Solution Manual for Trigonometry, 11th Edition

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SOLUTIONSMANUALTIMBRITTJackson State Community CollegeTRIGONOMETRY:AUNITCIRCLEAPPROACHELEVENTHEDITIONMichael SullivanChicago State University

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Table of ContentsPrefaceChapter 1Graphs and Functions1.1 The Distance and Midpoint Formulas......................................................................................... 11.2 Graphs of Equations in Two Variables; Circles........................................................................ 131.3 Functions and Their Graphs...................................................................................................... 371.4 Properties of Functions ............................................................................................................. 551.5 Library of Functions; Piecewise-defined Functions ................................................................. 701.6 Graphing Techniques: Transformations ................................................................................... 821.7 One-to-One Functions; Inverse Functions ................................................................................ 98Chapter Review.............................................................................................................................. 119Chapter Test................................................................................................................................... 129Chapter Projects............................................................................................................................. 132Chapter 2Trigonometric Functions2.1 Angles, Arc Length, and Circular Motion .............................................................................. 1352.2 Trigonometric Functions: Unit Circle Approach .................................................................... 1442.3 Properties of the Trigonometric Functions ............................................................................. 1622.4 Graphs of the Sine and Cosine Functions ............................................................................... 1762.5 Graphs of the Tangent, Cotangent, Cosecant, and Secant Functions...................................... 1962.6 Phase Shift; Sinusoidal Curve Fitting ..................................................................................... 206Chapter Review.............................................................................................................................. 217Chapter Test................................................................................................................................... 225Cumulative Review........................................................................................................................ 228Chapter Projects............................................................................................................................. 231Chapter 3Analytic Trigonometry3.1 The Inverse Sine, Cosine, and Tangent Functions.................................................................. 2343.2 The Inverse Trigonometric Functions (Continued) ................................................................ 2473.3 Trigonometric Equations ........................................................................................................ 2593.4 Trigonometric Identities ......................................................................................................... 2803.5 Sum and Difference Formulas ................................................................................................ 2923.6 Double-angle and Half-angle Formulas.................................................................................. 3173.7 Product-to-Sum and Sum-to-Product Formulas...................................................................... 343Chapter Review.............................................................................................................................. 356Chapter Test................................................................................................................................... 371Cumulative Review........................................................................................................................ 376Chapter Projects............................................................................................................................. 379Chapter 4Applications of Trigonometric Functions4.1 Right Triangle Trigonometry; Applications ........................................................................... 3834.2 The Law of Sines .................................................................................................................... 3974.3 The Law of Cosines ................................................................................................................ 4114.4 Area of a Triangle ................................................................................................................... 4244.5 Simple Harmonic Motion; Damped Motion; Combining Waves ........................................... 433Chapter Review.............................................................................................................................. 443Chapter Test................................................................................................................................... 449Cumulative Review........................................................................................................................ 453Chapter Projects............................................................................................................................. 456

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Chapter 5Polar Coordinates; Vectors5.1 Polar Coordinates.................................................................................................................... 4605.2 Polar Equations and Graphs.................................................................................................... 4695.3 The Complex Plane; De Moivre’s Theorem ........................................................................... 4985.4 Vectors.................................................................................................................................... 5115.5 The Dot Product...................................................................................................................... 5245.6 Vectors in Space ..................................................................................................................... 5305.7 The Cross Product................................................................................................................... 536Chapter Review.............................................................................................................................. 547Chapter Test................................................................................................................................... 556Cumulative Review........................................................................................................................ 560Chapter Projects............................................................................................................................. 562Chapter 6Analytic Geometry6.2 The Parabola ........................................................................................................................... 5666.3 The Ellipse .............................................................................................................................. 5816.4 The Hyperbola ........................................................................................................................ 5986.5 Rotation of Axes; General Form of a Conic ........................................................................... 6186.6 Polar Equations of Conics....................................................................................................... 6316.7 Plane Curves and Parametric Equations ................................................................................. 640Chapter Review.............................................................................................................................. 654Chapter Test................................................................................................................................... 664Cumulative Review........................................................................................................................ 668Chapter Projects............................................................................................................................. 670Chapter 7Exponential and Logarithmic Functions7.1 Exponential Functions ............................................................................................................ 6747.2 Logarithmic Functions............................................................................................................ 6957.3 Properties of Logarithms ........................................................................................................ 7177.4 Logarithmic and Exponential Equations................................................................................. 7267.5 Financial Models .................................................................................................................... 7457.6 Exponential Growth and Decay Models; Newton’s Law; Logistic Growthand Decay Models .................................................................................................................... 7537.7 Building Exponential, Logarithmic, and Logistic Models from Data..................................... 763Chapter Review.............................................................................................................................. 768Chapter Test................................................................................................................................... 777Cumulative Review........................................................................................................................ 780Chapter Projects............................................................................................................................. 782Appendix AReviewA.1 Algebra Essentials.................................................................................................................. 784A.2 Geometry Essentials............................................................................................................... 789A.3 Factoring Polynomials; Completing the Square..................................................................... 795A.4 Solving Equations .................................................................................................................. 799A.5 Complex Numbers; Quadratic Equations in the Complex Number System .......................... 813A.6 Interval Notation; Solving Inequalities .................................................................................. 818A.7nth Roots; Rational Exponents............................................................................................... 830A.8 Lines....................................................................................................................................... 840

