Solution Manual for Trigonometry, 12th Edition
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’ S S OLUTIONS M ANUAL B EVERLY F USFIELD T RIGONOMETRY T WELFTH E DITION Margaret L. Lial American River College John Hornsby University of New Orleans David I. Schneider University of Maryland Callie J. Daniels St. Charles Community CollegePage 2
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CONTENTS R Algebra Review R.1 Basic Concepts from Algebra .................................................................................... 1 R.2 Real Number Operations and Properties .................................................................... 3 R.3 Exponents, Polynomials, and Factoring ..................................................................... 8 R.4 Rational Expressions ................................................................................................ 14 R.5 Radical Expressions ................................................................................................. 20 R.6 Equations and Inequalities ....................................................................................... 27 R.7 Rectangular Coordinates and Graphs ....................................................................... 34 R.8 Functions .................................................................................................................. 42 R.9 Graphing Techniques ............................................................................................... 47 Chapter R Review Exercises ........................................................................................... 60 Chapter R Test ................................................................................................................ 68 1 Trigonometric Functions 1.1 Angles ....................................................................................................................... 72 1.2 Angle Relationships and Similar Triangles .............................................................. 79 Chapter 1 Quiz (Sections 1.1 −1.2).................................................................................. 84 1.3 Trigonometric Functions ........................................................................................... 85 1.4 Using the Definitions of the Trigonometric Functions ............................................. 98 Chapter 1 Review Exercises ......................................................................................... 106 Chapter 1 Test ............................................................................................................... 112 2 Acute Angles and Right Triangles 2.1 Trigonometric Functions of Acute Angles ............................................................. 115 2.2 Trigonometric Functions of Non-Acute Angles ..................................................... 122 2.3 Approximations of Trigonometric Function Values ............................................... 131 Chapter 2 Quiz (Sections 2.1−2.3)................................................................................ 138 2.4 Solutions and Applications of Right Triangles ....................................................... 139 2.5 Further Applications of Right Triangles ................................................................. 148 Chapter 2 Review Exercises ......................................................................................... 158 Chapter 2 Test ............................................................................................................... 164 3 Radian Measure and the Unit Circle 3.1 Radian Measure ...................................................................................................... 168 3.2 Applications of Radian Measure ............................................................................. 172 3.3 The Unit Circle and Circular Functions .................................................................. 181 Chapter 3 Quiz (Sections 3.1−3.3)................................................................................ 192 3.4 Linear and Angular Speed ...................................................................................... 192 Chapter 3 Review Exercises ......................................................................................... 197 Chapter 3 Test ............................................................................................................... 202Page 4
4 Graphs of the Circular Functions 4.1 Graphs of the Sine and Cosine Functions ............................................................... 205 4.2 Translations of the Graphs of the Sine and Cosine Functions ................................ 215 Chapter 4 Quiz (Sections 4.1−4.2)................................................................................ 230 4.3 Graphs of the Tangent and Cotangent Functions.................................................... 233 4.4: Graphs of the Secant and Cosecant Functions ....................................................... 244 Summary Exercises on Graphing Circular Functions ................................................... 253 4.5 Harmonic Motion .................................................................................................... 256 Chapter 4 Review Exercises ......................................................................................... 262 Chapter 4 Test ............................................................................................................... 271 5 Trigonometric Identities 5.1 Fundamental Identities ............................................................................................ 276 5.2 Verifying Trigonometric Identities ......................................................................... 284 5.3 Sum and Difference Identities for Cosine ............................................................... 295 5.4 Sum and Difference Identities for Sine and Tangent .............................................. 302 Chapter 5 Quiz (Sections 5.1−5.4)................................................................................ 313 5.5 Double-Angle Identities .......................................................................................... 314 5.6 Half-Angle Identities .............................................................................................. 323 Summary Exercises on Verifying Trigonometric Identities ......................................... 332 Chapter 5 Review Exercises ......................................................................................... 337 Chapter 5 Test ............................................................................................................... 349 6 Inverse Circular Functions and Trigonometric Equations 6.1 Inverse Circular Functions ...................................................................................... 352 6.2 Trigonometric Equations I ...................................................................................... 365 6.3 Trigonometric Equations II ..................................................................................... 375 Chapter 6 Quiz (Sections 6.1−6.3)................................................................................ 385 6.4 Equations Involving Inverse Trigonometric Functions .......................................... 