figure shows thedepth, y, measured in feet, at such a point as a function ofx, measured in hours after midnight at the beginning ofJanuary 1. The particular equation of the sinusoid is

11. What is the deepest the water gets? What is the firsttime on January 1 at which the water is this deep?What is the period of this function?

12. Where the graph dips below the x-axis, the water iscompletely gone, leaving the point on the beach outof the water. At what time does the lowest tide firstoccur on January 1? How deep a hole would youhave to dig in the sand so that water would flow intoit at that time?

13. Calculate the depth of the water at 4:00 p.m. onJanuary 1. Show that the answer agrees with thegraph.

x

y

2412

5

y = 3 + 4 cos π

5.8 (x D 1)

14. There is a high tide close to midnight at thebeginning of January 2 Is this high point onJanuary 1 or on January 2? Show calculations tojustify your answer.

15. Find graphically the first interval of times onJanuary 1 for which the water is completely gone.

16. Calculate algebraically the first time on January 3(i.e., ) at which the depth of the water isexactly zero.


x ≥ 48

(x = 24).


Exploration 3-9a: Rehearsal for Sinusoids Test continued Date:

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In this activity you will use

a point on the unit circle to

construct dilated images of

circular functions.

SKETCH AND INVESTIGATE1. Open the sketch Circular Transforms.gsp. The sketch contains a parameter k

that currently equals 2. Use the Calculator to multiply k by the angle measure

of DC�.

2. Mark point A as the center of rotation using TransformMark Center.Similarly, mark the calculation from step 1 as the angle of rotation.

3. Rotate point D by selecting it and choosing TransformRotate. Label the

rotated point F, and construct segment AF.

Q1 Press the Animate Point C button. What is the relation of ∠DAC to ∠DAF ?

Q2 For every complete trip that point C makes around the circle, how many times

does point F travel around the circle?

Q3 Double-click parameter k, and change its value to 3. Answer Q1 and Q2 again

for this new value.

4. Press the Show Point E button. This point, which you built in the activity

Trigonometry Tracers, traces out sin(mDC�). Press the Animate Point C button

to watch point E in action.

Q4 You’re about to create the graph of sin(k � mDC�). Before you do, make a

prediction: Based on your answers to Q2 and Q3, what do you predict the

graph will look like?

5. Measure yF by selecting point F and choosing MeasureOrdinate (y).

6. Plot the point �mDC�, yF� by selecting in order mDC� and yF, and then

choosing GraphPlot As (x,y).

7. Label the plotted point G, and turn on tracing for it.

Q5 Animate C, and observe the trace of G. Is your prediction about the graph

of sin(k � mDC�) correct?

8. Change the value of parameter k to draw new sine curves.

Q6 By taking new measurements, create the graphs of cos(k � mDC�) and

tan(k � mDC�). Describe the appearance of each of these functions.

2

F

DA

C

Precalculus with Trigonometry: Instructor’s Resource Book The Geometer’s Sketchpad® Activities 353© 2007 Key Curriculum Press

Transformations of Circular Functions

To mark the calculationas the angle of rotation,select it and chooseTransformMark Angle.

To turn on tracing, selectthe point and chooseDisplayTrace Point.

Choose DisplayEraseTraces to erase existingtraces.

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354 The Geometer’s Sketchpad® Activities Precalculus with Trigonometry: Instructor’s Resource Book© 2007 Key Curriculum Press

Objective: Students use a point on the unit circle to

define and plot transformed circular functions.

Prerequisites: Students should have completed the

activity Trigonometry Tracers earlier in the chapter.

Sketchpad Proficiency: Beginner. Students start with

pre-made sketches and are given detailed instructions

to complete the construction.

Class Time: 20–30 minutes

Required Sketch: Circular Transforms.gsp

Presentation Sketch: Circular Transforms Present.gsp

SKETCH AND INVESTIGATE

Q1 ∠DAF is twice as large as ∠DAC.

Q2 Point F travels twice around the circle for every

revolution of point C.

Q3 When k � 3, ∠DAF is three times as large as ∠DAC,

and F travels three times around the circle for every

revolution of C.

