Complete Solution Manual Trigonometry 12th Edition Questions & Answers with rationales (Chapter 1-8)

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CONTENTS R Algebra Review R.1 Basic Concepts from Algebra ............................... ..................................................... 1 R.2 Real Number Operat ions 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 Ci rcular 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 Copyright © 2021 Pearson Education, Inc. 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 a nd Cotangent Functions ............. ....................................... 233 4.4: Graphs of the Secant and Cosecant Functions .............. ......................................... 244 Summary Exercises on Graphi ng 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 Trigonomet ric Identities ........................ ................................................. 284 5.3 Sum and Difference Iden tities for Cosine .................. ............................................. 295 5.4 Sum and Difference Identitie s 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 Verifyi ng 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 Tr igonometric Functions ....... ................................... 387 Chapter 6 Review Exercises .................................... ..................................................... 395 Chapter 6 Test ................................................ ............................................................... 403 7 Applications of Trigonometry and Vectors 7.1 Oblique Triangles an d 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 V ectors and Applications ............ ...................................... 434 7.5 Algebraically Defined Vect ors and the Dot Product ......... ..................................... 443 Summary Exercises on Applicati ons of Trigonometry and Vectors . ........................... 449 Chapter 7 Review Exercises .................................... ..................................................... 451 Chapter 7 Test ................................................ ............................................................... 459 Copyright © 2021 Pearson Education, Inc. 8 Complex Numbers, Polar Equat ions, 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 a nd 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, Gra phs, and Applications ............ ....................................... 507 Chapter 8 Review Exercises .................................... ..................................................... 518 Chapter 8 Test ................................................ ............................................................... 525 Copyright © 2021 Pearson Education, Inc. 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
100 252.16 is a rational number and a real number. 2.16 belongs to D, F. 7. The set 11 1
392 71, , , ,  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. 11  10. (a) The additive inverse of 10 is –10. (b) The absolute valu e of 10 is 10. 11. The elements in the set | is a whole number less than 6xx are 0, 1, 2, 3, 4, 5 . 12. The elements in the set | is a whole number less than 9mm are 0, 1, 2, 3, 4, 5, 6, 7,8 . 13. The elements in the set | is a natural number greater than 4zz are 5, 6, 7,8, .  14. The elements in the set | is a natural number greater than 8yy are 9, 10, 11, 12, .  15. The elements in the set | is an integer less than or equal to 4zz are , 1 ,0 ,1 ,2 ,3 ,4 . 16. The elements in the set | is an integer less than 3pp are ,2 ,1 , 0 , 1 , 2 , . 17. The elements in the set | is an even integer greater than 8aa are 10, 12, 14, 16, .  18. The elements in the set | is an odd integer less than 1kk are ,7 ,5 ,3 ,1 . 19. The elements in the set | is a number whose absolute value is 4pp are 4, 4 . 20. The elements in the set | is a number whose absolute value is 7ww are 7, 7 .