Solution Manual for Precalculus: Concepts Through Functions, A Unit Circle Approach to Trigonometry, 4th Edition

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SOLUTIONSMANUALTIMBRITTJackson State Community CollegePRECALCULUS:CONCEPTSTHROUGHFUNCTIONS,AUNITCIRCLEAPPROACHTOTRIGONOMETRYFOURTH EDITIONMichael SullivanChicago State UniversityMichael Sullivan, IIIJoliet Junior College

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Table of ContentsChapter F: Foundations: A Prelude to functionsF.1 The Distance and Midpoint Formulas...........................................................................................1F.2 Graphs of Equations in Two Variables; Intercepts; Symmetry ..................................................12F.3 Lines............................................................................................................................................29F.4 Circles .........................................................................................................................................45Chapter Project ...................................................................................................................................56Chapter 1: Functions and Their Graphs1.1 Functions .....................................................................................................................................571.2 The Graph of a Function .............................................................................................................721.3 Properties of Functions................................................................................................................801.4 Library of Functions; Piecewise-defined Functions....................................................................971.5 Graphing Techniques: Transformations....................................................................................1091.6 Mathematical Models: Building Functions ...............................................................................1261.7 Building Mathematical Models Using Variation ......................................................................132Chapter Review ................................................................................................................................136Chapter Test .....................................................................................................................................143Chapter Projects ...............................................................................................................................147Chapter 2: Linear and Quadratic Functions2.1 Properties of Linear Functions and Linear Models ...................................................................1492.2 Building Linear Models from Data ...........................................................................................1612.3 Quadratic Functions and Their Zeros........................................................................................1662.4 Properties of Quadratic Functions.............................................................................................1852.5 Inequalities Involving Quadratic Functions ..............................................................................2072.6 Building Quadratic Models from Verbal Descriptions and from Data .....................................2262.7 Complex Zeros of a Quadratic Function ...................................................................................2342.8 Equations and Inequalities Involving the Absolute Value Function .........................................239Chapter Review ................................................................................................................................246Chapter Test .....................................................................................................................................258Cumulative Review ..........................................................................................................................262Chapter Projects ...............................................................................................................................265Chapter 3: Polynomial and Rational Functions3.1 Polynomial Functions and Models ............................................................................................2693.2 The Real Zeros of a Polynomial Function.................................................................................2933.3 Complex Zeros; Fundamental Theorem of Algebra..................................................................3233.4 Properties of Rational Functions ...............................................................................................3323.5 The Graph of a Rational Function .............................................................................................3413.6 Polynomial and Rational Inequalities........................................................................................398Chapter Review ................................................................................................................................421Chapter Test .....................................................................................................................................437Cumulative Review ..........................................................................................................................441Chapter Projects ...............................................................................................................................446

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Chapter 4: Exponential and Logarithmic Functions4.1 Composite Functions.................................................................................................................4484.2 One-to-One Functions; Inverse Functions ................................................................................4654.3 Exponential Functions...............................................................................................................4874.4 Logarithmic Functions ..............................................................................................................5074.5 Properties of Logarithms ...........................................................................................................5284.6 Logarithmic and Exponential Equations ...................................................................................5364.7 Financial Models .......................................................................................................................5564.8 Exponential Growth and Decay Models; Newton’s Law; Logistic Growthand Decay Models .............................................................................................................5644.9 Building Exponential, Logarithmic, and Logistic Models from Data.......................................573Chapter Review ................................................................................................................................579Chapter Test .....................................................................................................................................591Cumulative Review ..........................................................................................................................595Chapter Projects ...............................................................................................................................599Chapter 5: Trigonometric Functions5.1 Angles and Their Measure ........................................................................................................6025.2 Trigonometric Functions: Unit Circle Approach ......................................................................6105.3 Properties of the Trigonometric Functions................................................................................6285.4 Graphs of the Sine and Cosine Functions .................................................................................6415.5 Graphs of the Tangent, Cotangent, Cosecant, and Secant Functions........................................6615.6 Phase Shift; Sinusoidal Curve Fitting .......................................................................................670Chapter Review ................................................................................................................................684Chapter Test .....................................................................................................................................692Cumulative Review ..........................................................................................................................696Chapter Projects ...............................................................................................................................700Chapter 6: Analytic Trigonometry6.1 The Inverse Sine, Cosine, and Tangent Functions ....................................................................7046.2 The Inverse Trigonometric Functions (Continued)...................................................................7166.3 Trigonometric Equations...........................................................................................................7286.4 Trigonometric Identities ............................................................................................................7476.5 Sum and Difference Formulas...................................................................................................7606.6 Double-angle and Half-angle Formulas ....................................................................................7846.7 Product-to-Sum and Sum-to-Product Formulas ........................................................................812Chapter Review ................................................................................................................................821Chapter Test .....................................................................................................................................836Cumulative Review ..........................................................................................................................841Chapter Projects ...............................................................................................................................846