Q4 Predictions will vary. The important thing is that

students make a prediction.

Q5 The graph is a sine graph compressed in the x

direction. It has an amplitude of 1 and a period of 2�/3

so that it shows 3 complete cycles between 0 and 2�.

8. When you change k, the period becomes 2�/k, and

the graph shows k complete cycles between 0 and 2�.

Q6 The graphs of these functions resemble the graphs

produced in Trigonometry Tracers, but (like the sine

plot) compressed in the x direction so that they

show k cycles between 0 and 2�.

EXTENSION

You could challenge students to find a way to modify the

construction to produce vertical dilation in the resulting

graph. One method would be to put a point on segment AF

and plot the point’s y-coordinate as a function of the angle.

If segment AF is constructed as a ray, it’s possible to

produce both compression and stretching. Alternatively, you

could dilate point F toward or away from center point A.

Transformations of Circular FunctionsActivity Notes

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Overview of AssessmentResources

The Precalculus with Trigonometry: Concepts and Applications Assessment Resourcescontains a complete set of testing materials with solutions. These blackline masters canbe reproduced to use with your students.

You can also modify the electronic files of these tests to customize them. The tests areavailable in Microsoft Word format on the Instructor’s Resource CD that accompaniesthe Instructor’s Guide and also at www.keypress.com/keyonline, where you can becomea registered user of Precalculus with Trigonometry: Concepts and Applications. Equationsare set in MathType.

• The Assessment Suggestions are based on the National Council of Teachers ofMathematics’ Assessment Standards for School Mathematics. This section providessupplementary material to meet these standards, as well as specific suggestions andexamples from the author’s classroom on how to use the text.

• The Tests include up to three tests for each chapter. There are two forms of each test.Each Section Test covers material from one or more sections, and the Chapter Testcovers material from the whole chapter.

• Three Cumulative Tests are also provided, covering Chapters 1–6, Chapter 7–9, andChapters 10–15.

• Complete solutions to each test are provided.



Precalculus with Trigonometry: Assessment Resources Section, Chapter, and Cumulative Tests 33© 2007 Key Curriculum Press

Part 1: No calculators allowed (1–9)1. Darken exactly one cycle of this sinusoid.

2. Show the period of this sinusoid.

3. Show a critical point. Show a point of inflection.Identify which is which.

4. Show the phase displacement (for cosine).

5. At the point shown, is this graph concave up orconcave down?

6. What is the period of y = 2 + 3 sin 6(θ D 4−)?

θ

y

θ

y

θ

y

θ

y

θ

y

7. The dashed graph shows y H cos θ. Use this graphand the fact that to sketch the graph of y H sec θ.

8. The dashed graph is y H cos θ and the solid graph isy H sin θ. Use these graphs and the quotient propertyto draw the asymptotes and θ-intercepts of y H cot θ.(You don’t need to draw the graph.)

(quotient property)

9. Sketch: y H 5 C 2 cos 4(θ D 20−)

Show units on the two axes.

y

θ1

y

θ

540°450°360°270°180°90°−90°

1

cot θ =cos θ

sin θ

y

θ

540°450°360°270°180°90°−90°1

sec θ = 1cos θ

Name: Date:

Test 7, Sections 3-1 to 3-3 Form AObjective: Draw graphs of sinusoids and of tangent and secant functions.

(Hand in this page to get the rest of the test.)

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Part 2: Graphing calculators allowed(10–24)10. For y H D7 C 3 cos 5(θ D 13−), identify the:

• Vertical dilation

• Vertical translation

• Horizontal dilation

• Horizontal translation

11. For the sinusoid in Problem 10, identify the:

• Amplitude

• Period

• Phase displacement (for cosine)

• Sinusoidal axis location

12. The graph shows a half-cycle of a sinusoid. Sketch atleast one complete cycle of the sinusoid.

13. Write a particular equation of the sinusoid inProblem 12 using the cosine function.

14. For the sinusoid in Problem 12, write anotherparticular equation using the sine function.

15. If the sinusoid in Problem 12 were extended to θ H 5461−, what would y equal?

16. For y H tan θ, in the figure,

• Darken one complete cycle.

