Trigonometry Workbook for Dummies

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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

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

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