Trigonometry Workbook For Dummies

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

• Introduction 1About This Book 1Conventions Used in This Book 1Foolish Assumptions 2How This Book is Organized 2Part I: Trying Out Trig: Starting at the Beginning 2Part II: Trigonometric Functions 3Part III: Trigonometric Identities and Equations 3Part IV: Graphing the Trigonometric Functions 3Part V: The Part of Tens 4Icons Used in This Book 4Where to Go from Here 4Part I: Trying Out Trig: Starting at the Beginning 5Chapter 1: Tackling Technical Trig 7Getting Angles Labeled by Size 7Naming Angles Where Lines Intersect 9Writing Angle Names Correctly 10Finding Missing Angle Measures in Triangles 11Determining Angle Measures along Lines and outside Triangles 12Dealing with Circle Measurements 14Tuning In with the Right Chord 15Sectioning Off Sectors of Circles 16Answers to Problems on Tackling Technical Trig 17Chapter 2: Getting Acquainted with the Graph 21Plotting Points 21Identifying Points by Quadrant 23Working with Pythagoras 24Keeping Your Distance 26Finding Midpoints of Segments 27Dealing with Slippery Slopes 28Writing Equations of Circles 30Graphing Circles 32Answers to Problems on Graphing 33Chapter 3: Getting the Third Degree 37Recognizing First-Quadrant Angles 37Expanding Angles to Other Quadrants 39Expanding Angles beyond 360 Degrees 40Coordinating with Negative Angle Measures 41Dealing with Coterminal Angles 42Answers to Problems on Measuring in Degrees 43Chapter 4: Recognizing Radian Measure 45Becoming Acquainted with Graphed Radians 45Changing from Degrees to Radians 47Changing from Radians to Degrees 49Measuring Arcs 50Determining the Area of a Sector 52Answers to Problems on Radian Measure 53Chapter 5: Making Things Right with Right Triangles 57Naming the Parts of a Right Triangle 57Completing Pythagorean Triples 59Completing Right Triangles 61Working with the 30-60-90 Right Triangle 62Using the Isosceles Right Triangle 64Using Right Triangles in Applications 65Answers to Problems on Right Triangles 68Part II: Trigonometric Functions 75Chapter 6: Defining Trig Functions with a Right Triangle 77Defining the Sine Function 78Cooperating with the Cosine Function 79Sunning with the Tangent Definition 80Hunting for the Cosecant Definition 81Defining the Secant Function 82Coasting Home with the Cotangent 83Establishing Trig Functions for Angles in Special Right Triangles 85Applying the Trig Functions 86Answers to Problems on Defining Trig Functions 88Chapter 7: Discussing Properties of the Trig Functions 93Defining a Function and Its Inverse 93Deciding on the Domains 95Reaching Out for the Ranges 97Closing In on Exact Values 98Determining Exact Values for All Functions 99Answers to Problems in Properties of Trig Functions 102Chapter 8: Going Full Circle with the Circular Functions 105Finding Points on the Unit Circle 105Determining Reference Angles 108Assigning the Signs of Functions by Quadrant 111Figuring Out Trig Functions around the Clock 113Answers to Problems in Going Full Circle 115Part III: Trigonometric Identities and Equations 119Chapter 9: Identifying the Basic Identities 121Using the Reciprocal Identities 121Creating the Ratio Identities 123Playing Around with Pythagorean Identities 124Solving Identities Using Reciprocals, Ratios, and Pythagoras 127Answers to Problems on Basic Identities 130Chapter 10: Using Identities Defined with Operations 135Adding Up the Angles with Sum Identities 135Subtracting Angles with Difference Identities 138Doubling Your Pleasure with Double Angle Identities 140Multiplying the Many by Combining Sums and Doubles 142Halving Fun with Half-Angle Identities 144Simplifying Expressions with Identities 146Solving Identities 148Answers to Problems on Using Identities 151Chapter 11: Techniques for Solving Trig Identities 161Working on One Side at a Time 161Working Back and Forth on Identities 164Changing Everything to Sine and Cosine 165Multiplying by Conjugates 167Squaring Both Sides 168Finding Common Denominators 169Writing All Functions in Terms of Just One 171Answers to Problems Techniques for Solving Identities 173Chapter 12: Introducing Inverse Trig Functions 185Determining the Correct Quadrants 185Evaluating Expressions Using Inverse Trig Functions 187Solving Equations Using Inverse Trig Functions 189Creating Multiple Answers for Multiple and Half-Angles 191Answers to Problems on Inverse Trig Functions 193Chapter 13: Solving Trig Equations 195Solving for Solutions within One Rotation 195Solving Equations with Multiple Answers 197Special Factoring for a Solution 200Using Fractions and Common Denominators to Solve Equations 202Using the Quadratic Formula 205Answers to Problems on Solving Trig Equations 206Chapter 14: Revisiting the Triangle with New Laws 213Using the Law of Sines 213Adding the Law of Cosines 215Dealing with the Ambiguous Case 218Investigating the Law of Tangents 219Finding the Area of a Triangle the Traditional Way 220Flying In with Heron’s Formula 221Finding Area with an Angle Measure 222Applying Triangles 223Answers to Problems on Triangles 224Part IV: Graphing the Trigonometric Functions 231Chapter 15: Graphing Sine and Cosine 233Determining Intercepts and Extreme Values 233Graphing the Basic Sine and Cosine Curves 235Changing the Amplitude 236Adjusting the Period of the Curves 238Graphing from the Standard Equation 239Applying the Sine and Cosine Curves to Life 241Answers to Problems on Graphing Sine and Cosine 243Chapter 16: Graphing Tangent and Cotangent 249Establishing Vertical Asymptotes 249Graphing Tangent and Cotangent 250Altering the Basic Curves 252Answers to Problems on Graphing Tangent and Cotangent 253Chapter 17: Graphing Cosecant, Secant, and Inverse Trig Functions 255Determining the Vertical Asymptotes 255Graphing Cosecant and Secant 256Making Changes to the Graphs of Cosecant and Secant 257Analyzing the Graphs of the Inverse Trig Functions 258Answers to Problems on Cosecant, Secant, and Inverse Trig Functions 261Chapter 18: Transforming Graphs of Trig Functions 263Sliding the Graphs Left or Right 263Sliding the Graphs Up or Down 264Changing the Steepness 266Reflecting on the Situation — Horizontally 267Reflecting on Your Position — Vertically 268Putting It All Together 269Combining Trig Functions with Polynomials 270Answers to Problems on Transforming Trig Functions 272Part V: The Part of Tens 277Chapter 19: Ten Identities with a Negative Attitude 279Negative Angle Identities 279Complementing and Supplementing Identities 279Doing Fancy Factoring with Identities 280Chapter 20: Ten Formulas to Use in a Circle 281Running Around in Circles 281Adding Up the Area 281Defeating an Arc Rival 281Sectioning Off the Sector 282Striking a Chord 282Ringing True 283Inscribing and Radii 283Circumscribing and Radii 283Righting a Triangle 284Inscribing a Polygon 284Chapter 21: Ten Ways to Relate the Sides and Angles of Any Triangle 285Relating with the Law of Sines 285Hatching a Little Heron 286Summing Sines 286You Half It or You Don’t 286Cozying Up with Cosines 286Angling for an Angle 286Mixing It Up with Cosines 286Heron Again, Gone Tomorrow 287Divide and Conquer with the Tangent 287Heron Lies the Problem 287Appendix: Trig Functions Table 289Index 293