Copyright © 2021 Pearson Education, Inc. 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 .
