Overview Of Solutions Manual - High School Math
FPC2 CS B.qxd 1/10/06 3:22 AM Page 87Overview of Solutions ManualThe 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 inmind, though, that there may be more than one way to solve any given problem usinga correct method.As in the student text, exact answers are displayed using the ellipsis format. When realworld 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 MANUALSolutions are not provided for journal entries. Student responses are highly individualand will vary from student to student.vPRECALCULUS WITH TRIGONOMETRY COURSE SAMPLER87
FPC2 CS B.qxd 1/10/06 3:22 AM Page 8836. tanD1 5 1.3734 b.37. cotD1 3 0.3217 338. cscD1 1.001 1.5260 239. sinv2140. cos π D113π 3 32ur 1c. The arc length on the unit circle equals the radian measure.2. a. 1.3 m41. tanπ 3 6342. cotπ 0243. sec 2π 144. cscb. 2.6 m for r 2 m; 3.9 m for r 3 mc. 1.3r mSOLUTIONS MANUALd. a rθ3.60ππ 18034.π45π 18045.30ππ 18066.180π π1807.1202π π18038.4505π π1802D22559.π D π1804108010.π 6π18045. sinππ1 6 cos 1 6 423246. cscππsin 166π 2447. cos2 π sin2 π (D1)2 02 148. tan2ππD sec2 33( 3 )2 D 22 D149. y 5 7 cos 30(θ D 2 )50. y 5.5 0.5 cos 3617 (θ D 15 )51. x 17 sin 55 13.9255 cm11.37π 0.6457 18012.54π 0.9424 18013.123π 2.1467 18014.258π 4.5029 18053. θ cosD13 64.6230 754. θ tanD11 26.5650 215. 18 16. 90 17. 30 18. 45 19. 15 20. 120 Problem Set 3-521. 135 22. 180 Q1. πQ2. 360 23. 270 24. 150 180 Q3. 57.2957 πQ4.Q5. sin 47 0.7313 Q6. sin 47 0.1235 18025. 0.34 19.4805 π18026. 0.62 35.5233 πQ7. 72 Q9. 5 h34π 0.5934 180Q8. 7Q10. 5% 0.0527.180 1.26 72.1926 π1.πunits62.πunits328.180 1.57 89.9543 π3.πunits24.πunits429.180 57.2957 π30.180 3 171.8873 π31. sin 5 D0.9589 32. cos 2 D0.4161 33. tan(D2.3) 1.1192 34. sin 1066 D0.8415 35. sinD1 0.3 0.3046 Precalculus with Trigonometry: Solutions Manual 2007 Key Curriculum Press8852. x 100 sec 20 106.4177 cm5. 60 6. 30 7. 45 8. 90 π9.units210. π units11. 2 units12. 1.467 units13. tan 1 1.5574 14. sin 2 0.9092 Problem Set 3-531PRECALCULUS WITH TRIGONOMETRY COURSE SAMPLER
FPC2 CS B.qxd 1/10/06 3:22 AM Page 8915. sec 3 D1.0101 29. Period H 4Asymptotes at 4nPoints of inflection at 2 4n16. cot 4 0.8636 17. cosD1 0.3 1.2661 y18. tanD1 1.4 0.9505 419. cscD1 5 0.2013 x20. secD1 9 1.4594 21. sinπ 3 3222. cosπ 1 4 223. tanπ 1 6 3430. Period H12Asymptotes at1 1 n4 21Points of inflection at n224. csc π is undefined.y25. Period H 10Amplitude H 2Phase displacement H C4Sinusoidal axis H C34xySOLUTIONS MANUAL15x431. Period 2ππ nπ2Critical points at nπ, specifically ( 2nπ, 3) and (2n 1)π, 126. Period H 3Amplitude H 5Phase displacement H D1Sinusoidal axis H D4Asymptotes at()yyx24x2π–927. Period H 8Amplitude H 6Phase displacement H D1Sinusoidal axis H C232. Period 2πAsymptotes at nπ ππ 2nπ, 3 andCritical points at nπ, specifically22 π (2n 1)π, D32yyx2–43x2π28. Period H 6Amplitude H 4Phase displacement H C2Sinusoidal axis H C533. y 5 2 cos π3 (x D 1)y34. y 4 9 cos 10πxπ(x 5)35. y D2 5 cos 15x1236. y 0.25 0.05 cos π4 (x 1)37. y csc π6 x38. y cot π4 x32Problem Set 3-5PRECALCULUS WITH TRIGONOMETRY COURSE SAMPLERPrecalculus with Trigonometry: Solutions Manual 2007 Key Curriculum Press89
FPC2 CS B.qxd 1/10/06 3:22 AM Page 90Problem Set 3-639. y 3 tan x40. y D2 sec xQ1.41. z D8 2 sin 5π (t D 0.17)π42. E D2.4 7.2 cos 800(r D 100)43. z(0.4) D8 2 sin 5π (0.4 D 0.17) D8.9079 z(50) D8 2 sin 5π (50 D 0.17) D8.9079 z(50) is 0.9079 below the sinusoidal axis.Q6.1θθQ7. tanD190 3 23.1985 7Q8. Circle of radius 3 and center (0, 0)Q9. y abxQ10. PeriodicD11. cos 0.9 2πn 1.1592 , 5.1239 , 7.4424 , 11.4070 ,13.7256 d. A horizontal translation by a multiple of 2π results in agraph that coincides with itself. The period of the sinefunction is 2π.SOLUTIONS MANUALQ5.