387 Chapter 6 Review Exercises ......................................................................................... 395 Chapter 6 Test ............................................................................................................... 403 7 Applications of Trigonometry and Vectors 7.1 Oblique Triangles and the Law of Sines ................................................................. 406 7.2 The Ambiguous Case of the Law of Sines.............................................................. 414 7.3 The Law of Cosines ................................................................................................ 421 Chapter 7 Quiz (Sections 7.1−7.3)................................................................................ 433 7.4 Geometrically Defined Vectors and Applications .................................................. 434 7.5 Algebraically Defined Vectors and the Dot Product .............................................. 443 Summary Exercises on Applications of Trigonometry and Vectors ............................ 449 Chapter 7 Review Exercises ......................................................................................... 451 Chapter 7 Test ............................................................................................................... 459Page 5
8 Complex Numbers, Polar Equations, and Parametric Equations 8.1 Complex Numbers .................................................................................................. 462 8.2 Trigonometric (Polar) Form of Complex Numbers ................................................ 467 8.3 The Product and Quotient Theorems ...................................................................... 472 8.4 DeMoivre’s Theorem; Powers and Roots of Complex Numbers ........................... 478 Chapter 8 Quiz (Sections 8.1−8.4)................................................................................ 490 8.5 Polar Equations and Graphs .................................................................................... 492 8.6 Parametric Equations, Graphs, and Applications ................................................... 507 Chapter 8 Review Exercises ......................................................................................... 518 Chapter 8 Test ............................................................................................................... 525Page 6
1 Chapter R ALGEBRA REVIEW Section R.1 Basic Concepts from Algebra 1. The set 0, 1, 2, 3, describes the set of whole numbers. 2. The set containing no elements is the empty (or null) set, symbolized . 3. The opposite, or negative, of a number is its additive inverse. 4. The distance on a number line from a number to 0 is the absolute value of the number. 5. If the real number a is to the left of the real number b on a number line, then a < (or is less than) b . 6. (a) 0 is a whole number. Therefore, it is also an integer, a rational number, and a real number. 0 belongs to B, C, D, F. (b) 34 is a natural number. Therefore, it is also a whole number, an integer, a rational number, and a real number. 34 belongs to A, B, C, D, F. (c) 9 4 is a rational number and a real number. 9 4 belongs to D, F. (d) 36 6 is a natural number. Therefore, it is also a whole number, an integer, a rational number, and a real number. 36 belongs to A, B, C, D, F. (e) 13 is an irrational number and a real number. 13 belongs to E, F . (f) 216 54 100 25 2.16 is a rational number and a real number. 2.16 belongs to D, F. 7. The set 1 1 1 3 9 27 1, , , , is infinite . No, 3 is not an element of the set. 8. Using set notation, the set { x | x is a natural number less than 6} is {1, 2, 3, 4, 5}. 9. 1 1 10. (a) The additive inverse of 10 is –10. (b) The absolute value of 10 is 10. 11. The elements in the set | is a whole number less than 6 x x are 0, 1, 2, 3, 4, 5 . 12. The elements in the set | is a whole number less than 9 m m are 0, 1, 2, 3, 4, 5, 6, 7,8 . 13. The elements in the set | is a natural number greater than 4 z z are 5, 6, 7,8, . 14. The elements in the set | is a natural number greater than 8 y y are 9, 10, 11, 12, . 15. The elements in the set | is an integer less than or equal to 4 z z are , 1, 0, 1, 2, 3, 4 . 16. The elements in the set | is an integer less than 3 p p are , 2, 1, 0, 1, 2, . 17. The elements in the set | is an even integer greater than 8 a a are 10, 12, 14, 16, . 18. The elements in the set | is an odd integer less than 1 k k are , 7, 5, 3, 1 . 19. The elements in the set | is a number whose absolute value is 4 p p are 4, 4 . 20. The elements in the set | is a number whose absolute value is 7 w w are 7, 7 . Page 7
2 Chapter R Algebra Review 21. There are no elements in the set { x | x is an irrational number that is also rational}. Using set notation, this is . 22. There are no elements in the set { r | r is a number that is both positive and negative}. Using set notation, this is . For exercises 23–26, more than one description may be possible. 23. {2, 4, 6, 8} can be described as { x | x is an even natural number less than or equal to 8}. 24. {11, 12, 13, 14} can be described as { x | x is an integer between 10 and 15}. 25. {4, 8, 12, 16, …} can be described as { x | x is a positive multiple of 4}. 26. {…, –6, –3, 0, 3, 6, …} can be described as { x | x is an integer multiple of 3}. For Exercises 27 and 28, A = {1, 2, 3, 4, 5, 6}, B = {1, 3, 5}, C = {1, 6}, and D = {4}. 27. (a) 4 , or A D D (b) 1 B C (c) 1, 3, 5 , or B A B (d) 1, 6 , or C A C 28. (a) 1, 2, 3, 4, 5, 6 , or A B A (b) 1, 3, 4, 5 B D (c) 1, 3, 5, 6 B C (d) 1, 4, 6 C D For Exercises 29–34, 5 12 1 4 8 4 6, , , 3, 0, , 1, 2 , 3, 12 . A 29. 1 and 3 are natural numbers. 30. 0, 1, and 3 are whole numbers. 31. –6, 12 4 (or –3), 0, 1, and 3 are integers. 32. 5 12 1 4 8 4 6, (or 3), , 0, , 1, and 3 are rational numbers. 33. 3, 2 and 12 are irrational numbers. 34. All are real numbers. 35. 6 7 75 21 2 5 9, 6, 0.7, 0, , 7, 4.6, 8, , 13, X (a) Natural numbers: 8, 13, 75 5 (or 15) (b) Whole numbers: 0, 8, 13, 75 5 (or 15) (c) Integers: –9, 0, 8, 13, 75 5 (or 15) (d) Rational numbers: –9, –0.7, 0, 6 7 , 4.6, 8, 21 2 , 13, 75 5 (or 15) (e) Irrational numbers: 6, 7 (f) Real numbers: All are real numbers 36. 3 4 13 40 2 2 8, 5, 0.6, 0, , 3, , 5, , 17, X (a) Natural numbers: 5, 17, 40 2 (or 20) (b) Whole numbers: 0, 5, 17, 40 2 (or 20) (c) Integers: –8, 0, 5, 17, 40 2 (or 20) (d) Rational numbers: –8, –0.6, 0, 3 4 , 5, 13 2 , 17, 40 2 (or 20) (e) Irrational numbers: 5, 3, (f) Real numbers: All are real numbers 37. False. Some are whole numbers, but negative integers are not. 38. True 39. False. No irrational number is an integer. 40. True 41. True 42. False. No rational number is irrational. 43. True 44. True 45. True 46. True 47. 4 x when x = –4 and x = 4. 48 . (a) A. 4 4 (b) A. 4 4 (c) B. 4 4 Page 8