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Table of ContentsChapter 7: Applications of Trigonometric Functions7.1 Right Triangle Trigonometry; Applications..............................................................................8507.2 The Law of Sines.......................................................................................................................8627.3 The Law of Cosines...................................................................................................................8767.4 Area of a Triangle......................................................................................................................8897.5 Simple Harmonic Motion; Damped Motion; Combining Waves .............................................898Chapter Review ................................................................................................................................907Chapter Test .....................................................................................................................................914Cumulative Review ..........................................................................................................................917Chapter Projects ...............................................................................................................................924Chapter 8: Polar Coordinates; Vectors8.1 Polar Coordinates ......................................................................................................................9288.2 Polar Equations and Graphs ......................................................................................................9358.3 The Complex Plane; DeMoivre’s Theorem ..............................................................................9648.4 Vectors.......................................................................................................................................9758.5 The Dot Product ........................................................................................................................9878.6 Vectors in Space........................................................................................................................9938.7 The Cross Product .....................................................................................................................999Chapter Review ..............................................................................................................................1009Chapter Test ...................................................................................................................................1018Cumulative Review ........................................................................................................................1022Chapter Projects .............................................................................................................................1024Chapter 9: Analytic Geometry9.2 The Parabola............................................................................................................................10289.3 The Ellipse...............................................................................................................................10439.4 The Hyperbola.........................................................................................................................10599.5 Rotation of Axes; General Form of a Conic............................................................................10789.6 Polar Equations of Conics .......................................................................................................10909.7 Plane Curves and Parametric Equations..................................................................................1097Chapter Review ..............................................................................................................................1110Chapter Test ...................................................................................................................................1119Cumulative Review ........................................................................................................................1124Chapter Projects .............................................................................................................................1126Chapter 10: Systems of Equations and Inequalities10.1 Systems of Linear Equations: Substitution and Elimination.................................................113010.2 Systems of Linear Equations: Matrices.................................................................................115110.3 Systems of Linear Equations: Determinants .........................................................................117510.4 Matrix Algebra ......................................................................................................................118910.5 Partial Fraction Decomposition.............................................................................................120710.6 Systems of Nonlinear Equations ...........................................................................................122410.7 Systems of Inequalities..........................................................................................................125210.8 Linear Programming..............................................................................................................1266Chapter Review ..............................................................................................................................1278Chapter Test ...................................................................................................................................1294Cumulative Review ........................................................................................................................1302Chapter Projects .............................................................................................................................1306

