Trigonometry

1. Algebraic Prerequisites

• 1.1 Graphing Basics
  • 1.1.1 Using the Cartesian System
  • 1.1.2 Thinking Visually

• 1.2 Relationships between Two Points
  • 1.2.1 Finding the Distance between Two Points
  • 1.2.2 Finding the Second Endpoint of a Segment

• 1.3 Relationships among Three Points
  • 1.3.1 Collinearity and Distance
  • 1.3.2 Triangles

• 1.4 Circles
  • 1.4.1 Finding the Center-Radius Form of the Equation of a Circle
  • 1.4.2 Decoding the Circle Formula
  • 1.4.3 Finding the Center and Radius of a Circle
  • 1.4.4 Solving Word Problems Involving Circles

• 1.5 Graphing Equations
  • 1.5.1 Graphing Equations by Locating Points
  • 1.5.2 Finding the x- and y-Intercepts of an Equation

• 1.6 Function Basics
  • 1.6.1 Functions and the Vertical Line Test
  • 1.6.2 Identifying Functions
  • 1.6.3 Function Notation and Finding Function Values

• 1.7 Working with Functions
  • 1.7.1 Determining Intervals Over Which a Function Is Increasing
  • 1.7.2 Evaluating Piecewise-Defined Functions for Given Values
  • 1.7.3 Solving Word Problems Involving Functions

• 1.8 Function Domain and Range
  • 1.8.1 Finding the Domain and Range of a Function
  • 1.8.2 Domain and Range: One Explicit Example
  • 1.8.3 Satisfying the Domain of a Function

• 1.9 Linear Functions: Slope
  • 1.9.1 An Introduction to Slope
  • 1.9.2 Finding the Slope of a Line Given Two Points
  • 1.9.3 Interpreting Slope from a Graph
  • 1.9.4 Graphing a Line Using Point and Slope

• 1.10 Equations of a Line
  • 1.10.1 Writing an Equation in Slope-Intercept Form
  • 1.10.2 Writing an Equation Given Two Points
  • 1.10.3 Writing an Equation in Point-Slope Form
  • 1.10.4 Matching a Slope-Intercept Equation with Its Graph
  • 1.10.5 Slope for Parallel and Perpendicular Lines

• 1.11 Graphing Functions
  • 1.11.1 Graphing Some Important Functions
  • 1.11.2 Graphing Piecewise-Defined Functions
  • 1.11.3 Matching Equations with Their Graphs

• 1.12 Manipulating Graphs: Shifts and Stretches
  • 1.12.1 Shifting Curves along Axes
  • 1.12.2 Shifting or Translating Curves along Axes
  • 1.12.3 Stretching a Graph
  • 1.12.4 Graphing Quadratics Using Patterns

• 1.13 Manipulating Graphs: Symmetry and Reflections
  • 1.13.1 Determining Symmetry
  • 1.13.2 Reflections
  • 1.13.3 Reflecting Specific Functions

• 1.14 Quadratic Functions: Basics
  • 1.14.1 Deconstructing the Graph of a Quadratic Function
  • 1.14.2 Nice-Looking Parabolas
  • 1.14.3 Using Discriminants to Graph Parabolas
  • 1.14.4 Maximum Height in the Real World

• 1.15 Quadratic Functions: The Vertex
  • 1.15.1 Finding the Vertex by Completing the Square
  • 1.15.2 Using the Vertex to Write the Quadratic Equation
  • 1.15.3 Finding the Maximum or Minimum of a Quadratic
  • 1.15.4 Graphing Parabolas

• 1.16 Composite Functions
  • 1.16.1 Using Operations on Functions
  • 1.16.2 Composite Functions
  • 1.16.3 Components of Composite Functions
  • 1.16.4 Finding Functions That Form a Given Composite
  • 1.16.5 Finding the Difference Quotient of a Function
  • 1.16.6 Calculating the Average Rate of Change

• 1.17 Rational Functions
  • 1.17.1 Understanding Rational Functions
  • 1.17.2 Basic Rational Functions

• 1.18 Graphing Rational Functions
  • 1.18.1 Vertical Asymptotes
  • 1.18.2 Horizontal Asymptotes
  • 1.18.3 Graphing Rational Functions
  • 1.18.4 Graphing Rational Functions: More Examples
  • 1.18.5 Oblique Asymptotes
  • 1.18.6 Oblique Asymptotes: Another Example

• 1.19 Function Inverses
  • 1.19.1 Understanding Inverse Functions
  • 1.19.2 The Horizontal Line Test
  • 1.19.3 Are Two Functions Inverses of Each Other?
  • 1.19.4 Graphing the Inverse

• 1.20 Finding Function Inverses
  • 1.20.1 Finding the Inverse of a Function
  • 1.20.2 Finding the Inverse of a Function with Higher Powers