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Appendix BGraphing UtilitiesB.1 The Viewing Rectangle.......................................................................................................... 857B.2 Using a Graphing Utility to Graph Equations ........................................................................ 858B.3 Using a Graphing Utility to Locate Intercepts and Check for Symmetry .............................. 863B.5 Square Screens ....................................................................................................................... 865

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1Chapter 1Graphs and FunctionsSection 1.11.02.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 orordinate8.quadrants9.midpoint10.False; the distance between two points is nevernegative.11.False; points that lie in quadrant IV will have apositivex-coordinate and a negativey-coordinate.The point1, 4lies in quadrant II.12.True;1212,22xxyyM 13.b14.a15.(a)Quadrant II(b)x-axis(c)Quadrant III(d)Quadrant I(e)y-axis(f)Quadrant IV16.(a)Quadrant I(b)Quadrant III(c)Quadrant II(d)Quadrant I(e)y-axis(f)x-axis17.The points will be on a vertical line that is twounits to the right of they-axis.

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Chapter 1:Graphs and Functions218.The points will be on a horizontal line that isthree units above thex-axis.19.221222(,)(20)(10)21415d P P20.221222(,)( 20)(10)( 2)1415d P P21.221222(,)( 21)(21)( 3)19110d P P22.221222(,)2( 1)(21)319110d P P 23. 221222(,)(53)4428464682 17d P P 24. 221222(,)214034916255d P P 25.221222(,)4( 7)(03)11(3)1219130  d P P26.221222(,)422( 3)2542529d P P 27.221222(,)(65)1( 2)131910 d P P28.221222(,)6(4)2( 3)1051002512555d P P  29.221222(,)2.3( 0.2)1.1(0.3)2.50.86.250.646.892.62 d P P30.221222(,)0.31.21.12.3( 1.5)( 1.2)2.251.443.691.92 d P P31.22122222(,)(0)(0)()()d P Pababab 32.221222222(,)(0)(0)()()22d P Paaaaaaaa 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       

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Section 1.1:The Distance and Midpoint Formulas3Verifying 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, 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 2100 square units2Ad 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:

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Chapter 1:Graphs and Functions4222222(,)(,)(,)1042613010426130130130d A Cd B Cd A BThe area of a triangle is12Abh. In thisproblem, 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(,)(,)(,)29229145294 2914529116145145145d A Cd B Cd A BThe area of a triangle is12Abh. In thisproblem, 1(,)(,)2129 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 1.1: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 this problem, 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( ,),221840,2274,227 , 22     xxyyx y

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Chapter 1:Graphs and Functions642.The coordinates of the midpoint are:1212( ,),222432,2261,2213,2xxyyx y   43.The coordinates of the midpoint are:1212( ,),227951,22416 ,22(8,2)   xxyyx y44.The coordinates of the midpoint are:1212( ,),224232,2221,2211,2xxyyx y   45.The coordinates of the midpoint are:1212( ,),2200,22,22xxyyx yabab   46.The coordinates of the midpoint are:1212( ,),2200,22,22xxyyx yaaaa   47.The x coordinate would be235and the ycoordinate would be523. Thus the newpoint would be5,3.48.The new x coordinate would be123  andthe new y coordinate would be6410. Thusthe new point would be3,1049.a.If we use a right triangle to solve theproblem, we know the hypotenuse is 13 units inlength. One of the legs of the triangle will be2+3=5. Thus the other leg will be:222225132516914412bbbbThus the coordinates will have an y value of11213  and11211 . So the pointsare3,11and3,13.b.Consider points of the form3,ythat are adistance of 13 units from the point2,1. 2221212222223( 2)1512512226dxxyyyyyyyy     22222213226132261692260214301113yyyyyyyyyy11011yyor13013yy Thus, the points3,11and3,13are adistance of 13 units from the point2,1.

