Description
Book SynopsisCovers topics such as cartesian coordinates, sines and cosines, secants and cosecants, tangents and cotangents, transforming functions, operating on identities, adding Heron's formula, and determining domain and range.
Table of ContentsIntroduction 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
