Solution Manual for College Algebra and Trigonometry, 7th Edition

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SOLUTIONSMANUALBEVERLYFUSFIELDCOLLEGEALGEBRA&TRIGONOMETRYSEVENTHEDITIONandPRECALCULUSSEVENTHEDITIONMargaret L. LialAmerican River CollegeJohn HornsbyUniversity of New OrleansDavid I. SchneiderUniversity of MarylandCallie J. DanielsSt. Charles Community College

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CONTENTSRReview of Basic ConceptsR.1 Fractions, Decimals, and Percents..............................................................................1R.2 Sets and Real Numbers ............................................................................................12R.3 Real Number Operations and Properties..................................................................16R.4 Integer and Rational Exponents ...............................................................................23R.5 Polynomials..............................................................................................................28R.6 Factoring Polynomials..............................................................................................36R.7 Rational Expressions ................................................................................................45R.8 Radical Expressions .................................................................................................54Chapter R Review Exercises...........................................................................................61Chapter R Test ................................................................................................................671Equations and Inequalities1.1 Linear Equations .......................................................................................................691.2 Applications and Modeling with Linear Equations ..................................................731.3 Complex Numbers ....................................................................................................791.4 Quadratic Equations..................................................................................................85Chapter 1 Quiz (Sections 1.1−1.4)..................................................................................961.5 Applications and Modeling with Quadratic Equations.............................................961.6 Other Types of Equations and Applications ...........................................................106Summary Exercises on Solving Equations ...................................................................1271.7 Inequalities..............................................................................................................1311.8 Absolute Value Equations and Inequalities ............................................................157Chapter 1 Review Exercises .........................................................................................164Chapter 1 Test ...............................................................................................................1822Graphs and Functions2.1 Rectangular Coordinates and Graphs......................................................................1872.2 Circles .....................................................................................................................1962.3 Functions.................................................................................................................2052.4 Linear Functions .....................................................................................................212Chapter 2 Quiz (Sections 2.1−2.4)................................................................................2212.5 Equations of Lines and Linear Models ...................................................................221Summary Exercises on Graphs, Circles, Functions, and Equations .............................2302.6 Graphs of Basic Functions ......................................................................................2342.7 Graphing Techniques ..............................................................................................241Chapter 2 Quiz (Sections 2.5−2.7)................................................................................2552.8 Function Operations and Composition ...................................................................257Chapter 2 Review Exercises .........................................................................................267Chapter 2 Test ...............................................................................................................278

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3Polynomial and Rational Functions3.1 Quadratic Function and Models..............................................................................2833.2 Synthetic Division...................................................................................................3053.3 Zeros of Polynomial Functions...............................................................................3113.4 Polynomial Functions: Graphs, Applications, and Models ....................................333Summary Exercises on Polynomial Functions, Zeros, and Graphs ..............................3533.5 Rational Functions: Graphs, Applications, and Models .........................................367Chapter 3 Quiz (Sections 3.1−3.5)................................................................................3903.6 Polynomial and Rational Inequalities .....................................................................395Summary Exercises on Solving Equations and Inequalities.........................................4123.7 Variation .................................................................................................................420Chapter 3 Review Exercises .........................................................................................426Chapter 3 Test ...............................................................................................................4474Inverse, Exponential, and Logarithmic Functions4.1 Inverse Functions ....................................................................................................4544.2 Exponential Functions ............................................................................................4704.3 Logarithmic Functions ............................................................................................487Summary Exercises on Inverse, Exponential, and Logarithmic Functions ..................4994.4 Evaluating Logarithms and the Change-of-Base Theorem.....................................502Chapter 4 Quiz (Sections 4.1−4.4)................................................................................5084.5 Exponential and Logarithmic Equations.................................................................5094.6 Applications and Models of Exponential Growth and Decay ................................523Summary Exercises on Functions: Domains and Defining Equations .........................531Chapter 4 Review Exercises .........................................................................................535Chapter 4 Test ...............................................................................................................5425Trigonometric Functions5.1 Angles .....................................................................................................................5465.2 Trigonometric Functions.........................................................................................5535.3 Trigonometric Function Values and Angle Measures ............................................569Chapter 5 Quiz (Sections 5.1–5.3)................................................................................5845.4 Solutions and Applications of Right Triangles.......................................................585Chapter 5 Review Exercises .........................................................................................603Chapter 5 Test ...............................................................................................................611