Section R.2 Real Number Operations and Properties 3 (d) B. 4 4 49. (a) Additive inverse: 6 (b) Absolute value: 6 6 50. (a) Additive inverse: 12 12 (b) Absolute value: 12 12 51. (a) Additive inverse: 6 6 5 5 (b) Absolute value: 6 6 5 5 52. (a) Additive inverse: 0.16 (b) Absolute value: 0.16 0.16 53. 8 8 54. 19 19 55. 3 3 2 2 56. 3 3 4 4 57. 5 5 58. 12 12 59. 2 2 60. 6 6 61. 4.5 4.5 62. 12.4 12.4 63. Pacific Ocean, Indian Ocean, Caribbean Sea, South China Sea, Gulf of California 64. Point Success, Rainier, Matlalcueyetl, Steele, Denali 65. True. 14, 040 14, 040; 12,800 12,800 14,040 > 12,800 66. False. 2375 2375; 8448 8448 2375 is not greater than 8448. Use the following number line to answer exercises 67–74. 67. 6 1 True 68. 4 2 True 69. 4 3 False 70. 3 1 False 71. 3 2 True 72. 6 3 True 73. 3 3 False 74. 5 5 False 75. 2 6 76. 1 5 77. 4 9 78. 1 6 79. 10 5 80. 12 7 81. 7 1 82. 4 10 83. 5 5 84. 6 6 85. 5 0 0 86. 5 14 19 87. 0 5 False 88. 11 0 False 89. 7 7 True 90. 10 10 True 91. 6 7 3 6 10 True 92. 7 4 1 7 5 True 93. 2 5 4 6 10 10 True 94. 8 7 3 5 15 15 True 95. 3 3 3 3 True 96. 4 4 4 4 True 97. 8 6 8 6 False 98. 10 4 10 4 False Section R.2 Real Number Operations and Properties 1. The sum of two negative numbers is negative. 2. The product of two negative numbers is positive. 3. The quotient formed by any nonzero number divided by 0 is undefined, and the quotient formed by 0 divided by any nonzero number is 0. 4. The commutative property is used to change the order of two terms or factors, and the associative property is used to change the grouping of three terms or factors. 5. Like terms are terms with the same variables raised to the same powers. 6. The numerical coefficient in the term 2 7 y z is –7. 7. 3 10 1000 8. 2 5 10 2 10 5 5 9. 3 2 2 1 6 2 6 2 4 10. 7 4 7 28 x y x y 11. 6 ( 13) (6 13) 19 Page 9
4 Chapter R Algebra Review 12. 8 ( 16) (8 16) 24 13. 15 6 (15 6) 9 14. 17 9 (17 9) 8 15. 13 ( 4) 13 4 9 16. 19 ( 13) 19 13 6 17. 7 3 28 9 19 3 4 12 12 12 18. 5 4 15 8 7 6 9 18 18 18 19. The difference between 4.5 and 2.8 is 1.7. The number with the greater absolute value, 4.5 is positive, so the answer is positive. Thus, 2.8 4.5 1.7. 20. 3.8 6.2 6.2 3.8 2.4 21. 4 9 4 ( 9) 5 22. 3 7 3 ( 7) 4 23. 6 5 6 ( 5) (6 5) 11 24. 8 17 8 ( 17) (8 17) 25 25. 8 ( 13) 8 13 21 26. 12 ( 22) 12 22 34 27. 12.31 ( 2.13) 12.31 2.13 (12.31 2.13) 10.18 28. 15.88 ( 9.42) 15.88 9.42 (15.88 9.42) 6.46 29. 9 4 9 4 27 40 67 10 3 10 3 30 30 30 30. 3 3 3 3 6 21 27 14 4 14 4 28 28 28 31. 8 6 14 ( 14) 14 32. 7 15 22 ( 22) 22 33. 2 4 2 4 2 ( 4) 6 34. 16 13 16 13 16 ( 13) 3 35. The product of two numbers with the same sign is positive, so 8 ( 5) 40. 36. The product of two numbers with the same sign is positive, so 20 ( 4) 80. 37. The product of two numbers with different signs is negative, so 5 ( 7) 35. 38. The product of two numbers with different signs is negative, so 6 ( 9) 54. 39. 4(0) 4 0 0 The multiplication property of 0 states that the product of any real number and 0 is 0. 40. 0( 8) 0 8 0 The multiplication property of 0 states that the product of any real number and 0 is 0. 41. The product of two numbers with different signs is negative, so 5 12 5 2 6 6 . 2 25 2 5 5 5 42. The product of two numbers with different signs is negative, so 9 21 9 3 7 3 . 7 36 7 4 9 4 43. The product of two numbers with the same sign is positive, so 3 24 3 3 8 1. 8 9 8 9 44. The product of two numbers with the same sign is positive, so 2 22 2 2 11 1. 11 4 11 2 2 45. The product of two numbers with different signs is negative, so 0.06(0.4) 0.024. 46. The product of two numbers with different signs is negative, so 0.08(0.7) 0.056. 47. The quotient of two nonzero real numbers with the same sign is positive, so 24 1 1 24 6 4 6. 4 4 4 48. The quotient of two nonzero real numbers with the same sign is positive, so 45 1 1 45 5 9 5. 9 9 9 49. The quotient of two nonzero real numbers with different signs is negative, so 100 1 1 100 4 25 4. 25 25 25 Page 10
Section R.2 Real Number Operations and Properties 5 50. The quotient of two nonzero real numbers with different signs is negative, so 150 1 1 150 5 30 5. 30 30 30 51. 0 1 0 0 8 8 52. 0 1 0 0 14 14 53. Division by 0 is undefined, so 5 0 is undefined. 54. Division by 0 is undefined, so 13 0 is undefined. 55. The quotient of two nonzero real numbers with the same sign is positive, so 10 12 10 5 2 5 5 25 . 17 5 17 12 17 2 6 102 56. 22 33 22 5 2 11 5 10 23 5 23 33 23 3 11 69 57. 4 4 3 4 5 4 5 3 5 5 5 3 3 5 58. 12 12 5 12 13 12 13 5 13 13 13 5 5 13 59. 12 12 4 12 3 13 4 13 3 13 4 3 3 4 3 9 13 4 13 60. 7 7 2 7 3 6 2 6 3 6 2 3 7 3 7 2 3 2 4 61. 2 3 2 3 2 2 3 62. 3 3 2 6 1 2 63. 7.2 9 0.8 64. 4.5 5 0.9 65. , 1 4 1 4 3 3 d P Q or , 4 1 4 1 3 3 d P Q 66. , 8 4 8 4 12 12 d P R or , 4 8 12 12 d P R 67. , 8 1 8 1 9 9 d Q R or , 1 8 9 9 d Q R 68. , 12 1 13 13 d Q S or , 1 12 13 13 d Q S 69. (a) 2 8 64 (b) 2 8 (8 8) 64 (c) 2 ( 8) ( 8) ( 8) 64 (d) 2 ( 8) (64) 64 70. (a) 3 4 64 (b) 3 4 (4 4 4) 64 (c) 3 ( 4) ( 4) ( 4) ( 4) 64 (d) 3 ( 4) ( 64) 64 71. 4 2 2 2 2 2 16 72. 5 3 3 3 3 3 3 243 73. 4 2 2 2 2 2 16 74. 6 2 2 2 2 2 2 2 64 75. 5 3 3 3 3 3 3 243 76. 5 ( 2) ( 2)( 2)( 2)( 2)( 2) 32 77. 4 2 3 2 3 3 3 3 2 81 162 Page 11
6 Chapter R Algebra Review 78. 3 4 5 4 5 5 5 4 125 500 79. (a) Because 9 15 3 9 5 14, + ÷ = + = the grandson's answer was correct. (b) The reasoning was incorrect. Division must be done first, and then the addition follows. The grandson's “Order of Process rule” is not correct. It just happens coincidentally in this problem that he obtained the correct answer the wrong way. 80. 7 7 7 7 7 7 7 1 49 7 8 49 7 57 7 50 81. 12 3 4 12 12 24 82. 15 5 2 15 10 25 83. 6 3 12 4 18 12 4 18 3 15 84. 9 4 8 2 36 8 2 36 4 32 85. 10 30 2 3 10 15 3 10 45 55 86. 12 24 3 2 12 8 2 12 16 28 87. 3 5 7 3 ( 2) 5 7 3 ( 8) 5 21 8 16 8 8 88. 3 4 3 5 ( 3) 4 3 5 ( 27) 4 15 27 19 27 8 89. Simplify within parentheses. 2 2 2 18 4 5 (3 7) 18 4 5 ( 4) 18 4 5 4 18 16 5 4 2 5 4 7 4 11 − + − − = − + − − = − + + = − + + = + + = + = 90. Simplify within parentheses. 2 2 2 10 2 9 (1 8) 10 2 9 ( 7) 10 2 9 7 10 4 9 7 6 9 7 15 7 22 91. 3 3 4 9 8 7 2 4 1 7 2 4 1 7 8 4 7 8 4 56 60 92. 4 6 5 3 2 6 5 3 16 30 3 16 30 48 30 48 18 93. 8 4 6 12 8 24 12 4 3 4 3 8 2 7 6 6 7 7 94. 15 5 4 6 8 3 4 6 8 6 5 8 2 6 5 4 12 6 8 2 8 6 5 4 1 4 6 6 5 5 For Exercises 95–104, p = –4, q = 8, and r = –10. 95. 3 2 3 4 2 10 12 20 12 20 8 p r 96. 5 6 5 10 6 4 50 24 50 24 26 r p 97. 2 2 2 2 2 7 4 7 8 10 4 7 8 100 16 7 8 100 16 56 100 72 100 28 p q r 98. 2 2 2 4 2 8 10 16 2 8 10 16 16 10 32 10 42 p q r 99. 8 ( 10) 2 1 8 ( 4) 4 2 q r q p 100. 4 ( 10) 14 7 4 8 4 2 p r p q 101. 5 5( 10) 50 2 3 2( 4) 3( 10) 8 30 50 25 22 11 r p r Page 12
Section R.2 Real Number Operations and Properties 7 102. 3 3 8 24 3 2 3 4 2 10 12 2 10 24 24 24 3 12 20 12 20 8 q p r 103. 2 2 2 2 3 4 2 3 10 2 2 8 2 3 10 6 4 3 10 4 30 6 6 4 30 26 13 6 6 3 p r q 104. 2 2 2 6 2 8 6 2 4 4 4 4 2 8 4 8 4 4 8 4 1 8 2 q p p 105. distributive property 106. distributive property 107. inverse property 108. inverse property 109. identity property 110. identity property 111. commutative property 112. commutative property 113. associative property 114. associative property 115. closure property 116. closure property 117. Using the distributive property, 2( ) 2 2 . m p m p 118. 3( ) 3 3 a b a b 119. Using the distributive property, 12( ) 12[ ( )] 12( ) ( 12)( ) 12 12 . x y x y x y x y 120. 10( ) 10[ ( )] 10 10 p q p q p q 121. (2 ) 1(2 ) 1(2 ) ( 1) ( ) 2 d f d f d f d f 122. (3 ) 1(3 ) 1(3 ) ( 1)( ) 3 m n m n m n m n 123. Use the second form of the distributive property. 5 3 (5 3) 8 k k k k 124. 6 5 (6 5) 11 a a a a 125. 7 9 7 ( 9 ) [7 ( 9)] 2 r r r r r r 126. 4 6 4 ( 6 ) [4 ( 6)] 2 n n n n n n 127. Use the identity property, then the distributive property. 7 1 7 (1 7) 8 a a a a a a 128. 9 1 9 Identity property (1 9) 10 s s s s s s 129. 1 1 (1 1) 2 x x x x x x 130. 1 1 (1 1) 2 a a a a a a 131. 2( 3 2 ) 2 2( 3 ) 2(2 ) 2 6 4 x y z x y z x y z 132. 8(3 5 ) 8(3 ) 8 8( 5 ) 24 8 40 x y z x y z x y z 133. 3 16 32 40 8 9 27 9 3 16 3 32 3 40 8 9 8 27 8 9 3 16 3 32 5 8 9 8 27 3 2 4 5 3 9 3 y z y z y z y z 134. 1 20 8 32 4 1 1 1 20 8 32 4 4 4 1 1 1 20 8 32 4 4 4 5 + 2 8 = 5 2 8 m y z m y z m y z m y z m y z Page 13
