Table of Contents
Introduction 1
Chapter 1 Review of Numbers and Coordinate Systems 3
1.1 Review of Numbers, Including Natural, Whole, Integers, Zero, Rational, Irrational, Real, Complex, and Imaginary Numbers 3
1.2 Absolute Value 7
1.3 Significant Digits and Rounding Numbers and Decimals 7
1.4 Review of Coordinate Systems, Including Two- and Three-Dimensional Rectangular Coordinates, Polar Coordinates, Cylindrical Coordinates, and Spherical Coordinates 9
1.5 Chapter 1 Summary and Highlights 14
Chapter 2 Review of Geometry 17
2.1 Introduction 17
2.2 Lines and Angles 19
2.3 Triangles 23
2.4 Polygons and Quadrilaterals 28
2.5 Conic Sections, Including Circles, Arcs and Angles, Ellipses, Parabolas, and Hyperbolas 31
2.6 Three-Dimensional Objects, Including Cubes, Rectangular Solids, Cylinders, Spheres, Cones, and Pyramids 45
2.7 Chapter 2 Summary and Highlights 48
Chapter 3 Triangles and Trigonometric Functions 49
3.1 Right Triangles and the Trigonometric Functions 49
3.2 Solving Right Triangles 54
3.3 Examples and Applications of Right Triangles 55
3.4 Oblique Triangles and the Law of Sines and Law of Cosines 62
3.5 Solving Oblique Triangles 67
3.6 Examples and Applications of Oblique Triangles 72
3.7 Finding the Area of a Triangle 77
3.8 Chapter 3 Summary and Highlights 87
Chapter 4 Trigonometric Functions in a Coordinate System and Circular Functions 91
4.1 Review of Functions and Their Properties 92
4.2 Types of Functions, Including Composite, Inverse, Linear, Nonlinear, Even, Odd, Exponential, Logarithmic, Identity, Absolute Value, Squaring, Cubing, Square Root, Cube Root, Reciprocal, and Functions with More Than One Variable 94
4.3 CoordinateSystems, Radians, Degrees, and Arc Length 103
4.4 Angles in Standard Position and Coterminal Angles 107
4.5 The Trigonometric Functions Defined in a Coordinate System in Standard Position, Quadrant Signs, and Quadrantal Angles 108
4.6 Reference Angles and Reference Triangles 112
4.7 Negative Angles 118
4.8 Reciprocal Functions and Cofunction Relationships 119
4.9 Circular Functions and the Unit Circle 120
4.10 Linear and Angular Velocity 125
4.11 Chapter 4 Summary and Highlights 128
Chapter 5 Graphs of Trigonometric and Circular Functions and Their Periodic Nature 131
5.1 Circular Motion 131
5.2 Graphs of Sine and Cosine 137
5.3 Transforming Graphs of Sine and Cosine Through Changes in Amplitude, Period, and Vertical and Horizontal Shifting 144
5.4 Applications of Sinusoids 157
5.5 Graphs of Secant and Cosecant 164
5.6 Graphs of Tangent and Cotangent 169
5.7 Chapter 5 Summary and Highlights 173
Chapter 6 Inverse Trigonometric Functions 177
6.1 Review of General Inverse Functions 177
6.2 Inverse Trigonometric Functions 183
6.3 Inverse Sine and Inverse Cosine 188
6.4 Inverse Tangent 197
6.5 Inverse Cotangent, Inverse Secant, and Inverse Cosecant 204
6.6 Chapter 6 Summary and Highlights 214
Chapter 7 Trigonometric Identities 217
7.1 Summary of Identities 217
7.2 Quotient Identities and Reciprocal Identities 220
7.3 Pythagorean Identities 220
7.4 Negative Number/Angle Identities 222
7.5 Verifying Trigonometric Identities 225
7.6 Sum and Difference of Angles/Numbers Identities, Also Called Addition and Subtraction Identities 228
7.7 Cofunction Identities 234
7.8 Supplementary Angle Relations 237
7.9 Double-Angle/Number Identities 238
7.10 Half-Angle Identities 243
7.11 Product-To-Sum Identities 246
7.12 Sum/Difference-To-Product Identities 248
7.13 Squared Formulas 252
7.14 Chapter 7 Summary and Highlights 253
Chapter 8 Trigonometric Functions in Equations and Inequalities 257
8.1 Review of Solving Algebraic Equations 257
8.2 Review of Solving Algebraic Quadratic Equations 262
8.3 Review of Solving Algebraic Inequalities 269
8.4 Solving Algebraic Equations and Inequalities Using Graphing 270
8.5 Introduction to Solving Trigonometric Equations and Inequalities 273
8.6 Solving Simple Trigonometric Equations Using Standard Position Angles, Reference Triangles, and Identities 274
8.7 Solving Trigonometric Equations Involving Powers Using Factoring, a Unit Circle, and Identities 276
8.8 Solving Trigonometric Equations and Inequalities Using the Quadratic Formula, Identities, Unit Circles, Factoring, and Graphing 281
8.9 Estimating Solutions to Trigonometric Equations and Inequalities Using Graphing 287
8.10 Chapter 8 Summary and Highlights 290
Chapter 9 Trigonometric Functions and Vectors 293
9.1 Definitions of Vectors 293
9.2 Representing Vectors in Terms of Their Components in a Coordinate System 295
9.3 Representing Vectors in Terms of Their Components in a Coordinate System Using the Unit Vectors i, j, and k 298
9.4 Addition and Subtraction of Vectors 300
9.5 Simple Vector Problems 304
9.6 Multiplying a Vector with a Scalar 309
9.7 Dot or Scalar Products 310
9.8 Vector or Cross Product 314
9.9 Chapter 9 Summary and Highlights 318
Chapter 10 Trigonometric Functions in Polar Coordinates and Equations, and Parametric Equations 321
10.1 Polar Coordinates Defined 321
10.2 Converting Between Rectangular and Polar Coordinate Systems and Equations 325
10.3 Graphing Polar Equations 332
10.4 Parametric Equations 342
10.5 Chapter 10 Summary and Highlights 353
Chapter 11 Complex Numbers and the Complex Plane 355
11.1 Complex Numbers Defined 355
11.2 The Complex Plane in Rectangular Form 358
11.3 Addition and Subtraction of Complex Numbers in Rectangular Form 359
11.4 Complex Numbers in Polar Form and the Complex Plane 360
11.5 Converting Between Rectangular and Polar Form 362
11.6 Multiplication and Division of Complex Numbers in Rectangular and Polar Forms 364
11.7 Powers and Roots of Complex Numbers 372
11.8 Chapter 11 Summary and Highlights 378
Chapter 12 Relationships Between Trigonometric Functions, Exponential Functions, Hyperbolic Functions, and Series Expansions 381
12.1 Relationships Between Trigonometric and Exponential Functions 381
12.2 Background: Summary of Sequences, Progressions, and Series, and Expanding a Function into a Series 383
12.3 Hyperbolic Functions 389
12.4 Chapter 12 Summary and Highlights 393
Chapter 13 Spherical Trigonometry 395
13.1 Definitions and Properties 395
13.2 Measuring Spherical Triangles 399
13.3 The Law of Sines and the Law of Cosines for Spherical Triangles for Calculating Sides and Angles 400
13.4 Celestial Sphere 405
13.5 Chapter 13 Summary and Highlights 407
Index 409
