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## Summary

This book, intended for a graphing calculator optional trigonometry course, offers students the content and tools they will need to successfully master trigonometry. The authors have addressed the needs of students who will continue their study of mathematics, as well as those who are taking trigonometry as their final mathematics course. Emphasis is placed on exploring mathematical concepts by using real date, current applications and optional technology. Applied examples and exercises, allowing students to focus on real-life applications of mathematics. Selected examples feature traditional algebraic as well as optional graphing calculator solutions. We have taken great care to only use this format in examples where the graphing calculator can naturally be used to support and/or enhance the algebraic solution. For those interested in Mathematics.

## Table of Contents

Preface | p. viii |

Supplements Guide | p. xii |

Trigonometric Functions | p. 1 |

Angles | p. 2 |

Basic Terminology | |

Degree Measure | |

Standard Position | |

Coterminal Angles | |

Angle Relationships and Similar Triangles | p. 9 |

Geometric Properties | |

Triangles | |

Trigonometric Functions | p. 20 |

Trigonometric Functions | |

Quadrantal Angles | |

Using the Definitions of the Trigonometric Functions | p. 27 |

Reciprocal Identities | |

Signs and Ranges of Function Values | |

Pythagorean Identities | |

Quotient Identities | |

Summary | p. 37 |

Review Exercises | p. 39 |

Test | p. 42 |

Quantitative Reasoning | p. 44 |

Acute Angles and Right Triangles | p. 45 |

Trigonometric Functions of Acute Angles | p. 46 |

Right-Triangle-Based Definitions of the Trigonometric Functions | |

Cofunctions | |

Trigonometric Function Values of Special Angles | |

Trigonometric Functions of Non-Acute Angles | p. 55 |

Reference Angles | |

Special Angles as Reference Angles | |

Finding Angle Measures with Special Angles | |

Finding Trigonometric Function Values Using a Calculator | p. 62 |

Finding Function Values Using a Calculator | |

Finding Angle Measures Using a Calculator | |

Solving Right Triangles | p. 68 |

Significant Digits | |

Solving Triangles | |

Angles of Elevation or Depression | |

Further Applications of Right Triangles | p. 77 |

Bearing | |

Further Applications | |

Summary | p. 86 |

Review Exercises | p. 88 |

Test | p. 91 |

Quantitative Reasoning | p. 92 |

Radian Measure and Circular Functions | p. 93 |

Radian Measure | p. 94 |

Radian Measure | |

Converting Between Degrees and Radians | |

Finding Function Values for Angles in Radians | |

Applications of Radian Measure | p. 99 |

Arc Length on a Circle | |

Area of a Sector of a Circle | |

The Unit Circle and Circular Functions | p. 108 |

Circular Functions | |

Finding Values of Circular Functions | |

Determining a Number with a Given Circular Function Value | |

Applying Circular Functions | |

Linear and Angular Speed | p. 116 |

Linear Speed | |

Angular Speed | |

Summary | p. 122 |

Review Exercises | p. 124 |

Test | p. 128 |

Quantitative Reasoning | p. 129 |

Graphs of the Circular Functions | p. 131 |

Graphs of the Sine and Cosine Functions | p. 132 |

Periodic Functions | |

Graph of the Sine Function | |

Graph of the Cosine Function | |

Graphing Techniques, Amplitude, and Period | |

Using a Trigonometric Model | |

Translations of the Graphs of the Sine and Cosine Functions | p. 146 |

Horizontal Translations | |

Vertical Translations | |

Combinations of Translations | |

Determining a Trigonometric Model Using Curve Fitting | |

Graphs of the Other Circular Functions | p. 155 |

Graphs of the Cosecant and Secant Functions | |

Graphs of the Tangent and Cotangent Functions | |

Addition of Ordinates | |

Summary Exercises on Graphing Circular Functions | p. 168 |

Harmonic Motion | p. 168 |

Simple Harmonic Motion | |

Damped Oscillatory Motion | |

Summary | p. 173 |

Review Exercises | p. 175 |

Test | p. 178 |

Quantitative Reasoning | p. 779 |

Trigonometric Identities | p. 181 |

Fundamental Identities | p. 182 |

Negative-Angle Identities | |

Fundamental Identities | |

Using the Fundamental Identities | |

Verifying Trigonometric Identities | p. 188 |

Verifying Identities by Working with One Side | |

Verifying Identities by Working with Both Sides | |

Sum and Difference Identities for Cosine | p. 197 |

Difference Identity for Cosine | |

Sum Identity for Cosine | |

Cofunction Identities | |

Applying the Sum and Difference Identities | |

Sum and Difference Identities for Sine and Tangent | p. 205 |

Sum and Difference Identities for Sine | |

Sum and Difference Identities for Tangent | |

Applying the Sum and Difference Identities | |

Double-Angle Identities | p. 212 |

Double-Angle Identities | |

Product-to-Sum and Sum-to-Product Identities | |

Half-Angle Identities | p. 221 |

