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Trigonometry
Katherine Yoshiwara
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Front Matter
Colophon
Preface
1
Triangles and Circles
1.1
Angles and Triangles
Triangles
Angles
Section 1.1 Summary
Vocabulary
Concepts
Study Questions
Skills
Homework 1.1
1.2
Similar Triangles
Congruent Triangles
Similar Triangles
Using Proportions with Similar Triangles
Similar Right Triangles
Overlapping Triangles
Section 1.2 Summary
Vocabulary
Concepts
Study Questions
Skills
Homework 1.2
1.3
Circles
The Distance Formula
Equation for a Circle
Rational and Irrational Numbers
Circumference and Area
Section 1.3 Summary
Vocabulary
Concepts
Study Questions
Skills
Homework 1.3
1.4
Chapter 1 Summary and Review
Key Concepts
Chapter 1 Review Problems
2
The Trigonometric Ratios
2.1
Side and Angle Relationships
Introduction
The Triangle Inequality
Right Triangles: The Pythagorean Theorem
Section 2.1 Summary
Vocabulary
Concepts
Study Questions
Skills
Homework 2.1
2.2
Right Triangle Trigonometry
The Sine of an Angle
Using a Calculator
Using the Sine Ratio to Find an Unknown Side
The Cosine and the Tangent
The Three Trigonometric Ratios
Section 2.2 Summary
Vocabulary
Concepts
Study Questions
Skills
Homework 2.2
2.3
Solving Right Triangles
Introduction
Finding an Angle
The Special Angles
The Trigonometric Ratios for the Special Angles
Section 2.3 Summary
Vocabulary
Concepts
Study Questions
Skills
Homework 2.3
2.4
Chapter 2 Summary and Review
Key Concepts
Chapter 2 Review Problems
3
Laws of Sines and Cosines
3.1
Obtuse Angles
Angles in Standard Position
Trigonometric Ratios for Obtuse Angles
Using a Calculator
Trigonometric Ratios for Supplementary Angles
Supplements of the Special Angles
The Area of a Triangle
Section 3.1 Summary
Vocabulary
Concepts
Study Questions
Skills
Homework 3.1
3.2
The Law of Sines
Finding a Side
Solving Triangles with the Law of Sines
Finding an Angle
Applications
Measuring Astronomical Distances
Small Angles: Minutes and Seconds
Section 3.2 Summary
Vocabulary
Concepts
Study Questions
Skills
Homework 3.2
3.3
The Law of Cosines
Finding a Side
Finding an Angle
Using the Law of Cosines for the Ambiguous Case
Navigation
Which Law to Use
Section 3.3 Summary
Vocabulary
Concepts
Study Questions
Skills
Homework 3.3
3.4
Chapter 3 Summary and Review
Key Concepts
Chapter 3 Review Problems
3.5
Chapter 3 Activity
4
Trigonometric Functions
4.1
Angles and Rotation
Introduction
Angles in Standard Position
Trigonometric Ratios for All Angles
Reference Angles
Using Reference Angles
The Special Angles
Coterminal Angles
Section 4.1 Summary
Vocabulary
Concepts
Study Questions
Skills
Homework 4.1
4.2
Graphs of Trigonometric Functions
A Periodic Function of Angle
The Sine Function
The Cosine Function
Interlude: Review of Function Notation
The Tangent Function
Period, Midline and Amplitude
Section 4.2 Summary
Vocabulary
Concepts
Study Questions
Skills
Homework 4.2
4.3
Using Trigonometric Functions
Solving Trigonometric Equations
The Unit Circle
Location by Coordinates
Bearings
Angle of Inclination
Sinusoidal Functions
Other Periodic Functions
Section 4.3 Summary
Vocabulary
Concepts
Study Questions
Skills
Homework 4.3
4.4
Chapter 4 Summary and Review
Key Concepts
Chapter 4 Review Problems
5
Equations and Identities
5.1
Algebra with Trigonometric Ratios
Evaluating Trigonometric Expressions
Simplifying Trigonometric Expressions
Powers of Trigonometric Ratios
Products
Factoring
Section 5.1 Summary
Vocabulary
Concepts
Study Questions
Skills
Homework 5.1
5.2
Solving Equations
Introduction
Trigonometric Equations
Solving Trigonometric Equations
Graphical Solutions
Equations with Squares of Trig Ratios
Snell’s Law
Section 5.2 Summary
Vocabulary
Concepts
Study Questions
Skills
Homework 5.2
5.3
Trigonometric Identities
What is an Identity?