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6The Circular Functions and Their Graphs6.1 Radian Measure ......................................................................................................6166.2 The Unit Circle and Circular Functions..................................................................6266.3 Graphs of the Sine and Cosine Functions ...............................................................6376.4 Translations of the Graphs of the Sine and Cosine Functions ................................648Chapter 6 Quiz (Sections 6.1−6.4)................................................................................6626.5 Graphs of the Tangent and Cotangent Functions....................................................6646.6: Graphs of the Secant and Cosecant Functions.......................................................676Summary Exercises on Graphing Circular Functions...................................................6856.7 Harmonic Motion....................................................................................................688Chapter 6 Review Exercises .........................................................................................694Chapter 6 Test ...............................................................................................................7077Trigonometric Identities and Equations7.1 Fundamental Identities............................................................................................7137.2 Verifying Trigonometric Identities .........................................................................7217.3 Sum and Difference Identities ...............................................................................732Chapter 7 Quiz (Sections 7.1−7.3)................................................................................7467.4 Double-Angle and Half-Angle Identities................................................................748Summary Exercises on Verifying Trigonometric Identities .........................................7607.5 Inverse Circular Functions......................................................................................7657.6 Trigonometric Equations ........................................................................................777Chapter 7 Quiz (Sections 7.5–7.6)................................................................................7957.7 Equations Involving Inverse Trigonometric Functions ..........................................797Chapter 7 Review Exercises .........................................................................................806Chapter 7 Test ...............................................................................................................8258Applications of Trigonometry8.1 The Law of Sines ....................................................................................................8308.2 The Law of Cosines ................................................................................................843Chapter 8 Quiz (Sections 8.1−8.2)................................................................................8568.3 Geometrically Defined Vectors and Applications ..................................................8578.4 Algebraically Defined Vectors and the Dot Product ..............................................865Summary Exercises on Applications of Trigonometry and Vectors ............................8718.5 Trigonometric (Polar) Form of Complex Numbers; Products and Quotients.........8738.6 DeMoivre’s Theorem; Powers and Roots of Complex Numbers ...........................883Chapter 8 Quiz (Sections 8.3−8.6)................................................................................8968.7 Polar Equations and Graphs....................................................................................8978.8 Parametric Equations, Graphs, and Applications ...................................................912Chapter 8 Review Exercises .........................................................................................923Chapter 8 Test ...............................................................................................................936

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9Systems and Matrices9.1 Systems of Linear Equations ..................................................................................9429.2 Matrix Solution of Linear Systems .........................................................................9709.3 Determinant Solution of Linear Systems................................................................9949.4 Partial Fractions ....................................................................................................1018Chapter 9 Quiz (Sections 9.1−9.4)..............................................................................10339.5 Nonlinear Systems of Equations...........................................................................1037Summary Exercises on Systems of Equations ............................................................10529.6 Systems of Inequalities and Linear Programming ................................................10619.7 Properties of Matrices ...........................................................................................10799.8 Matrix Inverses .....................................................................................................1092Chapter 9 Review Exercises .......................................................................................1113Chapter 9 Test .............................................................................................................113410Analytic Geometry10.1 Parabolas .............................................................................................................114210.2 Ellipses................................................................................................................1154Chapter 10 Quiz (Sections 10.1−10.2)........................................................................116710.3 Hyperbolas ..........................................................................................................117010.4 Summary of the Conic Sections..........................................................................1185Chapter 10 Review Exercises .....................................................................................1193Chapter 10 Test ...........................................................................................................120311Further Topics in Algebra11.1 Sequences and Series ..........................................................................................120711.2 Arithmetic Sequences and Series........................................................................122111.3 Geometric Sequences and Series ........................................................................1230Summary Exercises on Sequences and Series ............................................................124211.4 The Binomial Theorem .......................................................................................124411.5 Mathematical Induction .....................................................................................1251Chapter 11 Quiz (Sections 11.1−11.5)........................................................................126511.6 Basics of Counting Theory .................................................................................126611.7 Basics of Probability ...........................................................................................1274Chapter 11 Review Exercises .....................................................................................1282Chapter 11 Test ...........................................................................................................1294AppendicesAppendix A Polar Form of Conic Sections ................................................................1298Appendix B Rotation of Axes.....................................................................................1301

