You are previewing The Complete Idiot's Guide to Trigonometry.
O'Reilly logo
The Complete Idiot's Guide to Trigonometry

Book Description

- Follows a standard course curriculum.

Table of Contents

  1. Cover
  2. Title Page
  3. Dedication
  4. Copyright
  5. Contents
  6. Introduction
  7. Part 1: The Basic Tools
    1. 1 What Is Trigonometry?
      1. How It All Started
      2. Where Ancients Used Trigonometry
      3. Modern Uses of Trigonometry
      4. Trigonometry and Other Mathematics Disciplines
      5. Two Historical Perspectives of Trigonometry
    2. 2 Geometry Tools Needed to Study Trigonometry
      1. Right Triangles
        1. The Pythagorean Theorem
        2. Sample Problem 1
        3. Sample Problem 2
        4. Special Right Triangles
        5. Classifying Triangles
      2. More on Triangles
        1. Area of Triangles
        2. Conditions for Congruence
      3. Circles, Arcs, and Chords
        1. Chords and Arcs
        2. Area of Circles
    3. 3 Algebra Tools Needed to Study Trigonometry
      1. Cartesian Coordinates
        1. Writing Coordinates
        2. Four Quadrants
      2. What Are Functions?
        1. The Basics of Functions
        2. The Vertical Line Test
        3. Inverse Functions and the Horizontal Line Test
      3. Complex Numbers
        1. How They Came into Existence
        2. Operations with Complex Numbers
  8. Part 2: Triangle Trigonometry
    1. 4 Trigonometric Functions and Right Triangles
      1. The Tangent Ratio
        1. Sample Problem 1
        2. Sample Problem 2
      2. Introducing the Sine and Cosine Ratios
        1. Cofunction Identity Between the Sine and Cosine
        2. Pythagorean Identity with the Sine and Cosine
        3. Values of Trig Functions for 0° and 90° Angles
      3. Practical Applications of Trigonometric Ratios
        1. Sample Problem 3
      4. Practice Problems
    2. 5 Relations Among Trigonometric Ratios
      1. Six Trigonometric Relatives
        1. Reciprocal Functions
        2. Three Fundamental Identities
        3. Other Pythagorean Relationships
        4. Sample Problem 1
        5. Sample Problem 2
      2. Determining Values of Trigonometric Functions
      3. Solving Right Triangles
        1. Finding the Sides
        2. Sample Problem 3
        3. Sample Problem 4
        4. Finding the Angles
        5. Sample Problem 5
      4. Practice Problems
    3. 6 The Law of Sines
      1. Oblique Triangles
        1. The Area of a Triangle
        2. The Law of Sines
        3. Sample Problem 1
      2. The Sine of an Obtuse Angle
      3. The Ambiguous Case
        1. Sample Problem 2
        2. Sample Problem 3
      4. Solving Oblique Triangles Using the Law of Sines
        1. Sample Problem 4
      5. Practice Problems
    4. 7 The Law of Cosines
      1. The Law of Cosines
        1. Proof of the Law of Cosines
        2. The Law of Cosines for an Obtuse Triangle
        3. Finding Angles with the Law of Cosines
      2. Two Cases That the Law of Cosines Helps to Solve
        1. Sample Problem 1
        2. Sample Problem 2
      3. Application of Trigonometry to Navigation and Surveying
        1. Sample Problem 3
        2. Sample Problem 4
      4. Hero’s Formula for the Area of a Triangle
        1. Sample Problem 5
      5. Practice Problems
  9. Part 3: Trigonometric Functions and the Unit Circle
    1. 8 Angles and Rotations
      1. Measuring Angles
        1. Degree Measures
        2. Coterminal Angles
        3. Radian Measure
        4. Sample Problem 1
      2. Sectors of Circles
        1. Sample Problem 2
        2. Sample Problem 3
        3. Sample Problem 4
        4. Sample Problem 5
      3. Practice Problems
    2. 9 The Unit Circle Approach
      1. What Is the Unit Circle?
      2. Placing Points on the Unit Circle
      3. Making Sense of the Unit Circle
      4. Evaluating Trigonometric Functions of Real Numbers
        1. Sample Problem 1
        2. Sample Problem 2
        3. Sample Problem 3
        4. Sample Problem 4
      5. Practice Problems
    3. 10 Trigonometric Functions of Any Angle
      1. Calculating Values of Trigonometric Functions
      2. Reference Angles
      3. Evaluating Trigonometric Functions
        1. Sample Problem 1
        2. Sample Problem 2
        3. Sample Problem 3
        4. Sample Problem 4
      4. Looking Closely at Trigonometric Functions
        1. Periodic Functions
        2. Even and Odd Trigonometric Functions
      5. Practice Problems
  10. Part 4: Graphs of Trigonometric Functions
    1. 11 Graphs of Sine and Cosine Functions
      1. Basic Sine Curve
        1. Domain and Range of the Sine Curve
        2. Key Points on Basic Sine Curve
      2. Other Sinusoidal Curves
        1. The Amplitude and Vertical Stretching and Shrinking
        2. Sample Problem 1
        3. Sample Problem 2
        4. Horizontal Stretching and Shrinking
        5. Sample Problem 3
        6. Sample Problem 4
      3. Translations of the Sine Curve
        1. Horizontal Shift
        2. Sample Problem 5
        3. Vertical Shift
        4. Sample Problem 6
      4. The Cosine Curve
        1. Sample Problem 7
      5. Modeling Periodic Behavior
        1. Sample Problem 8
      6. Practice Problems
    2. 12 Graphs of Other Trigonometric Functions
      1. Graphs of Tangent Function
        1. Domain and Period of the Tangent
        2. Asymptotes
        3. Altering the Tangent Graph
        4. Altering the Tangent with Multiplication