8 Chapter R Algebra Review 135. 12 4 3 2 ( 12 4 3 2) 3 y y y y y y 136. 5 9 8 5 ( 5 9 8 5) 11 r r r r r r 137. 6 5 4 6 11 6 4 11 5 6 ( 6 4 11) 11 1 11 or 11 p p p p p p p p p 138. 8 12 3 5 9 8 3 5 12 9 ( 8 3 5) 3 10 3 x x x x x x x x 139. 3( 2) 5 6 3 3 6 5 6 3 3 5 6 6 3 (3 5) 6 6 3 2 15 k k k k k k k k 140. 5( 3) 6 2 4 5 15 6 2 4 5 6 2 15 4 (5 6 2) 11 9 11 r r r r r r r r r r r 141. 10 (4 8) 10 1(4 8) 10 4 8 4 2 y y y y 142. 6 (9 5) 6 1(9 5) 6 9 5 9 1 y y y y 143. 10 (3)( ) 10 (3) 30 30 x y x y x y xy 144. 8 (6)( ) 8 (6) 48 48 x y x y x y xy 145. 2 2 (12 )(7 ) (12 ) (7 ) 3 3 8 (7 ) 56 w z w z w z wz 146. 5 5 (18 )(5 ) (18 ) (5 ) 6 6 15 (5 ) 75 w z w z w z wz 147. 3( 4) 2( 1) 3 12 2 2 3 2 12 2 14 m m m m m m m 148. 6( 5) 4( 6) 6 30 4 24 6 4 30 24 2 54 a a a a a a a 149. 0.25(8 4 ) 0.5(6 2 ) 0.25(8) 0.25(4 ) ( 0.5)(6) ( 0.5)(2 ) 2 3 2 3 (1 1) 2 3 0 1 1 p p p p p p p p p p 150. 0.4(10 5 ) 0.8(5 10 ) 0.4(10) 0.4( 5 ) ( 0.8)(5) ( 0.8)(10 ) 4 2 4 8 10 x x x x x x x Section R.3 Exponents, Polynomials, and Factoring 1. The polynomial 5 2 4 x x is a trinomial of degree 5. 2. A polynomial containing exactly one term is a(n) monomial. 3. A polynomial containing exactly two terms is a(n) binomial). 4. When multiplying powers of like bases, as in 2 3 , y y keep the base and add the exponents. 5. To raise a power to a power, as in 3 2 , a multiply the exponents. 6. (a) B; 2 2 2 ( 5 ) 10 25 x y x xy y (b) C; 2 2 2 ( 5 ) 10 25 x y x xy y (c) A; 2 2 ( 5 )( 5 ) 25 x y x y x y (d) D; 2 2 (5 )(5 ) 25 y x y x y x 7. (a) B; 2 3 (2 3)(4 6 9) 8 27 x x x x . (b) C; 2 3 (2 3)(4 6 9) 8 27 x x x x (c) A; 2 3 (3 2 )(9 6 4 ) 27 8 x x x x 8. D. 2 6 12 3 4 2 3 x x x x 9. B: 4 2 1 ( 1)( 1)( 1) x x x x . Choice A is not a complete factorization, because 2 1 x can be factored as ( 1)( 1). x x The other choices are not correct factorizations of 4 1 x .Page 14
Section R.3 Exponents, Polynomials, and Factoring 9 10. C: 3 2 8 ( 2)( 2 4) x x x x . Use the pattern for factoring the sum of cubes, 3 3 2 2 ( )( ). x y x y x xy y We have 3 3 3 8 2 x x , so substitute 2 for y . 11. 5 2 5 2 5 2 7 4 4 4 4 16 16 x x x x x x 12. 4 3 4 3 4 3 7 3 6 3 6 18 18 y y y y y y 13. 6 4 6 4 1 11 n n n n n 14. 8 5 8 5 1 14 a a a a a 15. 3 5 3 5 8 9 9 9 9 16. 2 8 2 8 10 4 4 4 4 17. 4 2 5 4 2 5 4 2 5 11 3 6 4 3 6 4 72 72 m m m m m m m m 18. 3 6 4 3 6 4 3 6 4 13 8 2 5 8 2 5 80 80 t t t t t t t t 19. 2 3 4 2 3 4 2 3 1 4 5 5 5 3 5 3 15 15 x y x y x x yy x y x y 20. 3 2 2 3 1 2 3 1 3 4 4 7 4 7 28 28 xy x y xx y y x y x y 21. 2 2 2 2 1 2 1 2 3 3 1 1 8 8 2 2 4 4 mn m n mm nn m n m n 22. 4 2 4 2 4 1 1 2 5 3 2 2 35 35 7 7 10 10 m n mn m m nn m n m n 23. 5 2 2 5 10 2 2 2 24. 3 4 4 3 12 6 6 6 25. 3 3 3 3 2 2 6 6 6 6 6 216 x x x x 26. 5 5 5 5 5 5 25 25 2 2 2 32 x x x x 27. 3 0 2 2 3 2 0 2 2 3 2 0 2 2 6 0 2 6 2 6 6 (4 ) 4 ( ) ( ) 4 4 4 1 4 16 m n m n m n m n m m m 28. 0 4 3 3 0 3 4 3 3 0 3 4 3 3 0 12 12 12 (2 ) 2 ( ) ( ) 2 2 8 1 8 x y x y x y x y y y 29. 3 8 8 3 8 3 24 2 2 3 2 3 6 ( ) ( ) r r r r s s s s 30. 2 4 4 2 4 2 8 2 2 2 ( ) p p p p q q q q 31. 4 2 4 2 4 4 8 8 2 4 4 8 4 8 4 2 4 ( 4) ( ) ( 4) 256 m m m m tp t p t p t p 32. 3 4 3 4 3 3 12 12 2 2 3 6 6 5 ( 5) ( ) ( 5) 125 ( ) n n n n r r r r 33. 0 3 5 1 x y z 34. 0 2 3 3 1 p q r 35. (a) B; 0 6 1 (b) C; 0 6 1 (c) B; 0 6 1 (d) C; 0 6 1 36. (a) D; 0 3 3 1 3 p (b) E; 0 3 3 1 3 p (c) B; 0 3 1 p (d) B; 0 3 1 p 37. 2 2 2 2 2 5 4 7 4 3 5 5 4 4 3 7 5 1 1 2 2 x x x x x x x x x x 38. 3 2 3 2 3 2 3 2 3 2 3 3 4 2 6 3 2 3 1 4 6 1 4 10 4 10 m m m m m m m m m m Page 15
10 Chapter R Algebra Review 39. 2 2 2 2 2 2 2 2 12 8 6 4 3 4 2 2 12 2 8 2 6 4 3 4 4 4 2 24 16 12 12 16 8 12 4 y y y y y y y y y y y y y 40. 2 2 2 2 2 2 2 3 8 5 5 3 2 4 3 8 3 5 5 3 5 2 5 4 24 15 15 10 20 9 5 20 p p p p p p p p p p p p p p 41. 4 2 3 2 2 4 2 3 2 2 4 3 2 4 3 2 4 3 2 6 3 2 5 4 6 3 2 5 4 6 2 3 5 1 1 4 1 6 2 7 4 6 2 7 4 m m m m m m m m m m m m m m m m m m m m m m m m m m m m 42. 3 3 2 2 3 3 2 2 3 2 3 2 3 2 8 3 2 4 3 1 8 3 2 4 3 1 8 2 1 4 1 3 3 1 6 3 4 3 1 6 3 4 4 x x x x x x x x x x x x x x x x x x x x x 43. 2 3 2 2 3 2 2 2 2 5 4 3 2 4 3 2 5 1 4 3 4 2 4 5 4 1 12 8 20 4 x x x x x x x x x x x x x x x 44. 3 2 3 2 3 3 5 4 3 2 4 3 2 2 4 2 3 2 8 6 b b b b b b b b b b b 45. Use the FOIL method. 2 2 4 1 7 2 4 7 4 2 1 7 1 2 28 8 7 2 28 2 r r r r r r r r r r r 46. Use the FOIL method. 2 2 5 6 3 4 5 3 5 4 6 3 6 4 15 20 18 24 15 2 24 m m m m m m m m m m m 47. Use the FOIL method. 2 2 3 1 2 7 3 2 3 7 1 2 1 7 6 21 2 7 6 19 7 x x x x x x x x x x x 48. Use the FOIL method. 2 2 5 3 2 3 5 2 5 3 3 2 3 3 10 15 6 9 10 9 9 z z z z z z z z z z z 49. 2 2 2 (2 3)(2 3) (2 ) 3 4 9 m m m m 50. 2 2 2 2 (8 3 )(8 3 ) (8 ) (3 ) 64 9 s t s t s t s t 51. 2 2 2 2 2 4 2 (4 5 )(4 5 ) (4 ) (5 ) 16 25 x y x y x y x y 52. 3 3 3 2 2 6 2 (2 )(2 ) (2 ) 4 m n m n m n m n 53. 2 2 2 2 2 (4 2 ) (4 ) 2(4 )(2 ) (2 ) 16 16 4 m n m m n n m mn n 54. 2 2 2 2 2 ( 6 ) 2( )(6 ) (6 ) 12 36 a b a a b b a ab b 55. 2 2 2 2 2 2 2 2 4 (5 3 ) (5 ) 2(5 )(3 ) (3 ) 25 30 9 r t r r t t r rt t 56. 4 2 4 2 4 2 8 4 2 (2 3 ) (2 ) 2(2 )(3 ) (3 ) 4 12 9 z y z z y y z z y y 57. 2 2 2 2 2 2 4 3 2 3 2 2 4 2 1 1 1 1 1 1 1 1 2 1 2 1 2 1 2 2 1 1 2 1 2 2 4 2 2 1 2 1 x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x Alternatively, 2 2 4 2 1 1 1 1 1 1 1 1 1 1 2 1 x x x x x x x x x x x x Page 16
Section R.3 Exponents, Polynomials, and Factoring 11 58. 2 2 2 2 2 2 4 4 4 4 4 4 4 4 8 16 8 16 8 16 8 8 16 16 8 16 t t t t t t t t t t t t t t t t t t t t 4 3 2 3 2 2 4 2 8 16 8 64 128 16 128 256 32 256 t t t t t t t t t t Alternatively, 2 2 4 2 4 4 4 4 4 4 4 4 16 16 32 256 t t t t t t t t t t t t 59. 3 2 2 3 2 2 3 2 2 2 2 4 4 2 4 4 2 8 8 6 12 8 y y y y y y y y y y y y y y 60. 3 2 2 3 2 2 3 2 3 3 3 6 9 3 6 9 3 18 27 9 27 27 z z z z z z z z z z z z z z 61. 3 2 2 2 2 2 2 3 2 2 3 2 2 5 2 5 2 5 2 5 2 2 2 5 5 2 5 4 20 25 2 4 20 25 5 4 20 25 8 40 50 20 100 125 8 60 150 125 x x x x x x x x x x x x x x x x x x x x x x 62. 3 2 2 2 2 2 2 3 2 2 3 2 4 1 4 1 4 1 4 1 4 2 4 1 1 4 1 16 8 1 4 16 8 1 1 16 8 1 64 32 4 16 8 1 64 48 12 1 x x x x x x x x x x x x x x x x x x x x x x 63. 4 2 2 2 2 4 3 2 3 2 2 4 3 2 2 2 2 4 4 4 4 4 4 4 16 16 4 16 16 8 24 32 16 q q q q q q q q q q q q q q q q q q q 64. 4 2 2 2 2 4 3 2 3 2 2 4 3 2 3 3 3 6 9 6 9 6 9 6 36 54 9 54 81 12 54 108 81 r r r r r r r r r r r r r r r r r r r 65. The greatest common factor of both terms is 8 a . 40 16 8 5 8 2 8 5 2 ab a a b a a b 66. The greatest common factor of both terms is 5 y . 25 15 5 5 5 3 5 5 3 xy y y x y y x 67. The greatest common factor of all three terms is 3 4 . x y 3 4 2 3 4 3 2 3 2 3 3 2 2 8 12 36 4 2 4 3 4 9 4 2 3 9 x y x y xy xy x y xy x xy y xy x y x y 68. The greatest common factor of all three terms is 2 2 . m n 5 2 3 3 2 2 3 2 2 2 3 10 4 18 2 5 2 2 2 9 2 5 2 9 m n m n m n m n m m n n m n mn m n m n mn 69. The positive factors of 1 (the coefficient of the first term) are 1 and 1. The factors of –15 are –1 and 15, 1 and –15, –3 and 5, or 3 and –5. Try different combinations of these factors until the correct one is found. 2 2 15 5 3 x x x x 70. The positive factors of 1 (the coefficient of the first term) are 1 and 1. The factors of –12 are –1 and 12, 1 and –12, –2 and 6, 2 and –6, –3 and 4, or 3 and –4. Try different combinations of these factors until the correct one is found. 