π490 c. 2π or D2π, or any multiple of 2π2. cosD1 0.4 2πn 0.4510 , 5.8321 , 6.7342 , 12.1153 ,13.0173 46. a. Since the length of the hypotenuse H the radius of thev1v1oppcircle H 1, y sin x v1,hyp radius of circle 1oppv2v2and y sin 2x v2.hyp radius of circle 13. cosD1(D0.2) 2πn 1.7721 , 4.5110 , 8.0553 ,10.7942 , 14.3385 4. cosD1(D0.5) 2πn 2.0943 , 4.1887 , 8.3775 ,10.4719 , 14.6607 b. Answers will vary. The second angle measure is doublethe first, but the moving points always have the samex-values.5. a. x M 1, 5, 21, 25πb. y 2 5 cos 10(x D 3)c. As k increases, the period decreases, and vice versa. Theperiod is always 2πk .c. x 0.9516 , 5.0483 , 20.9516 , 25.0483 6D210cosD1 2πnd. x 3 5πx 0.9516., 5.0483 , 20.9516 , 25.0483 104e. x 3 DcosD1 D 10π 100.9516 π547. 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 y cos x ory sin x.b. A radian measure corresponds to an angle measure, usingπmR(θ) m (θ) 180 , but because a radian measure is apure number, it can represent something other than anangle in an application problem.c.Q4.1b. Horizontal translation of 2π; the graph would coincidewith itself and appear unchanged.b. The description says that the circle is a unit circle.BC BC oppHence BC sin x , and1 OB hypAD AD oppAD tan x .1OA adjQ3. 30 yπ45. a. Horizontal translation of ;2 πsin x cos x D2arcx48. a. m ( AOB) xradius 1Q2. 90 yπ44. E(1234) D2.4 7.2 cos 800(1234 D 100) D4.2452 π(10,000 D 100) 0.3553 E(10,000) D2.4 7.2 cos 800E(10,000) is 2.7553 above the sinusoidal axis.Rπ26. a. x M D0.4, 4.4, 11.6, 16.4, 23.6, 28.4b. y 4 3 cos π6 (x D 2)c. x D0.3509 , 4.3509 , 11.6490 , 16.3509 , 23.6490 ,28.3509 5D46cosD1 2πnd. x 2 3πx D0.3509 , 4.3509 , 11.6490 , 16.3509 , 23.6490 ,28.3509 61e. x 2 cosD1 16π 100.3509 π37. a. x M D2.9, D0.5, 1.1, 3.5, 5.1xsin xtan x0.10.0998 0.1003 0.010.0099 0.0100 0.0010.0009 0.0010 sin xtan x 1, but approaches 1 as x approaches 0; 1,d.xxbut also approaches 1 as x approaches 0.b. y D2 4 cos π2 (x D 0.3)c. x D2.8608 , D0.5391 , 1.1391 , 3.4608 , 5.1391 D1 22 2πncosD14πx D2.8608 , D0.5391 , 1.1391 , 3.4608 , 5.1391 21e. x 0.3 cosD1 50π 101.1391 π4d. x 0.3 49. Journal entries will vary.Precalculus with Trigonometry: Solutions Manual 2007 Key Curriculum Press90Problem Set 3-633PRECALCULUS WITH TRIGONOMETRY COURSE SAMPLER
FPC2 CS B.qxd 1/11/06 8:00 PM Page 91Overview of Instructor’sResource BookINSTRUCTOR’S RESOURCE BOOKPrecalculus 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 problemsvPRECALCULUS WITH TRIGONOMETRY COURSE SAMPLER91
FPC2 CS B.qxd 1/10/06 3:22 AM Page 92Name:Group Members:Trigonometric Ratios TableDate: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 Cosecant0ºINSTRUCTOR’S RESOURCE 240º270º300º315º330º360ºPrecalculus with Trigonometry: Instructor’s Resource Book 2007 Key Curriculum Press92Blackline Masters 15PRECALCULUS WITH TRIGONOMETRY COURSE SAMPLER
FPC2 CS B.qxd 1/10/06 3:22 AM Page 93Name:Group Members:Problem Set 3-2/Pages 103–1055.Date:9.yy2.56153 70 25 θ20 65 110 155 200 2 INSTRUCTOR’S RESOURCE BOOK6.θ0.34 16 10.yy18504 θ44 16 6 14 24 34 1054 θ0.3 27.5.3 11.yy25000θ10 70 θ 33 7 3 7 50008.12.yyθ 2 50008 θ 20 4016 30Blackline MastersPRECALCULUS WITH TRIGONOMETRY COURSE SAMPLER 5000Precalculus with Trigonometry: Instructor’s Resource Book 2007 Key Curriculum Press93
FPC2 CS B.qxd 1/10/06 3:22 AM Page 94Name:Group Members:Exploration 3-1a: Periodic Daily TemperaturesDate:Objective: Transform the cosine function so that it fits, approximately, data on the averagedaily temperatures for a city.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.INSTRUCTOR’S RESOURCE BOOKMonth Temperature ( F)Month Temperature ( 3Apr.80.3Oct.81.5May85.6Nov.70.7June91.8Dec.64.64. The temperature graph in Problem 1 has a high pointat x 7 months. What transformation would youapply to the sinusoid in Problem 2 (dashed in thenext figure) to make it have a high point at θ 7 (solid) instead of at θ 0 ? Write the equation andconfirm it by plotting it on your grapher.y1θ12 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. 