2. The Trigonometric Functions

• 2.1 Angles and Radian Measure
  • 2.1.1 Finding the Quadrant in Which an Angle Lies
  • 2.1.2 Finding Coterminal Angles
  • 2.1.3 Finding the Complement and Supplement of an Angle
  • 2.1.4 Converting between Degrees and Radians
  • 2.1.5 Using the Arc Length Formula

• 2.2 Right Angle Trigonometry
  • 2.2.1 An Introduction to the Trigonometric Functions
  • 2.2.2 Evaluating Trigonometric Functions for an Angle in a Right Triangle
  • 2.2.3 Finding an Angle Given the Value of a Trigonometric Function
  • 2.2.4 Using Trigonometric Functions to Find Unknown Sides of Right Triangles
  • 2.2.5 Finding the Height of a Building

• 2.3 The Trigonometric Functions
  • 2.3.1 Evaluating Trigonometric Functions for an Angle in the Coordinate Plane
  • 2.3.2 Evaluating Trigonometric Functions Using the Reference Angle
  • 2.3.3 Finding the Value of Trigonometric Functions Given Information about the Values of Other Trigonometric Functions
  • 2.3.4 Trigonometric Functions of Important Angles

• 2.4 Graphing Sine and Cosine Functions
  • 2.4.1 An Introduction to the Graphs of Sine and Cosine Functions
  • 2.4.2 Graphing Sine or Cosine Functions with Different Coefficients
  • 2.4.3 Finding Maximum and Minimum Values and Zeros of Sine and Cosine
  • 2.4.4 Solving Word Problems Involving Sine or Cosine Functions

• 2.5 Graphing Sine and Cosine Functions with Vertical and Horizontal Shifts
  • 2.5.1 Graphing Sine and Cosine Functions with Phase Shifts
  • 2.5.2 Fancy Graphing: Changes in Period, Amplitude, Vertical Shift, and Phase Shift

• 2.6 Graphing Other Trigonometric Functions
  • 2.6.1 Graphing the Tangent, Secant, Cosecant, and Cotangent Functions
  • 2.6.2 Fancy Graphing: Tangent, Secant, Cosecant, and Cotangent
  • 2.6.3 Identifying a Trigonometric Function from its Graph

• 2.7 Inverse Trigonometric Functions
  • 2.7.1 An Introduction to Inverse Trigonometric Functions
  • 2.7.2 Evaluating Inverse Trigonometric Functions
  • 2.7.3 Solving an Equation Involving an Inverse Trigonometric Function
  • 2.7.4 Evaluating the Composition of a Trigonometric Function and Its Inverse
  • 2.7.5 Applying Trigonometric Functions: Is He Speeding?

3. Trigonometric Identities

• 3.1 Basic Trigonometric Identities
  • 3.1.1 Fundamental Trigonometric Identities
  • 3.1.2 Finding All Function Values

• 3.2 Simplifying Trigonometric Expressions
  • 3.2.1 Simplifying a Trigonometric Expression Using Trigonometric Identities
  • 3.2.2 Simplifying Trigonometric Expressions Involving Fractions
  • 3.2.3 Simplifying Products of Binomials Involving Trigonometric Functions
  • 3.2.4 Factoring Trigonometric Expressions
  • 3.2.5 Determining Whether a Trigonometric Function Is Odd, Even, or Neither

• 3.3 Proving Trigonometric Identities
  • 3.3.1 Proving an Identity
  • 3.3.2 Proving an Identity: Other Examples

• 3.4 Solving Trigonometric Equations
  • 3.4.1 Solving Trigonometric Equations
  • 3.4.2 Solving Trigonometric Equations by Factoring
  • 3.4.3 Solving Trigonometric Equations with Coefficients in the Argument
  • 3.4.4 Solving Trigonometric Equations Using the Quadratic Formula
  • 3.4.5 Solving Word Problems Involving Trigonometric Equations

• 3.5 The Sum and Difference Identities
  • 3.5.1 Identities for Sums and Differences of Angles
  • 3.5.2 Using Sum and Difference Identities
  • 3.5.3 Using Sum and Difference Identities to Simplify an Expression

• 3.6 Double-Angle Identities
  • 3.6.1 Confirming a Double-Angle Identity
  • 3.6.2 Using Double-Angle Identities
  • 3.6.3 Solving Word Problems Involving Multiple-Angle Identities

• 3.7 Other Advanced Identities
  • 3.7.1 Using a Cofunction Identity
  • 3.7.2 Using a Power-Reducing Identity
  • 3.7.3 Using Half-Angle Identities to Solve a Trigonometric Equation

4. Applications of Trigonometry

• 4.1 The Law of Sines
  • 4.1.1 The Law of Sines
  • 4.1.2 Solving a Triangle Given Two Sides and One Angle
  • 4.1.3 Solving a Triangle (SAS): Another Example
  • 4.1.4 The Law of Sines: An Application

• 4.2 The Law of Cosines
  • 4.2.1 The Law of Cosines
  • 4.2.2 The Law of Cosines (SSS)
  • 4.2.3 The Law of Cosines (SAS): An Application
  • 4.2.4 Heron's Formula