2 12 4 3 x x x x ’ S S OLUTIONS M ANUAL B EVERLY F USFIELD T RIGONOMETRY T WELFTH E DITION Margaret L. Lial American River College John Hornsby University of New Orleans David I. Schneider University of Maryland Callie J. Daniels St. Charles Community College CONTENTS R Algebra Review R.1 Basic Concepts from Algebra .................................................................................... 1 R.2 Real Number Operations and Properties .................................................................... 3 R.3 Exponents, Polynomials, and Factoring ..................................................................... 8 R.4 Rational Expressions ................................................................................................ 14 R.5 Radical Expressions ................................................................................................. 20 R.6 Equations and Inequalities ....................................................................................... 27 R.7 Rectangular Coordinates and Graphs ....................................................................... 34 R.8 Functions .................................................................................................................. 42 R.9 Graphing Techniques ............................................................................................... 47 Chapter R Review Exercises ........................................................................................... 60 Chapter R Test ................................................................................................................ 68 1 Trigonometric Functions 1.1 Angles ....................................................................................................................... 72 1.2 Angle Relationships and Similar Triangles .............................................................. 79 Chapter 1 Quiz (Sections 1.1−1.2).................................................................................. 84 1.3 Trigonometric Functions ........................................................................................... 85 1.4 Using the Definitions of the Trigonometric Functions ............................................. 98 Chapter 1 Review Exercises ......................................................................................... 106 Chapter 1 Test ............................................................................................................... 112 2 Acute Angles and Right Triangles 2.1 Trigonometric Functions of Acute Angles ............................................................. 115 2.2 Trigonometric Functions of Non-Acute Angles ..................................................... 122 2.3 Approximations of Trigonometric Function Values ............................................... 131 Chapter 2 Quiz (Sections 2.1−2.3)................................................................................ 138 2.4 Solutions and Applications of Right Triangles ....................................................... 139 2.5 Further Applications of Right Triangles ................................................................. 148 Chapter 2 Review Exercises ......................................................................................... 158 Chapter 2 Test ............................................................................................................... 164 3 Radian Measure and the Unit Circle 3.1 Radian Measure ...................................................................................................... 168 3.2 Applications of Radian Measure ............................................................................. 172 3.3 The Unit Circle and Circular Functions .................................................................. 181 Chapter 3 Quiz (Sections 3.1−3.3)................................................................................ 192 3.4 Linear and Angular Speed ...................................................................................... 192 Chapter 3 Review Exercises ......................................................................................... 197 Chapter 3 Test ............................................................................................................... 202 4 Graphs of the Circular Functions 4.1 Graphs of the Sine and Cosine Functions ............................................................... 205 4.2 Translations of the Graphs of the Sine and Cosine Functions ................................ 215 Chapter 4 Quiz (Sections 4.1−4.2)................................................................................ 230 4.3 Graphs of the Tangent and Cotangent Functions.................................................... 233 4.4: Graphs of the Secant and Cosecant Functions ....................................................... 244 Summary Exercises on Graphing Circular Functions ................................................... 253 4.5 Harmonic Motion .................................................................................................... 256 Chapter 4 Review Exercises ......................................................................................... 262 Chapter 4 Test ............................................................................................................... 271 5 Trigonometric Identities 5.1 Fundamental Identities ............................................................................................ 276 5.2 Verifying Trigonometric Identities ......................................................................... 284 5.3 Sum and Difference Identities for Cosine ............................................................... 295 5.4 Sum and Difference Identities for Sine and Tangent .............................................. 302 Chapter 5 Quiz (Sections 5.1−5.4)................................................................................ 313 5.5 Double-Angle Identities .......................................................................................... 314 5.6 Half-Angle Identities .............................................................................................. 323 Summary Exercises on Verifying Trigonometric Identities ......................................... 332 Chapter 5 Review Exercises ......................................................................................... 337 Chapter 5 Test ............................................................................................................... 349 6 Inverse Circular Functions and Trigonometric Equations 6.1 Inverse Circular Functions ...................................................................................... 352 6.2 Trigonometric Equations I ...................................................................................... 365 6.3 Trigonometric Equations II ..................................................................................... 375 Chapter 6 Quiz (Sections 6.1−6.3)................................................................................ 385 6.4 Equations Involving Inverse Trigonometric Functions .......................................... 