Trigonometry Workbook For Dummies

Mary Jane Sterling has been a mathematics teacher for more than 30 years.

Introduction 1

About This Book 1

Conventions Used in This Book 1

Foolish Assumptions 2

How This Book is Organized 2

Part I: Trying Out Trig: Starting at the Beginning 2

Part II: Trigonometric Functions 3

Part III: Trigonometric Identities and Equations 3

Part IV: Graphing the Trigonometric Functions 3

Part V: The Part of Tens 4

Icons Used in This Book 4

Where to Go from Here 4

Part I: Trying Out Trig: Starting at the Beginning 5

Chapter 1: Tackling Technical Trig 7

Getting Angles Labeled by Size 7

Naming Angles Where Lines Intersect 9

Writing Angle Names Correctly 10

Finding Missing Angle Measures in Triangles 11

Determining Angle Measures along Lines and outside Triangles 12

Dealing with Circle Measurements 14

Tuning In with the Right Chord 15

Sectioning Off Sectors of Circles 16

Answers to Problems on Tackling Technical Trig 17

Chapter 2: Getting Acquainted with the Graph 21

Plotting Points 21

Identifying Points by Quadrant 23

Working with Pythagoras 24

Keeping Your Distance 26

Finding Midpoints of Segments 27

Dealing with Slippery Slopes 28

Writing Equations of Circles 30

Graphing Circles 32

Answers to Problems on Graphing 33

Chapter 3: Getting the Third Degree 37

Recognizing First-Quadrant Angles 37

Expanding Angles to Other Quadrants 39

Expanding Angles beyond 360 Degrees 40

Coordinating with Negative Angle Measures 41

Dealing with Coterminal Angles 42

Answers to Problems on Measuring in Degrees 43

Chapter 4: Recognizing Radian Measure 45

Becoming Acquainted with Graphed Radians 45

Changing from Degrees to Radians 47

Changing from Radians to Degrees 49

Measuring Arcs 50

Determining the Area of a Sector 52

Answers to Problems on Radian Measure 53

Chapter 5: Making Things Right with Right Triangles 57

Naming the Parts of a Right Triangle 57

Completing Pythagorean Triples 59

Completing Right Triangles 61

Working with the 30-60-90 Right Triangle 62

Using the Isosceles Right Triangle 64

Using Right Triangles in Applications 65

Answers to Problems on Right Triangles 68

Part II: Trigonometric Functions 75

Chapter 6: Defining Trig Functions with a Right Triangle 77

Defining the Sine Function 78

Cooperating with the Cosine Function 79

Sunning with the Tangent Definition 80

Hunting for the Cosecant Definition 81

Defining the Secant Function 82

Coasting Home with the Cotangent 83

Establishing Trig Functions for Angles in Special Right Triangles 85

Applying the Trig Functions 86

Answers to Problems on Defining Trig Functions 88

Chapter 7: Discussing Properties of the Trig Functions 93

Defining a Function and Its Inverse 93

Deciding on the Domains 95

Reaching Out for the Ranges 97

Closing In on Exact Values 98

Determining Exact Values for All Functions 99

Answers to Problems in Properties of Trig Functions 102

Chapter 8: Going Full Circle with the Circular Functions 105

Finding Points on the Unit Circle 105

Determining Reference Angles 108

Assigning the Signs of Functions by Quadrant 111

Figuring Out Trig Functions around the Clock 113

Answers to Problems in Going Full Circle 115

Part III: Trigonometric Identities and Equations 119

Chapter 9: Identifying the Basic Identities 121

Using the Reciprocal Identities 121

Creating the Ratio Identities 123

Playing Around with Pythagorean Identities 124

Solving Identities Using Reciprocals, Ratios, and Pythagoras 127

Answers to Problems on Basic Identities 130

Chapter 10: Using Identities Defined with Operations 135

Adding Up the Angles with Sum Identities 135

Subtracting Angles with Difference Identities 138

Doubling Your Pleasure with Double Angle Identities 140

Multiplying the Many by Combining Sums and Doubles 142