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Section 1.1:The Distance and Midpoint Formulas750.a.If we use a right triangle to solve theproblem, we know the hypotenuse is 17 units inlength. One of the legs of the triangle will be2+6=8. Thus the other leg will be:222228176428922515bbbbThus the coordinates will have an x value of11514 and11516. So the points are14,6and16,6.b.Consider points of the form,6xthat area distance of 17 units from the point1, 2. 2221212222221262182164265dxxyyxxxxxxx 22222217265172652892650222401416xxxxxxxxxx14014xx or16016xxThus, the points14,6and16,6are adistance of 13 units from the point1, 2.51.Points on thex-axis have ay-coordinate of 0. Thus,we consider points of the form, 0xthat are adistance of 6 units from the point4,3.22212122222243016831689825dxxyyxxxxxxx  222222268256825368250811( 8)( 8)4(1)( 11)2(1)864448108228634332xxxxxxxxx 433xor433xThus, the points433, 0and433, 0areon thex-axis and a distance of 6 units from thepoint4,3.52.Points on they-axis have anx-coordinate of 0.Thus, we consider points of the form0,ythatare a distance of 6 units from the point4,3.2221212222224034961696625dxxyyyyyyyyy 

Solution Manual for Trigonometry, 11th Edition - Page 13 preview image

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Chapter 1:Graphs and Functions8222222266256625366250611( 6)(6)4(1)( 11)2(1)63644680226453252yyyyyyyyy 325y or325y Thus, the points0,325and0,325are on they-axis and a distance of 6 units from thepoint4,3.53.a.To shift 3 units left and 4 units down, wesubtract 3 from thex-coordinate and subtract4 from they-coordinate.23,541,1b.To shift left 2 units and up 8 units, wesubtract 2 from thex-coordinate and add 8 tothey-coordinate.22,580,1354.Let the coordinates of pointBbe,xy. Usingthe midpoint formula, we can write182,3,22xy  .This leads to two equations we can solve.122145xxx  832862yyy PointBhas coordinates5,2.55.1212,,22xxyyMx y .111,( 3, 6)Pxy and( ,)( 1, 4)x y , so122222312231xxxxxx and122222642862yyyyyyThus,2(1, 2)P.56.1212,,22xxyyMx y .222,(7,2)Pxyand( ,)(5,4)x y, so1211127521073xxxxxxand121112( 2)428( 2)6yyyyyy  Thus,1(3,6)P.57.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 2222(,)26(20)(4)21642025d B E2222(,)(20)(50)2542529d A F

Solution Manual for Trigonometry, 11th Edition - Page 14 preview image

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Section 1.1:The Distance and Midpoint Formulas958.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.59.221222(,)(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.60.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.61.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.

Solution Manual for Trigonometry, 11th Edition - Page 15 preview image

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Chapter 1:Graphs and Functions1062.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.63.Using the Pythagorean Theorem:222229090810081001620016200902127.28 feetdddd90909090d64.Using the Pythagorean Theorem:222226060360036007200720060284.85 feetdddd60606060d65.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 c.Using the distance formula:2222(3000)(30090)30021013410030 149366.20 feetd66.a.First: (60, 0), Second: (60, 60)Third: (0, 60)(0,0)(0,60)(60,0)(60,60)xyb.Using the distance formula:2222(18060)(2060)120(40)1600040 10126.49 feetd c.Using the distance formula:2222(2200)(22060)2201607400020 185272.03 feetd

Solution Manual for Trigonometry, 11th Edition - Page 16 preview image

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Section 1.1:The Distance and Midpoint Formulas1167.The Focus heading east moves a distance60tafterthours. The truck heading south moves adistance40tafterthours. Their distance apartafterthours is:22222(60 )(45 )36002025562575milesdtttttt68.15 miles5280 ft1 hr22 ft/sec1 hr1 mile3600 sec2221002210000484feetdtt10022td69.a.The shortest side is between1(2.6, 1.5)Pand2(2.7, 1.7)P. The estimate for thedesired intersection point is:12122.62.71.51.7,,22225.33.2,222.65, 1.6xxyy  b.Using the distance formula:2222(2.651.4)(1.61.3)(1.25)(0.3)1.56250.091.65251.285 unitsd70.Let1(2013, 102.87)Pand2(2017, 126.17)P. The midpoint is:1212,,2220132017102.87126.17,224030229.04,222015, 114.52   xxyyx yThe estimate for 2010 is $114.52 billion. Theestimate net sales of Costco WholesaleCorporation in 2015 is $0.85 billion off from thereported value of $113.67 billion.71.For 2009 we have the ordered pair2009, 21756and for 2017 we have the orderedpair2017, 24858. The midpoint is200920172175624858year, $,22402646614,222013, 23307  Using the midpoint, we estimate the povertylevel in 2013 to be $23,307. This is lower thanthe actual value.72.Let10, 0P,2, 0Pa, and33,22aaP . Then22122121222,000dP Pxxyyaaa22232121222222,302234444dPPxxyyaaaaaaaad45t60t

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