387 Chapter 6 Review Exercises ......................................................................................... 395 Chapter 6 Test ............................................................................................................... 403 7 Applications of Trigonometry and Vectors 7.1 Oblique Triangles and the Law of Sines ................................................................. 406 7.2 The Ambiguous Case of the Law of Sines.............................................................. 414 7.3 The Law of Cosines ................................................................................................ 421 Chapter 7 Quiz (Sections 7.1−7.3)................................................................................ 433 7.4 Geometrically Defined Vectors and Applications .................................................. 434 7.5 Algebraically Defined Vectors and the Dot Product .............................................. 443 Summary Exercises on Applications of Trigonometry and Vectors ............................ 449 Chapter 7 Review Exercises ......................................................................................... 451 Chapter 7 Test ............................................................................................................... 459 8 Complex Numbers, Polar Equations, and Parametric Equations 8.1 Complex Numbers .................................................................................................. 462 8.2 Trigonometric (Polar) Form of Complex Numbers ................................................ 467 8.3 The Product and Quotient Theorems ...................................................................... 472 8.4 DeMoivre’s Theorem; Powers and Roots of Complex Numbers ........................... 478 Chapter 8 Quiz (Sections 8.1−8.4)................................................................................ 490 8.5 Polar Equations and Graphs .................................................................................... 492 8.6 Parametric Equations, Graphs, and Applications ................................................... 507 Chapter 8 Review Exercises ......................................................................................... 518 Chapter 8 Test ............................................................................................................... 525 1 Chapter R ALGEBRA REVIEW Section R.1 Basic Concepts from Algebra 1. The set 0, 1, 2, 3, describes the set of whole numbers. 2. The set containing no elements is the empty (or null) set, symbolized . 3. The opposite, or negative, of a number is its additive inverse. 4. The distance on a number line from a number to 0 is the absolute value of the number. 5. If the real number a is to the left of the real number b on a number line, then a < (or is less than) b . 6. (a) 0 is a whole number. Therefore, it is also an integer, a rational number, and a real number. 0 belongs to B, C, D, F. (b) 34 is a natural number. Therefore, it is also a whole number, an integer, a rational number, and a real number. 34 belongs to A, B, C, D, F. (c) 9 4 is a rational number and a real number. 9 4 belongs to D, F. (d) 36 6 is a natural number. Therefore, it is also a whole number, an integer, a rational number, and a real number. 36 belongs to A, B, C, D, F. (e) 13 is an irrational number and a real number. 13 belongs to E, F . (f) 216 54 100 25 2.16 is a rational number and a real number. 2.16 belongs to D, F. 7. The set 1 1 1 3 9 27 1, , , , is infinite . No, 3 is not an element of the set. 8. Using set notation, the set { x | x is a natural number less than 6} is {1, 2, 3, 4, 5}. 9. 1 1 10. (a) The additive inverse of 10 is –10. (b) The absolute value of 10 is 10. 11. The elements in the set | is a whole number less than 6 x x are 0, 1, 2, 3, 4, 5 . 12. The elements in the set | is a whole number less than 9 m m are 0, 1, 2, 3, 4, 5, 6, 7,8 . 13. The elements in the set | is a natural number greater than 4 z z are 5, 6, 7,8, . 14. The elements in the set | is a natural number greater than 8 y y are 9, 10, 11, 12, . 15. The elements in the set | is an integer less than or equal to 4 z z are , 1, 0, 1, 2, 3, 4 . 16. The elements in the set | is an integer less than 3 p p are , 2, 1, 0, 1, 2, . 17. The elements in the set | is an even integer greater than 8 a a are 10, 12, 14, 16, . 18. The elements in the set | is an odd integer less than 1 k k are , 7, 5, 3, 1 . 19. The elements in the set | is a number whose absolute value is 4 p p are 4, 4 . 20. The elements in the set | is a number whose absolute value is 7 w w are 7, 7 . 2 Chapter R Algebra Review 21. There are no elements in the set { x | x is an irrational number that is also rational}. Using set notation, this is . 22. There are no elements in the set { r | r is a number that is both positive and negative}. Using set notation, this is . For exercises 23–26, more than one description may be possible. 23. {2, 4, 6, 8} can be described as { x | x is an even natural number less than or equal to 8}. 24. {11, 12, 13, 14} can be described as { x | x is an integer between 10 and 15}. 25. {4, 8, 12, 16, …} can be described as { x | x is a positive multiple of 4}. 26. {…, –6, –3, 0, 3, 6, …} can be described as { x | x is an integer multiple of 3}. For Exercises 27 and 28, A = {1, 2, 3, 4, 5, 6}, B = {1, 3, 5}, C = {1, 6}, and D = {4}. 27. (a) 4 , or A D D (b) 1 B C (c) 1, 3, 5 , or B A B (d) 1, 6 , or C A C 28. (a) 1, 2, 3, 4, 5, 6 , or A B A (b) 1, 3, 4, 5 B D (c) 1, 3, 5, 6 B C (d) 1, 4, 6 C D For Exercises 29–34, 5 12 1 4 8 4 6, , , 3, 0, , 1, 2 , 3, 12 . A 29. 1 and 3 are natural numbers. 30. 0, 1, and 3 are whole numbers. 31. –6, 12 4 (or –3), 0, 1, and 3 are integers. 32. 5 12 1 4 8 4 6, (or 3), , 0, , 1, and 3 are rational numbers. 33. 3, 2 and 12 are irrational numbers. 34. All are real numbers. 35. 6 7 75 21 2 5 9, 6, 0.7, 0, , 7, 4.6, 8, , 13, X (a) Natural numbers: 8, 13, 75 5 (or 15) (b) Whole numbers: 0, 8, 13, 75 5 (or 15) (c) Integers: –9, 0, 8, 13, 75 5 (or 15) (d) Rational numbers: –9, –0.7, 0, 6 7 , 4.6, 8, 21 2 , 13, 75 5 (or 15) (e) Irrational numbers: 6, 7 (f) Real numbers: All are real numbers 36. 3 4 13 40 2 2 8, 5, 0.6, 0, , 3, , 5, , 17, X (a) Natural numbers: 5, 17, 40 2 (or 20) (b) Whole numbers: 0, 5, 17, 40 2 (or 20) (c) Integers: –8, 0, 5, 17, 40 2 (or 20) (d) Rational numbers: –8, –0.6, 0, 3 4 , 5, 13 2 , 17, 40 2 (or 20) (e) Irrational numbers: 5, 3, (f) Real numbers: All are real numbers 37. False. Some are whole numbers, but negative integers are not. 38. True 39. False. No irrational number is an integer. 40. True 41. True 42. False. No rational number is irrational. 43. True 44. True 45. True 46. True 47. 4 x when x = –4 and x = 4. 48 . (a) A. 4 4 (b) A. 4 4 (c) B. 4 4 Section R.2 Real Number Operations and Properties 3 (d) B. 4 4 49. (a) Additive inverse: 6 (b) Absolute value: 6 6 50. (a) Additive inverse: 12 12 (b) Absolute value: 12 12 51. (a) Additive inverse: 6 6 5 5 (b) Absolute value: 6 6 5 5 52. (a) Additive inverse: 0.16 (b) Absolute value: 0.16 0.16 53. 8 8 54. 19 19 55. 3 3 2 2 56. 3 3 4 4 57. 5 5 58. 12 12 59. 2 2 60. 6 6 61. 4.5 4.5 62. 12.4 12.4 63. Pacific Ocean, Indian Ocean, Caribbean Sea, South China Sea, Gulf of California 64. Point Success, Rainier, Matlalcueyetl, Steele, Denali 65. True. 14, 040 14, 040; 12,800 12,800 14,040 > 12,800 66. False. 