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1Chapter RREVIEW OF BASIC CONCEPTSSection R.1Fractions, Decimals, andPercents1.True2.True3.False. This is an improper fraction. Its valueis 1.4.False. The reciprocal of62is26or13.5.C6.C7.81 818111162 82822⋅==⋅=⋅=⋅8.1 414111123 434334⋅==⋅=⋅=⋅9.153 535551183 63666⋅==⋅=⋅=⋅10.164 44424415 450545⋅==⋅=⋅=⋅11.903 303303 11505 30530553⋅==⋅=⋅=⋅12.100514075 205205 17 207207⋅==⋅=⋅=⋅13.181 181 18111905 185 1855⋅==⋅=⋅=⋅14.161 161 16111644 1644416⋅==⋅=⋅=⋅15.12012 10105114412 1212616.777 1177113212 11121217.17 1275Therefore,1251.77=18.19 1697Therefore, 1671.99=19.612 77725Therefore,7756.1212=20.513 67652Therefore,6725.1313=21.Multiply the denominator of the fraction bythe natural number and then add the numeratorto obtain the numerator of the improperfraction.5 210⋅=and10313+=The denominator of the improper fraction isthe same as the denominator in the mixednumber. Thus,3132.55=22.Multiply the denominator of the fraction bythe natural number and then add the numeratorto obtain the numerator of the improperfraction.7 535⋅=and35641+=The denominator of the improper fraction isthe same as the denominator in the mixednumber. Thus,6415.77=23.Multiply the denominator of the fraction bythe natural number and then add the numeratorto obtain the numerator of the improperfraction.3 1236⋅=and36238+=The denominator of the improper fraction isthe same as the denominator in the mixednumber. Thus,23812.33=

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2Chapter R Review of Basic Concepts24.Multiply the denominator of the fraction bythe natural number and then add the numeratorto obtain the numerator of the improperfraction.5 1050⋅=and50151+=The denominator of the improper fraction isthe same as the denominator in the mixednumber. Thus,15110.55=25.464 624575 735⋅⋅==⋅26.25 21079 75963⋅⋅==⋅27.36115 8120222 31 615 820 60⋅⋅⋅====⋅⋅28.351512021423 51 1520 212808152⋅⋅⋅====⋅⋅29.1121 121 2 6610510 52 5 525⋅⋅⋅⋅===⋅⋅⋅30.1 101 101 2 558 72 4 72887⋅⋅⋅⋅===⋅⋅⋅31.15815 83 5 4 24254 254 5 53 261,or 1555⋅⋅⋅⋅⋅==⋅⋅⋅⋅==32.21421 43 7 431, or 178 74 2 7282⋅⋅⋅⋅===⋅⋅⋅33.321321 33 7 33 32197171 71 71⋅⋅⋅⋅⋅=⋅====⋅⋅34.436436 44 9 44 4361611 919919⋅⋅⋅⋅⋅=⋅====⋅⋅35.Change both mixed numbers to improperfractions.1213513 565531, or543434 31212⋅⋅=⋅==⋅36.Change both mixed numbers to improperfractions.23333888 864421, or 45551515⋅⋅=⋅==⋅37.To divide fractions, multiply by the reciprocalof the divisor.37 23772142392979⋅÷=⋅==⋅38.To divide fractions, multiply by the reciprocalof the divisor.65642416 411 51411555⋅÷=⋅==⋅39.To divide fractions, multiply by the reciprocalof the divisor.53585 85 4 25 248434 34 33101, or 333⋅⋅⋅⋅÷=⋅===⋅⋅=40.To divide fractions, multiply by the reciprocalof the divisor.37 107 107 2 51035 35 3755142, or 433⋅⋅⋅÷=⋅==⋅⋅=41.To divide fractions, multiply by the reciprocalof the divisor.32832 1532 158 4 3 5515585 81 5 84 3121⋅⋅⋅⋅÷=⋅==⋅⋅⋅⋅==42.To divide fractions, multiply by the reciprocalof the divisor.24242124 214 6 3 77 6672171 7 64 31261⋅⋅⋅⋅÷=⋅==⋅⋅⋅⋅==43.To divide fractions, multiply by the reciprocalof the divisor.3313 13 11244 124 124 3 4114 416⋅⋅÷=⋅⋅⋅⋅=⋅===44.To divide fractions, multiply by the reciprocalof the divisor.2212 12 11130305 305 2 155 155575==⋅⋅÷=⋅=⋅⋅⋅⋅=45.To divide fractions, multiply by the reciprocalof the divisor.3656 52 3 52 56105131 31 31==⋅⋅⋅⋅⋅=÷=⋅=⋅46.To divide fractions, multiply by the reciprocalof the divisor.4898 92 4 92 981811449141==⋅⋅⋅⋅⋅=÷=⋅=⋅