• Draw all vertical asymptotes in this window.

• Write the θ-value at the rightmost θ-intercept.

• Mark a point where the graph is concave down.

17. For y H csc θ, in the figure,

• Darken one complete cycle.

• Draw all vertical asymptotes in this window.

• Mark a critical point.

• Give the y-value at the critical point you marked.

y

θ

y

θ63°

3

−1 43°

18. The tangent function can have a vertical dilation,such as

y H 3 tan θ

But the tangent function does not have an amplitude.Explain why not.

Ferris Wheel Problem (19–23): Ella Vader (Darth’s otherdaughter) rides a Ferris wheel. As the wheel turns, herdistance, y, from the ground, measured in feet, variessinusoidally with time, t, measured in seconds, accordingto the equation

y H 25 C 20 sin 18(t C 3)

Angle θ, in degrees, shown in the figure, is equal to theargument of the sine function, θ H 18(t C 3).

19. The last seat was filled, and the Ferris wheel startedrotating at time t H 0 s. What was θ at this time? Howhigh above the ground was Ella at this time?

20. When t H 1 s, what was θ? By how many degrees didthe Ferris wheel rotate in this 1 s? Where do you findthis number of degrees per second in the equationfor y ?

21. How high above the ground was Ella at t H 1 s?

22. What is the period of the sinusoid? That is, how longdoes it take for the Ferris wheel to make a complete360− revolution?

23. Sketch two cycles of the sinusoid of y as a functionof t. Put enough numbers on the axes so that yourinstructor can tell that you understand the meaningsof the 25, the 20, and the 3 in the equation and theperiod of the sinusoid.

24. What did you learn as a result of taking this test thatyou did not know before?

θ

Seat

Ground

Rotation

Ferris wheel

y

y

θ

34 Section, Chapter, and Cumulative Tests Precalculus with Trigonometry: Assessment Resources© 2007 Key Curriculum Press

Name: Date:

Test 7, Sections 3-1 to 3-3 continued Form A

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Part 1: No calculators allowed (1–8)1. Sketch the parent circular sinusoid y H sin x. Show

scales on the two axes.

2. Sketch the parent circular sinusoid y H cos x. Showscales on the two axes.

3. Find the number of degrees in radians.

4. Find the number of radians in 120−.

π3

5. Write the general solution for arccos 0.3 in terms of

6. The figure represents a circle of radius 10 ft.

• On the circle, darken an arc of length 13 ft.

• Sketch the central angle that subtends the arc.

• Give the radian measure of the central angle.

7. The graph shows a circular function sinusoid. Findthe:

Period

Horizontal dilation

Amplitude

Vertical translation

Phase displacement (for cosine)

8. Write a particular equation of the sinusoid inProblem 7 using the cosine function.

y

x105

10

5

cosD1 0.3.

Name: Date:

Test 8, Sections 3-4 to 3-8 Form AObjective: Convert between radians and degrees, and draw graphs of circular functions.


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Part 2: Graphing calculators allowed (9–18)For Problems 9–14, the figure shows a sinusoid.

9. Draw a line at y H 6. Estimate from the graph thethree values of x at y H 6 for the line you drew.

10. The particular equation of the sinusoid is

Plot this function and the line y H 6 on the samescreen. Use the intersect feature to find the first twovalues of x shown in the figure for which y H 6.Round to three decimal places.

11. Starting at the leftmost value of x in Problem 10,write the values of x that are larger by n H 1, 2, and3 periods. Show that the value of x for n H 1 agreeswith the graph.

12. Starting with the middle value of x in Problem 10,write the values of x that are larger by n H 1, 2, and3 periods.

13. Show how to use the technique of Problems 11 and12 to find the value of x just to the left of the y-axisfor which y H 6.

14. Solve the equation

algebraically, using the arccos relation, to find thefirst three values of x for which y H 6. Show that theanswers agree with the numerical ones you found inProblems 10, 11, and 12.