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Chapter 11: Sequence; Induction; the Binomial Theorem11.1 Sequences ..............................................................................................................................130811.2 Arithmetic Sequences............................................................................................................131711.3 Geometric Sequences; Geometric Series ..............................................................................132411.4 Mathematical Induction.........................................................................................................133511.5 The Binomial Theorem .........................................................................................................1343Chapter Review ..............................................................................................................................1348Chapter Test ...................................................................................................................................1353Cumulative Review ........................................................................................................................1356Chapter Projects .............................................................................................................................1359Chapter 12: Counting and Probability12.1 Counting ................................................................................................................................136112.2 Permutations and Combinations............................................................................................136312.3 Probability .............................................................................................................................1367Chapter Review ..............................................................................................................................1373Chapter Test ...................................................................................................................................1374Cumulative Review ........................................................................................................................1376Chapter Projects .............................................................................................................................1379Chapter 13: A Preview of Calculus: The Limit, Derivative, and Integral of a Function13.1 Finding Limits Using Tables and Graphs..............................................................................138213.2 Algebra Techniques for Finding Limits ................................................................................138813.3 One-sided Limits; Continuous Functions..............................................................................139313.4 The Tangent Problem; The Derivative ..................................................................................140013.5 The Area Problem; The Integral............................................................................................1410Chapter Review ..............................................................................................................................1424Chapter Test ...................................................................................................................................1431Chapter Projects .............................................................................................................................1434Appendix A: ReviewA.1 Algebra Essentials ..................................................................................................................1440A.2 Geometry Essentials ...............................................................................................................1445A.3 Polynomials ............................................................................................................................1450A.4 Factoring Polynomials............................................................................................................1457A.5 Synthetic Division ..................................................................................................................1462A.6 Rational Expressions ..............................................................................................................1464A.7nth Roots; Rational Exponents ...............................................................................................1473A.8 Solving Equations...................................................................................................................1481A.9 Problem Solving: Interest, Mixture, Uniform Motion, Constant Rate Job Applications .......1491A.10 Interval Notation; Solving Inequalities ................................................................................1499A.11 Complex Numbers................................................................................................................1509Appendix B: Graphing UtilitiesB.1 The Viewing Rectangle ..........................................................................................................1515B.2 Using a Graphing Utility to Graph Equations ........................................................................1516B.3 Using a Graphing Utility to Locate Intercepts and Check for Symmetry ..............................1520B.5 Square Screens........................................................................................................................1522

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1Chapter FFoundations: A Prelude to FunctionsSection F.11.02.5388 3.22342554.22211601213600372161Since the sum of the squares of two of the sidesof the triangle equals the square of the third side,the triangle is a right triangle.5.12bh6.True7.x-coordinate, or abscissa; y-coordinate, orordinate.8.quadrants9.midpoint10.False. Distance is always a positive number.11.False. The point would lie in quadrant II.12.True13.b14.a15.(a)Quadrant II(b)Positive x-axis(c)Quadrant III(d)Quadrant I(e)Negative y-axis(f)Quadrant IV16.(a)Quadrant I(b)Quadrant III(c)Quadrant II(d)Quadrant I(e)Positive y-axis(f)Negative x-axis17.The points will be on a vertical line that is twounits to the right of the y-axis.18.The points will be on a horizontal line that isthree units above the x-axis.

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Chapter FFoundations: A Prelude to Functions219.2212(,)(20)(10)415d P P20.2212(,)(20)(10)415d P P21.2212(,)(21)(21)9110d P P22.2212(,)2( 1)(21)9110d P P 23. 222212(,)(53)4428464682 17d P P 24. 222212(,)214034916255d P P 25.221222(,)6( 3)(02)9(2)81485d P P  26.222212(,)422( 3)2542529d P P 27.221222(,)( 24)5( 2)( 6)( 3)3694535d P P   28.221222(,)6(4)2( 3)1051002512555d P P  29.221222(,)2.3(0.2(1.10.3)(2.5)(0.8)6.250.646.892.62d P P 30.221222(,)0.31.2(1.12.3)( 1.5)( 1.2)2.251.443.691.92d P P 31.222212(,)(0)(0)d P Pabab32.2222122(,)(0)(0)22d P Paaaaaa33.(2,5),(1,3),( 1, 0)ABC  222222222222(,)1(2)(35)3(2)9413(,)11(03)(2)(3)4913(,)1(2)(05)1(5)12526d A Bd B Cd A C       Verifying that ∆ ABC is a right triangle by thePythagorean Theorem:222222(,)(,)(,)1313261313262626d A Bd B Cd A CThe area of a triangle is12Abh. In thisproblem,