2375 2375; 8448 8448 2375 is not greater than 8448. Use the following number line to answer exercises 67–74. 67. 6 1 True 68. 4 2 True 69. 4 3 False 70. 3 1 False 71. 3 2 True 72. 6 3 True 73. 3 3 False 74. 5 5 False 75. 2 6 76. 1 5 77. 4 9 78. 1 6 79. 10 5 80. 12 7 81. 7 1 82. 4 10 83. 5 5 84. 6 6 85. 5 0 0 86. 5 14 19 87. 0 5 False 88. 11 0 False 89. 7 7 True 90. 10 10 True 91. 6 7 3 6 10 True 92. 7 4 1 7 5 True 93. 2 5 4 6 10 10 True 94. 8 7 3 5 15 15 True 95. 3 3 3 3 True 96. 4 4 4 4 True 97. 8 6 8 6 False 98. 10 4 10 4 False Section R.2 Real Number Operations and Properties 1. The sum of two negative numbers is negative. 2. The product of two negative numbers is positive. 3. The quotient formed by any nonzero number divided by 0 is undefined, and the quotient formed by 0 divided by any nonzero number is 0. 4. The commutative property is used to change the order of two terms or factors, and the associative property is used to change the grouping of three terms or factors. 5. Like terms are terms with the same variables raised to the same powers. 6. The numerical coefficient in the term 2 7 y z is –7. 7. 3 10 1000 8. 2 5 10 2 10 5 5 9. 3 2 2 1 6 2 6 2 4 10. 7 4 7 28 x y x y 11. 6 ( 13) (6 13) 19 4 Chapter R Algebra Review 12. 8 ( 16) (8 16) 24 13. 15 6 (15 6) 9 14. 17 9 (17 9) 8 15. 13 ( 4) 13 4 9 16. 19 ( 13) 19 13 6 17. 7 3 28 9 19 3 4 12 12 12 18. 5 4 15 8 7 6 9 18 18 18 19. The difference between 4.5 and 2.8 is 1.7. The number with the greater absolute value, 4.5 is positive, so the answer is positive. Thus, 2.8 4.5 1.7. 20. 3.8 6.2 6.2 3.8 2.4 21. 4 9 4 ( 9) 5 22. 3 7 3 ( 7) 4 23. 6 5 6 ( 5) (6 5) 11 24. 8 17 8 ( 17) (8 17) 25 25. 8 ( 13) 8 13 21 26. 12 ( 22) 12 22 34 27. 12.31 ( 2.13) 12.31 2.13 (12.31 2.13) 10.18 28. 15.88 ( 9.42) 15.88 9.42 (15.88 9.42) 6.46 29. 9 4 9 4 27 40 67 10 3 10 3 30 30 30 30. 3 3 3 3 6 21 27 14 4 14 4 28 28 28 31. 8 6 14 ( 14) 14 32. 7 15 22 ( 22) 22 33. 2 4 2 4 2 ( 4) 6 34. 16 13 16 13 16 ( 13) 3 35. The product of two numbers with the same sign is positive, so 8 ( 5) 40. 36. The product of two numbers with the same sign is positive, so 20 ( 4) 80. 37. The product of two numbers with different signs is negative, so 5 ( 7) 35. 38. The product of two numbers with different signs is negative, so 6 ( 9) 54. 39. 4(0) 4 0 0 The multiplication property of 0 states that the product of any real number and 0 is 0. 40. 0( 8) 0 8 0 The multiplication property of 0 states that the product of any real number and 0 is 0. 41. The product of two numbers with different signs is negative, so 5 12 5 2 6 6 . 2 25 2 5 5 5 42. The product of two numbers with different signs is negative, so 9 21 9 3 7 3 . 7 36 7 4 9 4 43. The product of two numbers with the same sign is positive, so 3 24 3 3 8 1. 8 9 8 9 44. The product of two numbers with the same sign is positive, so 2 22 2 2 11 1. 11 4 11 2 2 45. The product of two numbers with different signs is negative, so 0.06(0.4) 0.024. 46. The product of two numbers with different signs is negative, so 0.08(0.7) 0.056. 47. The quotient of two nonzero real numbers with the same sign is positive, so 24 1 1 24 6 4 6. 4 4 4 48. The quotient of two nonzero real numbers with the same sign is positive, so 45 1 1 45 5 9 5. 9 9 9 49. The quotient of two nonzero real numbers with different signs is negative, so 100 1 1 100 4 25 4. 25 25 25 Section R.2 Real Number Operations and Properties 5 50. The quotient of two nonzero real numbers with different signs is negative, so 150 1 1 150 5 30 5. 30 30 30 51. 0 1 0 0 8 8 52. 0 1 0 0 14 14 53. Division by 0 is undefined, so 5 0 is undefined. 54. Division by 0 is undefined, so 13 0 is undefined. 55. The quotient of two nonzero real numbers with the same sign is positive, so 10 12 10 5 2 5 5 25 . 17 5 17 12 17 2 6 102 56. 22 33 22 5 2 11 5 10 23 5 23 33 23 3 11 69 57. 4 4 3 4 5 4 5 3 5 5 5 3 3 5 58. 12 12 5 12 13 12 13 5 13 13 13 5 5 13 59. 12 12 4 12 3 13 4 13 3 13 4 3 3 4 3 9 13 4 13 60. 7 7 2 7 3 6 2 6 3 6 2 3 7 3 7 2 3 2 4 61. 2 3 2 3 2 2 3 62. 3 3 2 6 1 2 63. 7.2 9 0.8 64. 4.5 5 0.9 65. , 1 4 1 4 3 3 d P Q or , 4 1 4 1 3 3 d P Q 66. , 8 4 8 4 12 12 d P R or , 4 8 12 12 d P R 67. , 8 1 8 1 9 9 d Q R or , 1 8 9 9 d Q R 68. , 12 1 13 13 d Q S or , 1 12 13 13 d Q S 69. (a) 2 8 64 (b) 2 8 (8 8) 64 (c) 2 ( 8) ( 8) ( 8) 64 (d) 2 ( 8) (64) 64 70. (a) 3 4 64 (b) 3 4 (4 4 4) 64 (c) 3 ( 4) ( 4) ( 4) ( 4) 64 (d) 3 ( 4) ( 64) 64 71. 4 2 2 2 2 2 16 72. 5 3 3 3 3 3 3 243 73. 4 2 2 2 2 2 16 74. 6 2 2 2 2 2 2 2 64 75. 5 3 3 3 3 3 3 243 76. 5 ( 2) ( 2)( 2)( 2)( 2)( 2) 32 77. 4 2 3 2 3 3 3 3 2 81 162 6 Chapter R Algebra Review 78. 3 4 5 4 5 5 5 4 125 500 79. (a) Because 9 15 3 9 5 14, + ÷ = + = the grandson's answer was correct. (b) The reasoning was incorrect. Division must be done first, and then the addition follows. The grandson's “Order of Process rule” is not correct. It just happens coincidentally in this problem that he obtained the correct answer the wrong way. 80. 7 7 7 7 7 7 7 1 49 7 8 49 7 57 7 50 81. 12 3 4 12 12 24 82. 15 5 2 15 10 25 83. 6 3 12 4 18 12 4 18 3 15 84. 9 4 8 2 36 8 2 36 4 32 85. 10 30 2 3 10 15 3 10 45 55 86. 12 24 3 2 12 8 2 12 16 28 87. 3 5 7 3 ( 2) 5 7 3 ( 8) 5 21 8 16 8 8 88. 3 4 3 5 ( 3) 4 3 5 ( 27) 4 15 27 19 27 8 89. Simplify within parentheses. 2 2 2 18 4 5 (3 7) 18 4 5 ( 4) 18 4 5 4 18 16 5 4 2 5 4 7 4 11 − + − − = − + − − = − + + = − + + = + + = + = 90. Simplify within parentheses. 2 2 2 10 2 9 (1 8) 10 2 9 ( 7) 10 2 9 7 10 4 9 7 6 9 7 15 7 22 91. 3 3 4 9 8 7 2 4 1 7 2 4 1 7 8 4 7 8 4 56 60 92. 4 6 5 3 2 6 5 3 16 30 3 16 30 48 30 48 18 93. 8 4 6 12 8 24 12 4 3 4 3 8 2 7 6 6 7 7 94. 15 5 4 6 8 3 4 6 8 6 5 8 2 6 5 4 12 6 8 2 8 6 5 4 1 4 6 6 5 5 For Exercises 95–104, p = –4, q = 8, and r = –10. 95. 3 2 3 4 2 10 12 20 12 20 8 p r 96. 5 6 5 10 6 4 50 24 50 24 26 r p 97. 2 2 2 2 2 7 4 7 8 10 4 7 8 100 16 7 8 100 16 56 100 72 100 28 p q r 98. 2 2 2 4 2 8 10 16 2 8 10 16 16 10 32 10 42 p q r 99. 8 ( 10) 2 1 8 ( 4) 4 2 q r q p 100. 4 ( 10) 14 7 4 8 4 2 p r p q 101. 5 5( 10) 50 2 3 2( 4) 3( 10) 8 30 50 25 22 11 r p r Section R.2 Real Number Operations and Properties 7 102. 3 3 8 24 3 2 3 4 2 10 12 2 10 24 24 24 3 12 20 12 20 8 q p r 103. 2 2 2 2 3 4 2 3 10 2 2 8 2 3 10 6 4 3 10 4 30 6 6 4 30 26 13 6 6 3 p r q 104. 2 2 2 6 2 8 6 2 4 4 4 4 2 8 4 8 4 4 8 4 1 8 2 q p p 105. distributive property 106. distributive property 107. inverse property 108. inverse property 109. identity property 110. identity property 111. commutative property 112. commutative property 113. associative property 114. associative property 115. closure property 116. closure property 117. Using the distributive property, 2( ) 2 2 . m p m p 118. 3( ) 3 3 a b a b 119. Using the distributive property, 12( ) 12[ ( )] 12( ) ( 12)( ) 12 12 . x y x y x y x y 120. 10( ) 10[ ( )] 10 10 p q p q p q 121. (2 ) 1(2 ) 1(2 ) ( 1) ( ) 2 d f d f d f d f 122. (3 ) 1(3 ) 1(3 ) ( 1)( ) 3 m n m n m n m n 123. Use the second form of the distributive property. 5 3 (5 3) 8 k k k k 124. 6 5 (6 5) 11 a a a a 125. 7 9 7 ( 9 ) [7 ( 9)] 2 r r r r r r 126. 4 6 4 ( 6 ) [4 ( 6)] 2 n n n n n n 127. Use the identity property, then the distributive property. 7 1 7 (1 7) 8 a a a a a a 128. 9 1 9 Identity property (1 9) 10 s s s s s s 129. 1 1 (1 1) 2 x x x x x x 130. 1 1 (1 1) 2 a a a a a a 131. 2( 3 2 ) 2 2( 3 ) 2(2 ) 2 6 4 x y z x y z x y z 132. 8(3 5 ) 8(3 ) 8 8( 5 ) 24 8 40 x y z x y z x y z 133. 