5 + 4 cos π

7(x D 2) = 6

y = 5 + 4 cos π

7(x D 2)

y

x252015105

10

5

Iridium Layer Problem (15–17): Sixty-five million years ago,a meteor is believed to have landed in the Gulf of Mexico.The materials thrown up into the atmosphere as a resultare believed to have been responsible for the extinction ofthe dinosaurs and other species. A thin layer of materialrich in the element iridium covered most of Earth. Sincethen, the iridium layer has been covered with othermaterials and warped into wavy shapes, parts have beeneroded away by rivers, and so forth. The figure shows apart of the iridium layer in the cliff on the left bank of ariver. From measurements on this layer, geologists figurethat the layer is sinusoidal, with equation

where x and y are in feet in the coordinates shown.

15. What are the x- and y-coordinates of the high pointto the left of the left vertical cliff?

16. What is the period of the sinusoid?

17. Geologists want to find the iridium layer in the faceof the vertical cliff on the right side of the river.Based on the sinusoidal model, how high above theriver should they look on this cliff face to find theiridium layer? How do you find this number?


y

Vertical cliff

Vertical cliff

Eroded awayIridium

layer

River x

637 feet

y = 60 + 20 cos π

110(x + 140)


Name: Date:

Test 8, Sections 3-4 to 3-8 continued Form A

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Part 1: No calculators allowed (1–11)1. The figure shows an x-axis drawn tangent to a unit

circle in a uv-coordinate system. If the number line iswrapped around the unit circle, show where thepoint x H 2.7 on the line maps onto the circle.

2. On the figure in Problem 1 sketch an angle of 2.7radians in standard position.

3. What steps are needed to find a decimalapproximation for the degree measure of an angle of 2.7 radians? In which quadrant will the angle terminate?

4. Find the exact radian measure of 30−.

5. Find the degree measure of radians.D2π3

u

1

1

v x

6. Draw a sketch illustrating the angle

7. is one value of arccos On your sketch inProblem 6, show another angle equal to arccosterminating in a different quadrant.

8. Write the general solution for arccos in terms of

9. Sketch the graph of the parent trigonometricfunction y H sin θ. Show scales on both axes.

10. Sketch the graph of the parent circular function y H cos x. Show scales on both axes.

11. Sketch the graph of y = 4 + 3 cos π8 (x D 5).

cosD1 D35 .

D35

D35

D35 .cosD1 D3

5

cosD1 D35 .

Name: Date:

Test 9, Chapter 3 Form AObjective: Analyze sinusoids.


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Part 2: Graphing calculators allowed(12–24)12. Draw a sketch representing a circle of radius 5 ft.

Sketch a central angle subtending an arc of 7 ft.What is the radian measure of this angle?

13. Find the degree measure of 3.5 radians.

14. Plot the graph of y H sec x using the window shownin this figure. Sketch the graph here, showing thevertical asymptotes.

15. Plot the graph of y H cot x on your grapher usingthe same window as in Problem 14. Sketch thegraph here.

16. Based on the relationship between cotangent andtangent, explain why the cotangent function hasvertical asymptotes where it does.

17. For this sinusoid, write the particular equationusing cosine.

18. Write the particular equation of the sinusoid inProblem 17 using the sine function.

y

x40302010

5

10

y

x4π2π

4

−4

y

x4π2π

4

−4

Earth and Mars Problem (19–23): The distance betweenEarth and Mars is a periodic function of time. Assumethat the function is sinusoidal and that its particularequation is

where x is in days after today and y is in millions of miles.The figure shows the graph of this function.

19. Identify the:

• Horizontal dilation

• Period

• Amplitude

• Phase displacement

• Vertical translation

20. Based on the equation, on what day will Earth andMars first be closest to each other? Show how youfind the answer.

21. There are intervals of time when Earth and Mars arewithin 100 million miles of each other. Estimate onthe given graph the first such time interval.

22. Find the time interval in Problem 21 numerically. Youmay use either the solver or the intersect feature onyour grapher.