Half-Angle Identities | |

Applying the Half-Angle Identities | |

Summary Exercises on Verifying Trigonometric Identities | p. 227 |

Summary | p. 229 |

Review Exercises | p. 231 |

Test | p. 233 |

Quantitative Reasoning | p. 234 |

Inverse Circular Functions and Trigonometric Equations | p. 235 |

Inverse Circular Functions | p. 236 |

Inverse Functions | |

Inverse Sine Function | |

Inverse Cosine Function | |

Inverse Tangent Function | |

Remaining Inverse Circular Functions | |

Inverse Function Values | |

Trigonometric Equations I | p. 249 |

Solving by Linear Methods | |

Solving by Factoring | |

Solving by Quadratic Methods | |

Solving by Using Trigonometric Identities | |

Trigonometric Equations II | p. 256 |

Equations with Half-Angles | |

Equations with Multiple Angles | |

Equations Involving Inverse Trigonometric Functions | p. 262 |

Solving for x in Terms of y Using Inverse Functions | |

Solving Inverse Trigonometric Equations | |

Summary | p. 269 |

Review Exercises | p. 271 |

Test | p. 273 |

Quantitative Reasoning | p. 274 |

Applications of Trigonometry and Vectors | p. 275 |

Oblique Triangles and the Law of Sines | p. 276 |

Congruency and Oblique Triangles | |

Derivation of the Law of Sines | |

Solving SAA and ASA Triangles (Case 1) | |

Area of a Triangle | |

The Ambiguous Case of the Law of Sines | p. 287 |

Description of the Ambiguous Case | |

Solving SSA Triangles (Case 2) | |

Analyzing Data for Possible Number of Triangles | |

The Law of Cosines | p. 293 |

Derivation of the Law of Cosines | |

Solving SAS and SSS Triangles (Cases 3 and 4) | |

Heron's Formula for the Area of a Triangle | |

Vectors, Operations, and the Dot Product | p. 305 |

Basic Terminology | |

Algebraic Interpretation of Vectors | |

Operations with Vectors | |

Dot Product and the Angle Between Vectors | |

Applications of Vectors | p. 315 |

The Equilibrant | |

Incline Applications | |

Navigation Applications | |

Summary | p. 322 |

Review Exercises | p. 325 |

Test | p. 329 |

Quantitative Reasoning | p. 330 |

Complex Numbers, Polar Equations, and Parametric Equations | p. 331 |

Complex Numbers | p. 332 |

Basic Concepts of Complex Numbers | |

Complex Solutions of Equations | |

Operations on Complex Numbers | |

Trigonometric (Polar) Form of Complex Numbers | p. 341 |

The Complex Plane and Vector Representation | |

Trigonometric (Polar) Form | |

Converting Between Trigonometric and Polar Forms | |

An Application of Complex Numbers to Fractals | |

The Product and Quotient Theorems | p. 347 |

Products of Complex Numbers in Trigonometric Form | |

Quotients of Complex Numbers in Trigonometric Form | |

De Moivre's Theorem; Powers and Roots of Complex Numbers | p. 352 |

Powers of Complex Numbers (De Moivre's Theorem) | |

Roots of Complex Numbers | |

Polar Equations and Graphs | p. 359 |

Polar Coordinate System | |

Graphs of Polar Equations | |

Converting from Polar to Rectangular Equations | |

Classifying Polar Equations | |

Parametric Equations, Graphs, and Applications | p. 371 |

Basic Concepts | |

Parametric Graphs and Their Rectangular Equivalents | |

The Cycloid | |

Applications of Parametric Equations | |

Summary | p. 379 |

Review Exercises | p. 382 |

Test | p. 385 |

Quantitative Reasoning | p. 386 |

Exponential and Logarithmic Functions | p. 387 |

Exponential Functions | p. 388 |

Exponents and Properties | |

Exponential Functions | |

Exponential Equations | |

Compound Interest | |

The Number e and Continuous Compounding | |

Exponential Models and Curve Fitting | |

Logarithmic Functions | p. 402 |

Logarithms | |

Logarithmic Equations | |

Logarithmic Functions | |

Properties of Logarithms | |

Evaluating Logarithms; Equations and Applications | p. 413 |

Common Logarithms | |

Natural Logarithms | |

Logarithms to Other Bases | |

Exponential and Logarithmic Equations | |

Summary | p. 426 |

Review Exercises | p. 428 |

Test | p. 431 |

Quantitative Reasoning | p. 432 |

Equations and Inequalities | p. 433 |

Equations | |

Solving Linear Equations | |

Solving Quadratic Equations | |

Inequalities | |

Solving Linear Inequalities | |

Interval Notation | |

Three-Part Inequalities | |

Graphs of Equations | p. 442 |

The Rectangular Coordinate System | |

The Pythagorean Theorem and the Distance Formula | |

The Midpoint Formula | |

Graphing Equations | |

Circles | |

Functions | p. 450 |

Relations and Functions | |

Domain and Range | |

Determining Whether a Relation Is a Function | |

Function Notation | |

Increasing, Decreasing, and Constant Functions | |

Graphing Techniques | p. 460 |

Stretching and Shrinking | |

Reflecting | |

Symmetry | |

Translations | |

Glossary | p. 469 |

Solutions to Selected Exercises | p. S-1 |

Answers to Selected Exercises | p. A-1 |

Index of Applications | p. I-1 |

Index | p. I-3 |

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