3 16 32 40 8 9 27 9 3 16 3 32 3 40 8 9 8 27 8 9 3 16 3 32 5 8 9 8 27 3 2 4 5 3 9 3 y z y z y z y z 134. 1 20 8 32 4 1 1 1 20 8 32 4 4 4 1 1 1 20 8 32 4 4 4 5 + 2 8 = 5 2 8 m y z m y z m y z m y z m y z 8 Chapter R Algebra Review 135. 12 4 3 2 ( 12 4 3 2) 3 y y y y y y 136. 5 9 8 5 ( 5 9 8 5) 11 r r r r r r 137. 6 5 4 6 11 6 4 11 5 6 ( 6 4 11) 11 1 11 or 11 p p p p p p p p p 138. 8 12 3 5 9 8 3 5 12 9 ( 8 3 5) 3 10 3 x x x x x x x x 139. 3( 2) 5 6 3 3 6 5 6 3 3 5 6 6 3 (3 5) 6 6 3 2 15 k k k k k k k k 140. 5( 3) 6 2 4 5 15 6 2 4 5 6 2 15 4 (5 6 2) 11 9 11 r r r r r r r r r r r 141. 10 (4 8) 10 1(4 8) 10 4 8 4 2 y y y y 142. 6 (9 5) 6 1(9 5) 6 9 5 9 1 y y y y 143. 10 (3)( ) 10 (3) 30 30 x y x y x y xy 144. 8 (6)( ) 8 (6) 48 48 x y x y x y xy 145. 2 2 (12 )(7 ) (12 ) (7 ) 3 3 8 (7 ) 56 w z w z w z wz 146. 5 5 (18 )(5 ) (18 ) (5 ) 6 6 15 (5 ) 75 w z w z w z wz 147. 3( 4) 2( 1) 3 12 2 2 3 2 12 2 14 m m m m m m m 148. 6( 5) 4( 6) 6 30 4 24 6 4 30 24 2 54 a a a a a a a 149. 0.25(8 4 ) 0.5(6 2 ) 0.25(8) 0.25(4 ) ( 0.5)(6) ( 0.5)(2 ) 2 3 2 3 (1 1) 2 3 0 1 1 p p p p p p p p p p 150. 0.4(10 5 ) 0.8(5 10 ) 0.4(10) 0.4( 5 ) ( 0.8)(5) ( 0.8)(10 ) 4 2 4 8 10 x x x x x x x Section R.3 Exponents, Polynomials, and Factoring 1. The polynomial 5 2 4 x x is a trinomial of degree 5. 2. A polynomial containing exactly one term is a(n) monomial. 3. A polynomial containing exactly two terms is a(n) binomial). 4. When multiplying powers of like bases, as in 2 3 , y y keep the base and add the exponents. 5. To raise a power to a power, as in 3 2 , a multiply the exponents. 6. (a) B; 2 2 2 ( 5 ) 10 25 x y x xy y (b) C; 2 2 2 ( 5 ) 10 25 x y x xy y (c) A; 2 2 ( 5 )( 5 ) 25 x y x y x y (d) D; 2 2 (5 )(5 ) 25 y x y x y x 7. (a) B; 2 3 (2 3)(4 6 9) 8 27 x x x x . (b) C; 2 3 (2 3)(4 6 9) 8 27 x x x x (c) A; 2 3 (3 2 )(9 6 4 ) 27 8 x x x x 8. D. 2 6 12 3 4 2 3 x x x x 9. B: 4 2 1 ( 1)( 1)( 1) x x x x . Choice A is not a complete factorization, because 2 1 x can be factored as ( 1)( 1). x x The other choices are not correct factorizations of 4 1 x . Section R.3 Exponents, Polynomials, and Factoring 9 10. C: 3 2 8 ( 2)( 2 4) x x x x . Use the pattern for factoring the sum of cubes, 3 3 2 2 ( )( ). x y x y x xy y We have 3 3 3 8 2 x x , so substitute 2 for y . 11. 5 2 5 2 5 2 7 4 4 4 4 16 16 x x x x x x 12. 4 3 4 3 4 3 7 3 6 3 6 18 18 y y y y y y 13. 6 4 6 4 1 11 n n n n n 14. 8 5 8 5 1 14 a a a a a 15. 3 5 3 5 8 9 9 9 9 16. 2 8 2 8 10 4 4 4 4 17. 4 2 5 4 2 5 4 2 5 11 3 6 4 3 6 4 72 72 m m m m m m m m 18. 3 6 4 3 6 4 3 6 4 13 8 2 5 8 2 5 80 80 t t t t t t t t 19. 2 3 4 2 3 4 2 3 1 4 5 5 5 3 5 3 15 15 x y x y x x yy x y x y 20. 3 2 2 3 1 2 3 1 3 4 4 7 4 7 28 28 xy x y xx y y x y x y 21. 2 2 2 2 1 2 1 2 3 3 1 1 8 8 2 2 4 4 mn m n mm nn m n m n 22. 4 2 4 2 4 1 1 2 5 3 2 2 35 35 7 7 10 10 m n mn m m nn m n m n 23. 5 2 2 5 10 2 2 2 24. 3 4 4 3 12 6 6 6 25. 3 3 3 3 2 2 6 6 6 6 6 216 x x x x 26. 5 5 5 5 5 5 25 25 2 2 2 32 x x x x 27. 3 0 2 2 3 2 0 2 2 3 2 0 2 2 6 0 2 6 2 6 6 (4 ) 4 ( ) ( ) 4 4 4 1 4 16 m n m n m n m n m m m 28. 0 4 3 3 0 3 4 3 3 0 3 4 3 3 0 12 12 12 (2 ) 2 ( ) ( ) 2 2 8 1 8 x y x y x y x y y y 29. 3 8 8 3 8 3 24 2 2 3 2 3 6 ( ) ( ) r r r r s s s s 30. 2 4 4 2 4 2 8 2 2 2 ( ) p p p p q q q q 31. 4 2 4 2 4 4 8 8 2 4 4 8 4 8 4 2 4 ( 4) ( ) ( 4) 256 m m m m tp t p t p t p 32. 3 4 3 4 3 3 12 12 2 2 3 6 6 5 ( 5) ( ) ( 5) 125 ( ) n n n n r r r r 33. 0 3 5 1 x y z 34. 0 2 3 3 1 p q r 35. (a) B; 0 6 1 (b) C; 0 6 1 (c) B; 0 6 1 (d) C; 0 6 1 36. (a) D; 0 3 3 1 3 p (b) E; 0 3 3 1 3 p (c) B; 0 3 1 p (d) B; 0 3 1 p 37. 2 2 2 2 2 5 4 7 4 3 5 5 4 4 3 7 5 1 1 2 2 x x x x x x x x x x 38. 3 2 3 2 3 2 3 2 3 2 3 3 4 2 6 3 2 3 1 4 6 1 4 10 4 10 m m m m m m m m m m 10 Chapter R Algebra Review 39. 2 2 2 2 2 2 2 2 12 8 6 4 3 4 2 2 12 2 8 2 6 4 3 4 4 4 2 24 16 12 12 16 8 12 4 y y y y y y y y y y y y y 40. 2 2 2 2 2 2 2 3 8 5 5 3 2 4 3 8 3 5 5 3 5 2 5 4 24 15 15 10 20 9 5 20 p p p p p p p p p p p p p p 41. 4 2 3 2 2 4 2 3 2 2 4 3 2 4 3 2 4 3 2 6 3 2 5 4 6 3 2 5 4 6 2 3 5 1 1 4 1 6 2 7 4 6 2 7 4 m m m m m m m m m m m m m m m m m m m m m m m m m m m m 42. 3 3 2 2 3 3 2 2 3 2 3 2 3 2 8 3 2 4 3 1 8 3 2 4 3 1 8 2 1 4 1 3 3 1 6 3 4 3 1 6 3 4 4 x x x x x x x x x x x x x x x x x x x x x 43. 2 3 2 2 3 2 2 2 2 5 4 3 2 4 3 2 5 1 4 3 4 2 4 5 4 1 12 8 20 4 x x x x x x x x x x x x x x x 44. 3 2 3 2 3 3 5 4 3 2 4 3 2 2 4 2 3 2 8 6 b b b b b b b b b b b 45. Use the FOIL method. 2 2 4 1 7 2 4 7 4 2 1 7 1 2 28 8 7 2 28 2 r r r r r r r r r r r 46. Use the FOIL method. 2 2 5 6 3 4 5 3 5 4 6 3 6 4 15 20 18 24 15 2 24 m m m m m m m m m m m 47. Use the FOIL method. 2 2 3 1 2 7 3 2 3 7 1 2 1 7 6 21 2 7 6 19 7 x x x x x x x x x x x 48. Use the FOIL method. 2 2 5 3 2 3 5 2 5 3 3 2 3 3 10 15 6 9 10 9 9 z z z z z z z z z z z 49. 2 2 2 (2 3)(2 3) (2 ) 3 4 9 m m m m 50. 2 2 2 2 (8 3 )(8 3 ) (8 ) (3 ) 64 9 s t s t s t s t 51. 2 2 2 2 2 4 2 (4 5 )(4 5 ) (4 ) (5 ) 16 25 x y x y x y x y 52. 3 3 3 2 2 6 2 (2 )(2 ) (2 ) 4 m n m n m n m n 53. 2 2 2 2 2 (4 2 ) (4 ) 2(4 )(2 ) (2 ) 16 16 4 m n m m n n m mn n 54. 2 2 2 2 2 ( 6 ) 2( )(6 ) (6 ) 12 36 a b a a b b a ab b 55. 2 2 2 2 2 2 2 2 4 (5 3 ) (5 ) 2(5 )(3 ) (3 ) 25 30 9 r t r r t t r rt t 56. 4 2 4 2 4 2 8 4 2 (2 3 ) (2 ) 2(2 )(3 ) (3 ) 4 12 9 z y z z y y z z y y 57. 2 2 2 2 2 2 4 3 2 3 2 2 4 2 1 1 1 1 1 1 1 1 2 1 2 1 2 1 2 2 1 1 2 1 2 2 4 2 2 1 2 1 x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x Alternatively, 2 2 4 2 1 1 1 1 1 1 1 1 1 1 2 1 x x x x x x x x x x x x Section R.3 Exponents, Polynomials, and Factoring 11 58. 2 2 2 2 2 2 4 4 4 4 4 4 4 4 8 16 8 16 8 16 8 8 16 16 8 16 t t t t t t t t t t t t t t t t t t t t 4 3 2 3 2 2 4 2 8 16 8 64 128 16 128 256 32 256 t t t t t t t t t t Alternatively, 2 2 4 2 4 4 4 4 4 4 4 4 16 16 32 256 t t t t t t t t t t t t 59. 3 2 2 3 2 2 3 2 2 2 2 4 4 2 4 4 2 8 8 6 12 8 y y y y y y y y y y y y y y 60. 3 2 2 3 2 2 3 2 3 3 3 6 9 3 6 9 3 18 27 9 27 27 z z z z z z z z z z z z z z 61. 3 2 2 2 2 2 2 3 2 2 3 2 2 5 2 5 2 5 2 5 2 2 2 5 5 2 5 4 20 25 2 4 20 25 5 4 20 25 8 40 50 20 100 125 8 60 150 125 x x x x x x x x x x x x x x x x x x x x x x 62. 3 2 2 2 2 2 2 3 2 2 3 2 4 1 4 1 4 1 4 1 4 2 4 1 1 4 1 16 8 1 4 16 8 1 1 16 8 1 64 32 4 16 8 1 64 48 12 1 x x x x x x x x x x x x x x x x x x x x x x 63. 4 2 2 2 2 4 3 2 3 2 2 4 3 2 2 2 2 4 4 4 4 4 4 4 16 16 4 16 16 8 24 32 16 q q q q q q q q q q q q q q q q q q q 64. 4 2 2 2 2 4 3 2 3 2 2 4 3 2 3 3 3 6 9 6 9 6 9 6 36 54 9 54 81 12 54 108 81 r r r r r r r r r r r r r r r r r r r 65. The greatest common factor of both terms is 8 a . 40 16 8 5 8 2 8 5 2 ab a a b a a b 66. The greatest common factor of both terms is 5 y . 25 15 5 5 5 3 5 5 3 xy y y x y y x 67. The greatest common factor of all three terms is 3 4 . x y 3 4 2 3 4 3 2 3 2 3 3 2 2 8 12 36 4 2 4 3 4 9 4 2 3 9 x y x y xy xy x y xy x xy y xy x y x y 68. The greatest common factor of all three terms is 2 2 . m n 5 2 3 3 2 2 3 2 2 2 3 10 4 18 2 5 2 2 2 9 2 5 2 9 m n m n m n m n m m n n m n mn m n m n mn 69. The positive factors of 1 (the coefficient of the first term) are 1 and 1. The factors of –15 are –1 and 15, 1 and –15, –3 and 5, or 3 and –5. Try different combinations of these factors until the correct one is found. 2 2 15 5 3 x x x x 70. The positive factors of 1 (the coefficient of the first term) are 1 and 1. The factors of –12 are –1 and 12, 1 and –12, –2 and 6, 2 and –6, –3 and 4, or 3 and –4. Try different combinations of these factors until the correct one is found. 2 12 4 3 x x x x