Trigonometry A Circular Function Approach

by Aratari, Marie

• ISBN13: 9780201771749
• ISBN10: 0201771748
• Edition: 1st
• Format: Paperback
• Copyright: 2003-10-29
• Publisher: Pearson

Summary

The text presents a circular function approach to trigonometry by demonstrating connections between the familiar algebra and the new language of trigonometry. This methodalong with foreshadowingis used throughout the text to provide students with a comfortable base for learning something new from something old or familiar. With just a few connections to algebra, students have the tools to understand the circular functions, their domains and ranges, and the relationship between the circular functions and the functional values. The approach immediately launches the student into the concept of periodic functions, their applications, graphs, and use in modeling many periodic phenomena. Beginning with this approach provides the student with a common thread that can be used to discover, connect and understand the remaining concepts of trigonometry.

Table of Contents

Preface	xi
Circular Functions	1	(72)
Fundamentals for Trigonometry	2	(2)
The Coordinate Plane
Graphing
Solving Equations
Circular Functions: The Cosine and Sine Functions	4	(12)
Special Arcs on the Unit Circle
Circular Functions
Positive and Negative Values of cos t and sin t
Basic Identity
Functional Values for Common Arcs	16	(11)
Circular Functional Values for t = 0, π/2, π, 3π/2
Circular Functional Values for t = π/4
Circular Functional Values for t = π/6
Circular Functional Values for t = π/3
Noncommon Arcs
Using a Calculator to Find Functional Values
Choice of Variable
Reference Arcs and Functional Values for cos x and sin x, x ER	27	(13)
Reference Arc
Least Positive Coterminal Arc
Integer Multiples of Common Arcs
Functional Values for Arcs with Common Reference Arcs
Functional Values for Arcs without Common Reference Arcs
Four Additional Circular Functions: Tangent, Secant, Cosecant, and Cotangent Functions	40	(13)
Tangent Function
Definitions of the Reciprocal Functions
Negative Identities and Periods for the Circular Functions	53	(20)
Periodic Property of the Circular Functions
Period of cos x and sin x
Period of sec x and csc x
Period of tan x and cot x
Application of Circular Functions
Simple Harmonic Motion
Chapter Summary	63	(3)
Review Exercises	66	(5)
Chapter Test	71	(2)
Graphs of the Circular Functions	73	(86)
Graphs of the Sine and Cosine Functions	74	(18)
Graph of the Sine Function
Graph of the Cosine Function
Important Characteristics of the Graphs of y = sin x and y = cos x
Period, Phase Shift, and Other Translations	92	(17)
Period Change (Horizontal Shrink or Stretch)
Combining Period and Amplitude Changes
Translations
Combining Translations with Period and Amplitude Changes
Finding Equations of Sinusoidal Graphs
Applications and Modeling with Sinusoidal Functions	109	(10)
Simple Harmonic Motion
Models Involving Sinusoidal Functions
Graphs of the Tangent, Cotangent, Secant, and Cosecant Functions	119	(14)
Graph of the Tangent Function
Graph of the Cotangent Function
Important Characteristics of the Graphs of y = tan x and y = cot x
Graphs of the Secant and Cosecant Functions
Inverses of the Circular Functions	133	(26)
Inverse Sine
Inverse Cosine and Inverse Tangent
Using a Calculator for Inverse Functional Values
Chapter Summary	147	(4)
Review Exercises	151	(4)
Chapter Test	155	(4)
The Trigonometric Functions	159	(76)
Angles and Their Measure	160	(13)
Angles
Measure of an Angle
Radians
Degrees
Special Angles Degree and Radian Measure
Coterminal Angles
Applications
Trigonometric and Circular Functions	173	(12)
Functional Values for Common Angles
Solving Right Triangles and Applications	185	(19)
Cofunctions
Solving Right Triangles
Applications of Right Triangles
Angles of Depression or Elevation
Bearing
Solution of Triangles Using Law of Sines	204	(14)
Law of Sines
Ambiguous Case SSA of the Law of Sines
Solving the Ambiguous Case
Application of the Law of Sines
SSS or SAS
Solution of Triangles Using Law of Cosines	218	(17)
Law of Cosines
Using the Law of Cosines to Solve the Ambiguous Case (Optional)
Bearing
Chapter Summary	227	(3)
Review Exercises	230	(3)
Chapter Test	233	(2)
Identities	235	(56)
Proving Identities	236	(12)
Simplify Expressions
Proving Identities
Identities and Graphing
Sum and Difference Identities for Cosine	248	(10)
Cosine Difference Identity
Cosine Sum Identity
Using Cosine Sum or Difference Identities to Find Exact Functional Values
Sum and Difference Identities for Sine and Tangent	258	(8)
Sine Sum and Difference Identities
Tangent Sum and Difference Identities
Using Sine and Tangent Sum or Difference Identities to Find Exact Functional Values
Double-Angle Identities	266	(7)
Double-Angle Identities for Sine and Cosine
Using Double-Angle Identities to Find Exact Functional Values
Using Double-Angle Identities to Rewrite Expressions
Half-Angle and Additional Identities	273	(18)
Half-Angle Identities (Formulas)
Using Half-Angle Identities to Find Exact Functional Values
Using Half-Angle Identities to Prove Identities and Simplify Expressions
Product-to-Sum and Sum-to-Product Identities
Chapter Summary	284	(2)
Review Exercises	286	(3)
Chapter Test	289	(2)
Trigonometric Equations	291	(36)
Solving Conditional Equations I	292	(6)
Solving Trigonometric Conditional Equations II	298	(6)
More Trigonometric Equations, Multiple-Angle Equations	304	(9)
Solving Trigonometric Equations Using Identities and Other Strategies
Multiple-Angle Equations
Using Technology to Solve Trigonometric Equations
Systems of Equations
Parametric Equations	313	(14)
Graphing Parametric Equations
Application of Parametric Equations
Using Technology to Graph Parametric Equations
Finding Parametric Equations
Chapter Summary	322	(1)
Review Exercises	323	(2)
Chapter Test	325	(2)
Vectors, Polar Equations, and Complex Numbers	327	(66)
Geometric Vectors and Applications	328	(11)
Vectors Viewed Geometrically
Equivalent Vectors
Applications of Vectors
Algebraic Vectors	339	(11)
Standard Position and Component Form of a Vector
Magnitude, Direction and Horizontal and Vertical Components
Equivalent Vectors
i,j Form of a Vector
Dot Product
Finding the Angle Between Two Vectors
Application of Dot Product
Work
Polar Coordinate System	350	(14)
Polar Coordinates
Relationship Between Polar Coordinates and Rectangular Coordinates
Polar Equations and Graphs
Complex Numbers	364	(8)
Standard Form of a Complex Number
Operations with Complex Numbers
Powers of i
Geometric Representation of a Complex Number
Trigonometric Form for Complex Numbers	372	(21)
Trigonometric Form
Powers and Roots of Complex Numbers in Trigonometric Form
Solving Algebraic Equations Using Trigonometry
Chapter Summary	385	(4)
Review Exercises	389	(2)
Chapter Test	391	(2)
Cumulative Review	393	(6)
Appendix A Algebra Review	399	(26)
Appendix B Geometry Review	425
Answers to Selected Exercises	1	(1)
Index	1