        5. Altering the Tangent with Multiplication of the Angle
        6. Sample Problem 1
        7. Altering the Tangent with Addition
      2. Graphs of Cotangent Function
        1. Sample Problem 2
      3. Graphs of the Reciprocal Functions
        1. Graph of the Cosecant Function
        2. Graph of the Secant Function
      4. Graphing Combinations of Functions
      5. Practice Problems
    3. 13 Graphs of Inverse Trigonometric Functions
      1. Inverse Sine Function
        1. Sketching the Graph of Arcsine Function
        2. Evaluating the Inverse Sine Function
        3. Sample Problem 1
      2. Other Inverse Trigonometric Functions
        1. Inverse Cosine and Tangent Functions
        2. Inverses of Reciprocal Trigonometric Functions
        3. Sample Problem 2
        4. Sample Problem 3
        5. Compositions of Trig and Inverse Trig Functions
          1. Sample Problem 4
          2. Sample Problem 5
        6. Practice Problems
  11. Part 5: Trigonometric Identities and Equations
    1. 14 Kaleidoscope of Identities
      1. Basic Identities
        1. Reciprocal Identities
        2. Relationships with Negatives
        3. Revisiting Pythagorean Identities
        4. Cofunction Identities
      2. Many Uses of the Fundamental Identities
        1. Sample Problem 1
        2. Sample Problem 2
        3. Sample Problem 3
        4. Sample Problem 4
      3. Practice Problems
    2. 15 Verifying Trigonometric Identities
      1. Important Guidelines
      2. Different Tactics for Verifying Identities
        1. Working with the More Complicated Side
        2. Sample Problem 1
        3. Sample Problem 2
        4. Combining Fractions Before Using Identities
        5. Sample Problem 3
        6. Converting to Sines and Cosines
        7. Sample Problem 4
        8. Sample Problem 5
        9. Working with Each Side Separately
        10. Sample Problem 6
        11. Other Useful Techniques
        12. Sample Problem 7
        13. Sample Problem 8
      3. Practice Problems
    3. 16 Solving Trigonometric Equations
      1. What It Means to Solve a Trigonometric Equation
      2. Solving Techniques
        1. Sample Problem 1
        2. Sample Problem 2
        3. Sample Problem 3
      3. Equations of Quadratic Type
        1. Sample Problem 4
        2. Sample Problem 5
        3. Sample Problem 6
      4. When Multiple Angles Are Involved
        1. Sample Problem 7
        2. Sample Problem 8
      5. Practice Problems
    4. 17 Sum and Difference Formulas
      1. Sum and Difference Formulas for Sine and Cosine
        1. Sample Problem 1
        2. Sample Problem 2
        3. Sum and Difference Formulas for the Tangent
        4. Sample Problem 3
        5. Sample Problem 4
      2. Applying Formulas to Various Trig Problems
        1. Sample Problem 5
        2. Sample Problem 6
        3. Sample Problem 7
        4. Sample Problem 8
      3. Practice Problems
    5. 18 Double-Angle and Power-Reducing Formulas
      1. Double-Angle Formulas
        1. Sample Problem 1
        2. Sample Problem 2
        3. Sample Problem 3
        4. Sample Problem 4
        5. Deriving a Triple-Angle Formula
      2. Power-Reducing Formulas
        1. Sample Problem 5
        2. Sample Problem 6
      3. Practice Problems
    6. 19 Half-Angle and Product-to-Sum Formulas
      1. Half-Angle Formulas
        1. Sample Problem 1
        2. Sample Problem 2
      2. Product–to-Sum Formulas
        1. Sample Problem 3
      3. Sum-to-Product Formulas
        1. Sample Problem 4
        2. Sample Problem 5
        3. Sample Problem 6
      4. How to Avoid Frustration While Working with Formulas
      5. Practice Problems
  12. Part 6: Polar Coordinates and Complex Numbers
    1. 20 Polar Coordinates
      1. Introducing Polar Coordinates
        1. Plotting Points in the Polar Coordinate System
        2. Sample Problem 1
      2. Coordinate Conversions
        1. Polar-to-Rectangular Conversions
        2. Sample Problem 2
        3. Sample Problem 3
        4. Rectangular-to-Polar Conversions
        5. Sample Problem 4
        6. Sample Problem 5
      3. Converting Polar Equations to Rectangular Form
        1. Sample Problem 6
        2. Sample Problem 7
        3. Sample Problem 8
      4. Graphs of Polar Equations
        1. Sample Problem 9
      5. Practice Problems
    2. 21 Complex Numbers and Operations with Them
      1. Geometric Representation of Complex Numbers
      2. Converting from One Form to Another
        1. Sample Problem 1
        2. Sample Problem 2
        3. Sample Problem 3
      3. Product of Two Complex Numbers in Polar Form
        1. Sample Problem 4
        2. Sample Problem 5
      4. De Moivre’s Theorem and Powers of Complex Numbers
        1. Sample Problem 6
        2. Sample Problem 7
      5. Roots of Complex Numbers
        1. Sample Problem 8
        2. Sample Problem 9
      6. Practice Problems
    3. 22 Trigonometry and Calculators?
      1. Calculators That Can Help You with Trigonometry
      2. Evaluating Trigonometric Functions
        1. Sample Problem 1
        2. Sample Problem 2
      3. Evaluating Inverse Trigonometric Functions
        1. Sample Problem 3
      4. Sketching Graphs of Trigonometric Functions
        1. Sample Problem 4
      5. Solving Trig Equations with a Calculator
  13. Appendixes
    1. A Glossary
    2. B Practice Problems and Answers
    3. C Rules and Formulas Review
    4. D Trigonometric Tables
  14. Index