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Section R.1 Fractions, Decimals, and Percents347.Change the first number to an improperfraction, and then multiply by the reciprocal ofthe divisor.3327327827 86 4848434 33 9 2 49 2184 31⋅===÷=÷=⋅⋅⋅⋅⋅⋅=⋅48.Change the first number to an improperfraction, and then multiply by the reciprocal ofthe divisor.3728728 1028 105105155 74 7 2 54 285 75710⋅÷=÷=⋅⋅⋅⋅⋅⋅⋅====49.Change both mixed numbers to improperfractions, and then multiply by the reciprocalof the divisor.15512575 72127272 122 123511, or 12424=⋅=⋅÷=÷⋅=50.Change both mixed numbers to improperfractions, and then multiply by the reciprocalof the divisor.2095959722207520 5219 710037, or631 6351.Change both mixed numbers to improperfractions, and then multiply by the reciprocalof the divisor.5152147213221 32218 4721 8 421 48437, or 18 4832747478473284752.Change both mixed numbers to improperfractions, and then multiply by the reciprocalof the divisor.3423923523 5211092 5 9235, or 1 81051051853.7474111515151554.25257999955.717182 42121212123 4356.53511616138616257.Because93 3,and 3 is prime, the LCD(least common denominator) is3 39.11333339Now add the two fractions with the samedenominator.515389399958.To add415and1 ,5first find the LCD. Since153 5and 5 is prime, the LCD is 15.414134343755315151515151559.Because82 2 2and62 3,the LCD is2 2 224.3333988324and5542066424Now add fractions with the same denominator.35920298624244 ,2or51 2460.Because62 3and 93 3,the LCD is2 3 318.5533156618and222992418Now add fractions with the same denominator.515,691818241918or111861.Because93 3and164 4,the LCD is3 3 4 4144.1680991556144and3392716169144Now add fractions with the same denominator.53802791614414410714462.Because42 2and255 5,the LCD is2 2 5 5100.3325754425100and6642425254100Now add fractions with the same denominator.3675249942510001001 0

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4Chapter R Review of Basic Concepts63.1124125338888811819224444411259328484Because82 2 2and42 2,the LCD is2 2 2or 8.1125923284842251888433, or 58864.2122144411121132223333366666Because 62 3,the LCD is 6.362114213462 63262813415,66or665.72725999966.838311111511167.133133102 52151515153 5368.1131132 42121212123 438−⋅−====⋅69.Because124 3=⋅(12 is a multiple of 3), theLCD is 12.1443412⋅=Now subtract fractions with the samedenominator.717431 311231212124 34⋅−=−===⋅70.Because63 2=⋅(6 is a multiple of 2), theLCDis 6.123336⋅=Now subtract fractions with the samedenominator.5132566661 212323⋅−=−===⋅71.Because122 2 3=⋅⋅and 93 3,=⋅the LCD is2 2 336.3⋅⋅⋅=773211212336=⋅=and 1449436⋅=Now subtract fractions with the samedenominator.7121417129363636−=−=72.The LCD of 1211111314and 16 is 48.161216312433429484848−=⋅−⋅=−=73.33163194444444225271155555=+=+==+=+=Because42 2,=⋅and 5 is prime, the LCD is2 2 520.⋅⋅=3219574414545549528677, or 320202020−=⋅−⋅=−=74.Change both numbers to improper fractionsthen add, using 45 as the commondenominator.44191991353159951716516, or 241359591064545455−=−=⋅−⋅=−=