124 7 5. The average of the highest and lowest temperaturesin the table is 94.9 2 61.7 78.3. Write an equation forthe transformation that would translate the graph inProblem 4 upward by 78.3 units.y10090Temperature ( F)8070605040302010x612Months18242. The graph of y cos θ 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.y1θ12 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.y94.978.361.7θ7 7. On your grapher, plot the points you plotted inProblem 1. How well does the sinusoidal equation inProblem 6 fit the points?24 13. Earth rotates 360 around the Sun in 12 months.How do these numbers relate to the dilation factoryou used in Problem 2?6894Exploration Masters8. What did you learn as a result of doing thisExploration that you did not know before?Precalculus with Trigonometry: Instructor’s Resource Book 2007 Key Curriculum PressPRECALCULUS WITH TRIGONOMETRY COURSE SAMPLER
FPC2 CS B.qxd 1/10/06 3:22 AM Page 95Name:Group Members:Exploration 3-1b: Sine and Cosine Graphs, ManuallyDate:Objective: Find the shape of sine and cosine graphs by plotting them on graph paper.1. On your grapher, make a table of values of y sin θ for each 10 from 0 to 90 . Set themode to round to 2 decimal places. Plot the values on this graph paper. Also plot y sin θfor each 90 through 720 . Connect the points with a smooth curve, observing the shapeyou plotted for 0 to 90 .y1180 270 360 450 540 630 θ720 540 630 θ720 INSTRUCTOR’S RESOURCE BOOK90 12. Plot the graph of y cos θ pointwise, the way you did for sine in Problem 1.y190 180 270 360 450 13. Find sin 45 and cos 65 . Show that thecorresponding points are on the graphs inProblems 1 and 2, respectively.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.4. Find the inverse trigonometric functionsθ sinD1 0.4 and θ cosD1 0.8. Show that thecorresponding points are on the graphs inProblems 1 and 2, respectively.Precalculus with Trigonometry: Instructor’s Resource Book 2007 Key Curriculum PressPRECALCULUS WITH TRIGONOMETRY COURSE SAMPLER7. What did you learn as a result of doing thisExploration that you did not know before?Exploration Masters 6995
FPC2 CS B.qxd 1/10/06 3:22 AM Page 96Name:Group Members:Exploration 3-2a: Transformed Sinusoid GraphsDate:Objective: Given the equation of a transformed sinusoid, sketch the graph, and vice versa.1. Write the horizontal dilation factor, period,amplitude, phase displacement, and verticaldisplacement, and sketch the graph.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.y 4 3 cos 2(θ D 70 )Horizontal dilation factor:Period:Period:INSTRUCTOR’S RESOURCE BOOKAmplitude:Horizontal dilation factor:Phase displacement:Amplitude:Vertical displacement:Phase displacement:yVertical displacement:Equation:yθ2θ10 2. Write the horizontal dilation factor, period,amplitude, phase displacement, and verticaldisplacement, and sketch the graph. 8y D2 4 sin 30(θ 1 )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?Horizontal dilation factor:Period:Amplitude:5. By the most time-efficient method possible, find yfor your equation in Problem 3 if θ 35 . Write theanswer to as many decimal places as your grapherwill give. Draw something on the given graph toshow that your answer is reasonable.Phase displacement:Vertical displacement:yθ7096Exploration Masters6. What did you learn as a result of doing thisExploration that you did not know before?Precalculus with Trigonometry: Instructor’s Resource Book 2007 Key Curriculum PressPRECALCULUS WITH TRIGONOMETRY COURSE SAMPLER
FPC2 CS B.qxd 1/10/06 3:22 AM Page 97Name:Group Members:Exploration 3-2b: Sinusoidal Equations from GraphsDate:Objective: Given the equation, sketch the sinusoid, and vice versa.5. This is a half-cycle of a sinusoid. Write a particularequation.1. Sketch two cycles of this sinusoid:y D3 5 sin 4(θ D 20 )yy4INSTRUCTOR’S RESOURCE BOOKθθ400 600 32. Write a particular equation (cos) for this sinusoid:6. This is a quarter-cycle of a sinusoid. Write aparticular equation.y100yθ24 10 3 θ36 6012 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? 1007. On the sinusoid in Problem 2, mark a point ofinflection. Mark another point at which the graph isincreasing but concave down.8. What did you learn as a result of doing thisExploration that you did not know before?Precalculus with Trigonometry: Instructor’s Resource Book 2007 Key Curriculum PressPRECALCULUS WITH TRIGONOMETRY COURSE SAMPLERExploration Masters 7197