23. Find the time interval in Problem 21 algebraically.Show that your answer agrees with Problem 22.


y

x

50

100

150

200

250

200015001000500

y = 141 + 93 cos π

390(x D 140)


Name: Date:

Test 9, Chapter 3 continued Form A

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Overview of Instructor’s Resource CD

The Instructor’s Resource CD that accompanies the Instructor’s Guide containsinstructor’s resources in electronic format. These resources are also available atwww.keypress.com/keyonline, where you can become a registered user of Precalculuswith Trigonometry: Concepts and Applications. All of these resources are intended foryou to use with students, or in many instances, for students to use on their own.

• Dynamic Precalculus Explorations are dynamic sketches that help students visualizekey concepts. They are all directly referenced in the Student Edition and can be usedby students on their own or by teachers as a visual aid during a lecture. They requireany Java-enabled browser.

• Presentation sketches are dynamic sketches that require The Geometer’s Sketchpadsoftware. They are specifically designed to use as teacher presentations and the lastpage of each sketch includes teaching notes. Presentation sketches are referenced inthe Instructor’s Guide.

• Calculator programs mentioned in the Student Edition and in the Explorations in theInstructor’s Resource Book are included and can be downloaded to TI-83 or TI-84graphing calculators using TI-Connect™ links.

• Data sets for problems in the Student Edition that include ten or more data points,as well as supplemental data sets mentioned in the Instructor’s Guide, are includedin TI List, Microsoft Excel, and Fathom formats.

• Sketchpad and Fathom files are provided for the technology activities in theInstructor’s Resource Book.

• All blackline masters, Exploration masters and solutions, and technology activitiesfrom the Instructor’s Resource Book are provided as PDF files.

• Both versions of all Section, Chapter, and Cumulative Tests from the AssessmentResources are provided as PDF files and also in Microsoft Word format, so that theymay be easily modified by teachers. Equations in the Word documents are set inMathType.



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Dynamic Precalculus Exploration

Sinusoidal Dilation

This exploration will help you study dilations of the sine function y = sin x. If you movea point outside of either sketch, click the Reload or Refresh button in your browserwindow.

Sketch

The following sketch shows a unit circle in the uv-coordinate system with anglesof measure x and 2x radians. The uv-coordinate system is superimposed on anxy-coordinate system with sinusoids y = sin x in red and y = sin 2x in blue.

Investigate

1. Drag the black point to change the value of x. What length on the unit circle does xcorrespond to? What length does 2x correspond to?

2. How do the (u, v) coordinates of each point correspond to the (x, y) coordinates ofpoints on the sinusoids?

3. Press the Show Lines button. Explain why the red and blue horizontal lines intersectthe sinusoids where they do. Does this confirm or change your answer to Problem 2?

4. Do the moving points on the sinusoids have the same value of x ?



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Sketch

The sketch below shows the graph of y = sin x in red and y = sin kx in blue, with slidersthat you can move to change the values of x and k.

Investigate

1. Move the sliders to change x and k, and observe how the sketch changes. How is theblue segment in the unit circle related to the red segment?

2. What happens to the period of the graph of y = sin kx as k increases? As k decreases?Explain.

3. What happens when k is negative? Can you explain why?



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Circular Function Presentation Sketch

The following teaching notes are included in the Notes page of the sketch.

Use the colored buttons to show or hide individual functions, and use the Animate Pointbuttons to move the point along the unit circle.

When you press a button to show a function, the following objects will appear:

• the value of the function evaluated at the given angle

• the graph of the function on the interval [Dπ, π], with a plotted point correspondingto the given angle

• a solid segment on the unit circle, whose length is equal to the value of the function.

This presentation sketch may be used to demonstrate graphs of circular functions, aswell as relationships between them, such as the reciprocal relation sec x = (1/cos x). Youmay choose to have any collection of functions showing. Use the same button forshowing and hiding a function.

The Show Scales button will display the controls for adjusting the x- and y-scales. Theunit circle has radius equal to the y-scale unit, for improved visualization of the functionvalues.


overview of solutions manual - prek...

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