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Section R.1 Fractions, Decimals, and Percents575.7488124227229999922214333334=+=+==+=+=Because 93 3,=⋅and 3 is prime, the LCD is3 39.⋅=1438433227474425, or 3 993999329−=−⋅=−=76.558458912121212125524529667644667=+=+==+=+=Since122 2 3=⋅⋅and 62 3,=⋅the LCD is2 2 312.⋅⋅=55898912612122925874621311227, or 2 12−=−⋅=−=77.Observe that there are 24 dots in the entirefigure, 6 dots in the triangle, 12 dots in therectangle, and 2 dots in the overlappingregion.(a)121242=of all the dots are in therectangle.(b)61244=of all the dots are in the triangle.(c)2163=of the dots in the triangle are inthe overlapping region.21126=of the dots in the rectangle are inthe overlapping region.78.(a)12 is 13 of 36, so Benita got a hit inexactly13of her at-bats.(b)1 is a little less than110of 11, so Chasegot a home run in just less than110of hisat-bats.(c)9 is a little less than14of 40, so Christinegot a hit in just less than14of her at-bats.(d)8 is12of 16, and 10 is12of 20, so Chinand Greg each got hits12of the time theywere at bat.79.367.9412(a)Tens: 6(b)Tenths:9(c)Thousandths:1(d)Ones: 7(e)Hundredths: 480.Answers will vary. One example is 5243.0164.81.46.249(a)46.25(b)46.2(c)46(d)5082.(a)0.889(b)0.444(c)0.976(d)0.86583.40.41084.60.61085.640.6410086.820.8210087.1380.138100088.1040.104100089.430.0431000

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6Chapter R Review of Basic Concepts90.870.087100091.80538053.80531000100092.16651665.1665 1000100093.25.320109.2008.574143.09494.90.52732.430589.800712.75795.28.733.1225.6196.46.8813.4533.4397.43.5028.1715.3398.345.1056.31288.7999.3.8715.002.9021.77100.8.201.0912.0021.29101.32.56047.3561.80081.716102.75.200123.9603.897203.057103.18.0002.78915.211104.29.0008.58220.418105.12.81 decimal place9.11 decimal place1281152112116.482 decimal places106.34.042 decimal places0.562 decimal places204241702022419.06244 decimal places107.22.412 decimal places330 decimal places67236723202739.532 decimal places108.55.762 decimal places720 decimal places11152390322024014.722 decimal places109.0.21 decimal place0.032 decimal places61230.0063 decimal places110.0.072 decimal places0.0043 decimal places282350.000285 decimal places

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Section R.1 Fractions, Decimals, and Percents7111.7.1511 78.6577161155550112.5.2414 73.3670332856560113.To change the divisor 11.6 into a wholenumber, move each decimal point one place tothe right. Move the decimal point straight upand divide as with whole numbers.2.8116 324.8232920000089280Therefore,32.4811.62.8.114.To change the divisor 17.4 into a wholenumber, move each decimal point one place tothe right. Move the decimal point straight upand divide as with whole numbers.4.9174 852.66961560006156 60Therefore,85.2617.44.9.115.To change the divisor 9.74 into a wholenumber, move each decimal point two placesto the right. Move the decimal point straightup and divide as with whole numbers.2.05974 1996.701948487048700Therefore,19.9679.742.05.116.To change the divisor 5.27 into a wholenumber, move each decimal point two placesto the right. Move the decimal point straightup and divide as with whole numbers.8.44527 4447.88421623182108210821080Therefore,44.47885.278.44.117.Move the decimal point one place to the right.123.26101232.6118.Move the decimal point one place to the right.785.91107859.1119.Move the decimal point two places to theright.57.1161005711.6120.Move the decimal point two places to theright.82.0531008205.3121.Move the decimal point three places to theright.0.094100094122.Move the decimal point three places to theright.0.025100025123.Move the decimal point one place to the left.1.62100.162124.Move the decimal point one place to the left.8.04100.804125.Move the decimal point two places to the left.124.031001.2403126.Move the decimal point two places to the left.490.351004.9035127.Move the decimal point three places to the left.23.291000023.2910000.02329128.Move the decimal point three places to the left.59.81000059.810000.0598