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Section F.1:The Distance and Midpoint Formulas3 1(,)(,)2111313132213 square units2Ad A Bd B C34.(2, 5),(12, 3),(10,11)ABC 222222222222(,)12(2)(35)14(2)1964200102(,)1012( 113)(2)( 14)4196200102(,)10(2)( 115)12(16)14425640020d A Bd B Cd A C       Verifying that ∆ ABC is a right triangle by thePythagorean Theorem:222222(,)(,)(,)10210220200200400400400d A Bd B Cd A CThe area of a triangle is12Abh. In thisproblem, 1(,)(,)21 102 10221 100 22100 square unitsAd A Bd B C35.(5,3),(6, 0),(5,5)ABC 222222222222(,)6(5)(03)11(3)1219130(,)56(50)(1)512526(,)5(5)(53)1021004104226d A Bd B Cd A C   Verifying that ∆ ABC is a right triangle by thePythagorean Theorem:222222(,)(,)(,)1042613010426130130130d A Cd B Cd A BThe area of a triangle is12Abh. In this

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Chapter FFoundations: A Prelude to Functions4problem, 1(,)(,)211042621 2262621 2 26226 square unitsAd A Cd B C36.(6, 3),(3,5),( 1, 5)ABC  222222222222(,)3(6)( 53)9(8)8164145(,)13(5( 5))(4)1016100116229(,)1(6)(53)5225429d A Bd B Cd A C       Verifying that ∆ ABC is a right triangle by thePythagorean Theorem:222222(,)(,)(,)2911614529116145145145d A Cd B Cd A BThe area of a triangle is12Abh. In thisproblem, 1(,)(,)21291162129 22921 2 29229 square unitsAd A Cd B C37.(4,3),(0,3),(4, 2)ABC222222222222(,)(04)3( 3)(4)0160164(,)402( 3)45162541(,)(44)2( 3)05025255d A Bd B Cd A C    Verifying that ∆ ABC is a right triangle by thePythagorean Theorem:

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Section F.1:The Distance and Midpoint Formulas5222222(,)(,)(,)45411625414141d A Bd A Cd B CThe area of a triangle is12Abh. In thisproblem, 1(,)(,)21 4 5210 square unitsAd A Bd A C38.(4,3),(4, 1),(2, 1)ABC222222222222(,)(44)1( 3)04016164(,)2411(2)04042(,)(24)1( 3)(2)44162025d A Bd B Cd A C  Verifying that ∆ ABC is a right triangle by thePythagorean Theorem:222222(,)(,)(,)4225164202020d A Bd B Cd A CThe area of a triangle is12Abh. In thisproblem, 1(,)(,)21 4 224 square unitsAd A Bd B C39.The coordinates of the midpoint are:1212( ,),224435 ,2280,22(4, 0)xxyyx y   40.The coordinates of the midpoint are:1212( ,),222204,2204,220, 2xxyyx y   41.The coordinates of the midpoint are:1212( ,),223620,2232,223 ,12xxyyx y    

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Chapter FFoundations: A Prelude to Functions642.The coordinates of the midpoint are:1212( ,),222432,2261,2213,2xxyyx y   43.The coordinates of the midpoint are:1212( ,),224( 2)2( 5),2227,2271, 2xxyyx y      44.The coordinates of the midpoint are:1212( ,),224232,2221,2211,2xxyyx y   45.The coordinates of the midpoint are:1212( ,),220.22.3 0.31.1,222.1 1.4,22(1.05, 0.7)xxyyx y   46.The coordinates of the midpoint are:1212( ,),221.2(0.3)2.31.1,220.93.4,220.45, 1.7xxyyx y    47.The coordinates of the midpoint are:1212( ,),2200,22,22xxyyx yabab   48.The coordinates of the midpoint are:1212( ,),2200,22,22xxyyx yaaaa   49.Consider points of the form2,ythat are adistance of 5 units from the point2,1.222121222222221411612217dxxyyyyyyyy    2222225217521725217028042yyyyyyyyyy404yy or202yy