Halving Fun with Half-Angle Identities 144

Simplifying Expressions with Identities 146

Solving Identities 148

Answers to Problems on Using Identities 151

Chapter 11: Techniques for Solving Trig Identities 161

Working on One Side at a Time 161

Working Back and Forth on Identities 164

Changing Everything to Sine and Cosine 165

Multiplying by Conjugates 167

Squaring Both Sides 168

Finding Common Denominators 169

Writing All Functions in Terms of Just One 171

Answers to Problems Techniques for Solving Identities 173

Chapter 12: Introducing Inverse Trig Functions 185

Determining the Correct Quadrants 185

Evaluating Expressions Using Inverse Trig Functions 187

Solving Equations Using Inverse Trig Functions 189

Creating Multiple Answers for Multiple and Half-Angles 191

Answers to Problems on Inverse Trig Functions 193

Chapter 13: Solving Trig Equations 195

Solving for Solutions within One Rotation 195

Solving Equations with Multiple Answers 197

Special Factoring for a Solution 200

Using Fractions and Common Denominators to Solve Equations 202

Using the Quadratic Formula 205

Answers to Problems on Solving Trig Equations 206

Chapter 14: Revisiting the Triangle with New Laws 213

Using the Law of Sines 213

Adding the Law of Cosines 215

Dealing with the Ambiguous Case 218

Investigating the Law of Tangents 219

Finding the Area of a Triangle the Traditional Way 220

Flying In with Heron’s Formula 221

Finding Area with an Angle Measure 222

Applying Triangles 223

Answers to Problems on Triangles 224

Part IV: Graphing the Trigonometric Functions 231

Chapter 15: Graphing Sine and Cosine 233

Determining Intercepts and Extreme Values 233

Graphing the Basic Sine and Cosine Curves 235

Changing the Amplitude 236

Adjusting the Period of the Curves 238

Graphing from the Standard Equation 239

Applying the Sine and Cosine Curves to Life 241

Answers to Problems on Graphing Sine and Cosine 243

Chapter 16: Graphing Tangent and Cotangent 249

Establishing Vertical Asymptotes 249

Graphing Tangent and Cotangent 250

Altering the Basic Curves 252

Answers to Problems on Graphing Tangent and Cotangent 253

Chapter 17: Graphing Cosecant, Secant, and Inverse Trig Functions 255

Determining the Vertical Asymptotes 255

Graphing Cosecant and Secant 256

Making Changes to the Graphs of Cosecant and Secant 257

Analyzing the Graphs of the Inverse Trig Functions 258

Answers to Problems on Cosecant, Secant, and Inverse Trig Functions 261

Chapter 18: Transforming Graphs of Trig Functions 263

Sliding the Graphs Left or Right 263

Sliding the Graphs Up or Down 264

Changing the Steepness 266

Reflecting on the Situation — Horizontally 267

Reflecting on Your Position — Vertically 268

Putting It All Together 269

Combining Trig Functions with Polynomials 270

Answers to Problems on Transforming Trig Functions 272

Part V: The Part of Tens 277

Chapter 19: Ten Identities with a Negative Attitude 279

Negative Angle Identities 279

Complementing and Supplementing Identities 279

Doing Fancy Factoring with Identities 280

Chapter 20: Ten Formulas to Use in a Circle 281

Running Around in Circles 281

Adding Up the Area 281

Defeating an Arc Rival 281

Sectioning Off the Sector 282

Striking a Chord 282

Ringing True 283

Inscribing and Radii 283

Circumscribing and Radii 283

Righting a Triangle 284

Inscribing a Polygon 284

Chapter 21: Ten Ways to Relate the Sides and Angles of Any Triangle 285

Relating with the Law of Sines 285

Hatching a Little Heron 286

Summing Sines 286

You Half It or You Don’t 286

Cozying Up with Cosines 286

Angling for an Angle 286

Mixing It Up with Cosines 286

Heron Again, Gone Tomorrow 287

Divide and Conquer with the Tangent 287

Heron Lies the Problem 287

Appendix: Trig Functions Table 289

Index 293