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8Chapter R Review of Basic Concepts129.Convert from a decimal to a percent.0.010.01 100%1%=⋅=Fraction in Lowest Terms(or Whole Number)DecimalPercent11000.011%130.Convert from a percent to a decimal.22%0.02100==Fraction in Lowest Terms(or Whole Number)DecimalPercent1500.022%131.Convert from a percent to a fraction.55%100=In lowest terms,51 5110020 520⋅==⋅Fraction in Lowest Terms(or Whole Number)DecimalPercent1200.055%132.Convert to a decimal first.Divide 1 by 10.Move the decimal point one place to the left.1100.1÷=Convert the decimal to a percent.0.10.1 100%10%=⋅=Fraction in Lowest Terms(or Whole Number)DecimalPercent1100.110%133.Convert the decimal to a percent.0.1250.125 100%12.5%=⋅=Fraction in Lowest Terms(or Whole Number)DecimalPercent180.12512.5%134.Convert the percent to a decimal first.20%0.20,=or 0.2Convert from a percent to a fraction.2020%100=In lowest terms,201 2011005 205⋅==⋅Fraction in Lowest Terms(or Whole Number)DecimalPercent150.220%135.Convert to a decimal first.Divide 1 by 4. Adda decimal point and as many 0s as necessary.0.254 1.00820200Convert the decimal to a percent.0.25 = 25%Fraction in Lowest Terms(or Whole Number)DecimalPercent140.2525%136.Convert to a decimal first.Divide 1 by 3. Adda decimal point and as many 0s as necessary.0.33...3 1.00...91091Note that the pattern repeats. Therefore,10.33Convert the decimal to a percent.0.333.3%or10.333%3Fraction in Lowest Terms(or Whole Number)DecimalPercent130.333.3%or133%3

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Section R.1 Fractions, Decimals, and Percents9137.Convert the percent to a decimal first.50% = 0.5Convert from a percent to a fraction.5050%100In lowest terms,50150%1002Fraction in Lowest Terms(or Whole Number)DecimalPercent120.550%138.Divide 2 by 3. Add a decimal point and asmany 0s as necessary.0.66...3 2.00...1820182Note that the pattern repeats. Therefore,20.63Fraction in Lowest Terms(or Whole Number)DecimalPercent230.6266%3or66.6%139.Convert the decimal to a percent first.0.75 = 75%Convert from a percent to a fraction.7575%100In lowest terms,75375%1004Fraction in Lowest Terms(or Whole Number)DecimalPercent340.7575%140.Convert the decimal to a percent.1.0 = 100%Fraction in Lowest Terms(or Whole Number)DecimalPercent11.0100%141.Divide 21 by 5. Add a decimal point and asmany 0s as necessary.4.25 21.02010100142.Divide 9 by 5. Add a decimal point and asmany 0s as necessary.1.85 9.0540400143.Divide 9 by 4. Add a decimal point and asmany 0s as necessary.2.254 9.00810820200144.Divide 15 by 4. Add a decimal point and asmany 0s as necessary.3.754 15.0012302820200

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Solution Manual for College Algebra and Trigonometry, 7th Edition - Page 16 preview image

10Chapter R Review of Basic Concepts145.Divide 3 by 8. Add a decimal point and asmany 0s as necessary.0.3758 3.00024605640400146.Divide 7 by 8. Add a decimal point and asmany 0s as necessary.0.8758 7.00064605640400147.Divide 5 by 9. Add a decimal point and asmany 0s as necessary.0.555...9 5.000...45504550455Note that the pattern repeats. Therefore,50.5,9=or about0.556.148.Divide 8 by 9. Add a decimal point and asmany 0s as necessary.0.888...9 8.000...72807280728Note that the pattern repeats. Therefore,80.8,9=or about0.889.149.Divide 1 by 6. Add a decimal point and asmany 0s as necessary.0.166...6 1.000...6403640364Note that the pattern repeats. Therefore,10.16,6=or about0.167.150.Divide 5 by 6. Add a decimal point and asmany 0s as necessary.0.833...6 5.000...48201820182Note that the pattern repeats. Therefore,50.83,6=or about0.833.151.54%0.54=152.39%0.39=153.7%07%0.07==154.4%04%0.04==155.117%1.17=156.189%1.89=157.2.4%02.4%0.024==158.3.1%03.1%0.031==159.16%6.25%06.25%0.06254===160.15%5.5%05.5%0.0552===161.0.7979%=162.0.8383%=163.0.022%=164.0.088%=165.0.0040.4%=

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