Using Trigonometric Ratios in Identities
Checking Identities Graphically
Pythagorean Identity
Tangent Identity
Trig Ratios are Related
Solving Equations
Proving Identities
Section 5.3 Summary
Vocabulary
Concepts
Study Questions
Skills
Homework 5.3
5.4
Chapter 5 Summary and Review
Key Concepts
Chapter 5 Review Problems
6
Radians
6.1
Arclength and Radians
Arclength
Measuring Angles in Radians
Converting Between Degrees and Radians
Arclength Formula
Unit Circle
Section 6.1 Summary
Vocabulary
Concepts
Study Questions
Skills
Homework 6.1
6.2
The Circular Functions
Trigonometric Functions of Angles in Radians
Using a Calculator
Sine and Cosine of Real Numbers
The Tangent Function
The Special Values
Section 6.2 Summary
Vocabulary
Concepts
Study Questions
Skills
Homework 6.2
6.3
Graphs of the Circular Functions
The Sine and Cosine Functions
The Tangent Function
Solving Equations
Modeling with Circular Functions
Domain and Range
Section 6.3 Summary
Vocabulary
Concepts
Study Questions
Skills
Homework 6.3
6.4
Chapter 6 Summary and Review
Key Concepts
Chapter 6 Review Problems
7
Circular Functions
7.1
Transformations of Graphs
Period, Midline, and Amplitude
Graphs of Sinusoidal Functions
Modeling with Sinusoidal Functions
The Tangent Function
Section 7.1 Summary
Vocabulary
Concepts
Study Questions
Skills
Homework 7-1
7.2
The General Sinusoidal Function
Horizontal Shifts
Combining Transformations
Modeling with Sinusoidal Functions
Section 7.2 Summary
Concepts
Study Questions
Skills
Homework 7-2
7.3
Solving Equations
Introduction
Multiple Solutions
Using a Calculator for Multiple Solutions
Using a Substitution
Applications
Section 7.3 Summary
Concepts
Study Questions
Skills
Homework 7-3
7.4
Chapter 7 Summary and Review
Key Concepts
Review Problems
8
More Functions and Identities
8.1
Sum and Difference Formulas
The Sum of Angles Identities
The Difference of Angles Identities
Sum and Difference Identities for Tangent
Double Angle Identities
Solving Equations
Section 8.1 Summary
Concepts
Study Questions
Skills
Homework 8-1
8.2
Inverse Trigonometric Functions
Inverse of a Function
The Graph of the Inverse
Does Every Function Have an Inverse?
Restricting the Domain: The Inverse Sine Function
The Inverse Cosine and Inverse Tangent Functions
Modeling with Inverse Functions
Alternate Notations
Simplifying Expressions
Section 8.2 Summary
Vocabulary
Concepts
Study Questions
Skills
Homework 8-2
8.3
The Reciprocal Functions
Three More Functions
Application to Right Triangles
Graphs of the Reciprocal Functions
Solving Equations
Using Identities
Section 8.3 Summary
Vocabulary
Concepts
Study Questions
Skills
Homework 8-3
8.4
Chapter Summary and Review
Key Concepts
Review Problems
9
Vectors
9.1
Geometric Form
Introduction
Notation for Vectors
Scalar Multiplication of Vectors
Addition of Vectors
Velocity
Components of a Vector
Using Components
Section 9.1 Summary
Vocabulary
Concepts
Study Questions
Skills
Homework 9-1
9.2
Coordinate Form
Introduction
Unit Vectors
Converting Between Geometric and Coordinate Form
Scalar Multiples of Vectors in Coordinate Form
Adding Vectors in Coordinate Form
Force
Section 9.2 Summary
Vocabulary
Concepts
Study Questions
Skills
Homework 9-2
9.3
The Dot Product
Components
Coordinate Form for Components
The Dot Product
Geometric Meaning of the Dot Product
Section 9.3 Summary
Vocabulary
Concepts
Study Questions
Skills
Homework 9-3
9.4
Chapter Summary and Review
Key Concepts
Review Problems
10
Polar Coordinates and Complex Numbers
10.1
Polar Coordinates
Plotting Points
Regions in the Plane
Converting Between Polar and Cartesian Coordinates
Equations in Polar Coordinates
Section 10.1 Summary
Vocabulary
Concepts
Study Questions
Skills
Homework 10-1
10.2
Polar Graphs
Graphing in Polar Coordinates
Using a Graphing Calculator
Sketching Familiar Equations
Finding Intersection Points
A Catalog of Polar Curves
The Coordinate Curves
Circles
Roses
Limaçons
Lemniscates
Spirals
Section 10.2 Summary
Vocabulary
Concepts
Study Questions
Skills
Homework 10-2
10.3
Complex Numbers
Imaginary Numbers
Complex Numbers
Arithmetic of Complex Numbers
Products of Complex Numbers
Quotients of Complex Numbers
Graphing Complex Numbers
Zeros of Polynomials
Section 10.3 Summary
Vocabulary
Concepts
Study Questions
Skills
Homework 10-3
10.4
Polar Form for Complex Numbers
Introduction
Polar Form
Products and Quotients in Polar Form
Powers and Roots of Complex Numbers
Section 10.4 Summary
Vocabulary
Concepts
Study Questions
Skills
Homework 10-4
10.5
Chapter Summary and Review
Key Concepts
Review Problems
Appendices
A
Answers to Selected Exercises and Homework Problems
B
GNU Free Documentation License
C
Glossary
Index
Colophon
Colophon
Colophon
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