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Section F.1:The Distance and Midpoint Formulas7Thus, the points2,4and2, 2are adistance of 5 units from the point2,1.50.Consider points of the form,3xthat are adistance of 13 units from the point1, 2. 2221212222221232152125226dxxyyxxxxxxx 22222213226132261692260214301311xxxxxxxxxx13013xxor11011xx Thus, the points13,3and11,3are adistance of 13 units from the point1, 2.51.Points on the x-axis have a y-coordinate of 0.Thus, we consider points of the form, 0xthatare a distance of 5 units from the point4,3.22212122222243016831689825dxxyyxxxxxxx  22222258255825258250808xxxxxxxxx x0xor808xxThus, the points0, 0and8, 0are on the x-axis and a distance of 5 units from the point4,3.52.Points on the y-axis have an x-coordinate of 0.Thus, we consider points of the form0,ythatare a distance of 5 units from the point4, 4.222121222222404416816168832dxxyyyyyyyyy2222225832583225832087071yyyyyyyyyy707yyor101yyThus, the points0, 7and0,1are on the y-axis and a distance of 5 units from the point4, 4.53.The midpoint of AB is:0600,223, 0D The midpoint of AC is:0404,222, 2E The midpoint of BC is:6404,225, 2F 2222(,)04(34)(4)(1)16117d C D 

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Chapter FFoundations: A Prelude to Functions82222(,)26(20)(4)21642025d B E2222(,)(20)(50)2542529d A F54.Let12(0, 0),(0, 4),( ,)PPPx y221222122222222222,(00)(40)164,(0)(0)416,(0)(4)(4)4(4)16dPPdPPxyxyxydPPxyxyxyTherefore,222248168162yyyyyyywhich gives2222161223xxx Two triangles are possible. The third vertex is23, 2or23, 2.55.Let10, 0P,20,Ps,3, 0Ps, and4,Ps s.(0,s)(s,0)(0,0)(s,s)XYThe points1Pand4Pare endpoints of onediagonal and the points2Pand3Pare theendpoints of the other diagonal.1,400,,2222ssssM2,300,,2222ssssMThe midpoints of the diagonals are the same.Therefore, the diagonals of a square intersect attheir midpoints.56.Let10, 0P,2, 0Pa, and33,22aaP . To show that these verticesform an equilateral triangle, we need to showthat the distance between any pair of points is thesame constant value.22122121222,000dP Pxxyyaaa22232121222222,302234444dPPxxyyaaaaaaaa22132121222222,3002234444dP PxxyyaaaaaaaSince all three distances have the same constantvalue, the triangle is an equilateral triangle.Now find the midpoints:

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Section F.1:The Distance and Midpoint Formulas91 22 31 3000,, 02223330,22,4422300322,,2244P PP PP PaaDMaaaaaEMaaaaFM  22222233,0424344316162aaadD Eaaaaa 2222223,0424344316162aaadD Faaaaa 22222333,44440242aaaadE FaaaSince the sides are the same length, the triangleis equilateral.57.221222(,)(42)(11)(6)0366d P P222322(,)4(4)( 31)0(4)164d PP   221322(,)(42)( 31)(6)(4)3616522 13d P P  Since222122313(,)(,)(,)d P Pd PPd P P,the triangle is a right triangle.58.221222(,)6( 1)(24)7(2)49453d P P  222322(,)46( 52)(2)(7)44953d PP  221322(,)4( 1)( 54)5(9)2581106d P P   Since222122313(,)(,)(,)d P Pd PPd P P,the triangle is a right triangle.Since1223,,dPPdPP, the triangle isisosceles.Therefore, the triangle is an isosceles righttriangle.

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Chapter FFoundations: A Prelude to Functions1059.221222(,)0(2)7( 1)28464682 17d P P  222322(,)30(27)3(5)92534d PP 221322(,)3( 2)2( 1)5325934d P P  Since2313(,)(,)d PPd P P, the triangle isisosceles.Since222132312(,)(,)(,)d P Pd PPd P P,the triangle is also a right triangle.Therefore, the triangle is an isosceles righttriangle.60.221222(,)4702( 11)(2)121412555d P P 222322(,)4(4)(60)86643610010d PP 221322(,)4762( 3)4916255d P PSince222132312(,)(,)(,)d P Pd PPd P P,the triangle is a right triangle.61.Using the Pythagorean Theorem:222229090810081001620016200902127.28 feetdddd90909090d62.Using the Pythagorean Theorem:222226060360036007200720060284.85 feetdddd60606060d63.a.First: (90, 0), Second: (90, 90)Third: (0, 90)(0,0)(0,90)(90,0)(90,90)XYb.Using the distance formula:2222(31090)(1590)220( 75)5402552161232.43 feetd 

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