FPC2 CS B.qxd 1/10/06 3:22 AM Page 98Name:Group Members:Exploration 3-3a: Tangent and Secant GraphsDate:Objective: Discover what the tangent and secant function graphs look like and how they relateto sine and cosine.No graphers allowed for Problems 1–7.4. Based on the quotient property, find out where theθ-intercepts are for the graph of y tan θ. Markthese intercepts on the figure in Problem 3.1. The reciprocal property states that1sec θ cos θINSTRUCTOR’S RESOURCE BOOKWithout your grapher, use this property to sketchthe graph of y sec θ on the same axes as the graphof the parent function y cos θ. In particular, showwhat happens to the secant graph wherevercos θ 0.5. At θ 45 , sin θ and cos θ are equal. Based on thisfact, what does tan 45 equal? Mark this point on thegraph in Problem 3. Mark all other points where sin θ cos θ .tan 45 H6. Use the points and asymptotes you have marked tosketch the graph of y tan θ on the figure inProblem 3. (No graphers allowed!)y7. Check your graphs with your instructor.Graphers allowed for the remaining problems.1θ 90 90 180 270 360 450 540 8. On your grapher, plot the graph of y csc θ. Sketchthe result here.9. On your grapher, plot the graph of y cot θ. Sketchthe result here.2. Write the quotient property expressing tan θ as aquotient of two other trigonometric functions.10. At what values of θ are the points of inflection fory tan θ? Explain why the tangent function has nocritical points.3. The next figure shows the parent functionsy sin θ and y cos θ. Based on the answer toProblem 2, determine where the asymptotes are forthe graph of y tan θ, and mark them on the figure.y11. Explain why the graph of y sec θ has no points ofinflection, even though the graph goes from concaveup to concave down at various places.1θ 90 90 180 270 360 450 540 12. What did you learn as a result of doing thisExploration that you did not know before?7298Exploration MastersPrecalculus with Trigonometry: Instructor’s Resource Book 2007 Key Curriculum PressPRECALCULUS WITH TRIGONOMETRY COURSE SAMPLER
FPC2 CS B.qxd 1/10/06 3:22 AM Page 99Name:Group Members:Exploration 3-3b: Transformed Tangentand Secant GraphsDate:Objective: Sketch transformed tangent, cotangent, secant, and cosecant graphs, and findequations from given graphs.1. For y 3 12 tan 5(θ D 7 ), state5. For y 1 3 csc 4(θ 10 ), giveThe hor
Overview of Solutions Manual The Precalculus with Trigonometry: Concepts and Applications Solutions Manualcontains one 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 .
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This solutions manual contains solutions for all odd numbered problems plus a large number of solutions for even numbered problems. Of the 624 exercises in Statistical Inference, Second Edition, this manual gives solutions for 484 (78%) of them. There is an obtuse pattern as to which solutions were included in this manual.
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Solutions Manual Fundamentals, Practice and Modeling Upper Saddle River, NJ. By Plummer, Deal and Griffin . Solutions Manual Fundamentals, Practice and Modeling Upper Saddle River, NJ. By Plummer, Deal and Griffin . Solutions Manual .
Susannah G Tringe*‡, Andreas Wagner† and Stephanie W Ruby* Addresses: *Department of Molecular Genetics and Microbiology, University of New Mexico Health Sciences Center, Albuquerque, NM 87131, USA. †Department of Biology, University of New Mexico, Albuquerque, NM 87131, USA. ‡Current address: DOE Joint Genome Institute, 2800