• 4.3 Vector Basics
  • 4.3.1 An Introduction to Vectors
  • 4.3.2 Finding the Magnitude and Direction of a Vector
  • 4.3.3 Vector Addition and Scalar Multiplication

• 4.4 Components of Vectors and Unit Vectors
  • 4.4.1 Finding the Components of a Vector
  • 4.4.2 Finding a Unit Vector
  • 4.4.3 Solving Word Problems Involving Velocity or Forces

5. Complex Numbers and Polar Coordinates

• 5.1 Complex Numbers
  • 5.1.1 Introducing and Writing Complex Numbers
  • 5.1.2 Rewriting Powers of i
  • 5.1.3 Adding and Subtracting Complex Numbers
  • 5.1.4 Multiplying Complex Numbers
  • 5.1.5 Dividing Complex Numbers

• 5.2 Complex Numbers in Trigonometric Form
  • 5.2.1 Graphing a Complex Number and Finding Its Absolute Value
  • 5.2.2 Expressing a Complex Number in Trigonometric or Polar Form
  • 5.2.3 Multiplying and Dividing Complex Numbers in Trigonometric or Polar Form

• 5.3 Powers and Roots of Complex Numbers
  • 5.3.1 Using DeMoivre's Theorem to Raise a Complex Number to a Power
  • 5.3.2 Roots of Complex Numbers
  • 5.3.3 More Roots of Complex Numbers
  • 5.3.4 Roots of Unity

• 5.4 Polar Coordinates
  • 5.4.1 An Introduction to Polar Coordinates
  • 5.4.2 Converting between Polar and Rectangular Coordinates
  • 5.4.3 Converting between Polar and Rectangular Equations
  • 5.4.4 Graphing Simple Polar Equations
  • 5.4.5 Graphing Special Polar Equations

6. Exponential and Logarithmic Functions

• 6.1 Exponential Functions
  • 6.1.1 An Introduction to Exponential Functions
  • 6.1.2 Graphing Exponential Functions: Useful Patterns
  • 6.1.3 Graphing Exponential Functions: More Examples

• 6.2 Applying Exponential Functions
  • 6.2.1 Using Properties of Exponents to Solve Exponential Equations
  • 6.2.2 Finding Present Value and Future Value
  • 6.2.3 Finding an Interest Rate to Match Given Goals

• 6.3 The Number e
  • 6.3.1 e
  • 6.3.2 Applying Exponential Functions

• 6.4 Logarithmic Functions
  • 6.4.1 An Introduction to Logarithmic Functions
  • 6.4.2 Converting between Exponential and Logarithmic Functions

• 6.5 Solving Logarithmic Functions
  • 6.5.1 Finding the Value of a Logarithmic Function
  • 6.5.2 Solving for x in Logarithmic Equations
  • 6.5.3 Graphing Logarithmic Functions
  • 6.5.4 Matching Logarithmic Functions with Their Graphs

• 6.6 Properties of Logarithms
  • 6.6.1 Properties of Logarithms
  • 6.6.2 Expanding a Logarithmic Expression Using Properties
  • 6.6.3 Combining Logarithmic Expressions

• 6.7 Evaluating Logarithms
  • 6.7.1 Evaluating Logarithmic Functions Using a Calculator
  • 6.7.2 Using the Change of Base Formula

• 6.8 Applying Logarithmic Functions
  • 6.8.1 The Richter Scale
  • 6.8.2 The Distance Modulus Formula

• 6.9 Solving Exponential and Logarithmic Equations
  • 6.9.1 Solving Exponential Equations
  • 6.9.2 Solving Logarithmic Equations
  • 6.9.3 Solving Equations with Logarithmic Exponents

• 6.10 Applying Exponents and Logarithms
  • 6.10.1 Compound Interest
  • 6.10.2 Predicting Change

• 6.11 Word Problems Involving Exponential Growth and Decay
  • 6.11.1 An Introduction to Exponential Growth and Decay
  • 6.11.2 Half-Life
  • 6.11.3 Newton's Law of Cooling
  • 6.11.4 Continuously Compounded Interest

7. Conic Sections

• 7.1 Conic Sections: Parabolas
  • 7.1.1 An Introduction to Conic Sections
  • 7.1.2 An Introduction to Parabolas
  • 7.1.3 Determining Information about a Parabola from Its Equation
  • 7.1.4 Writing an Equation for a Parabola

• 7.2 Conic Sections: Ellipses
  • 7.2.1 An Introduction to Ellipses
  • 7.2.2 Finding the Equation for an Ellipse
  • 7.2.3 Applying Ellipses: Satellites
  • 7.2.4 The Eccentricity of an Ellipse

• 7.3 Conic Sections: Hyperbolas
  • 7.3.1 An Introduction to Hyperbolas
  • 7.3.2 Finding the Equation for a Hyperbola
  • 7.3.3 Applying Hyperbolas: Navigation

• 7.4 Conic Sections
  • 7.4.1 Identifying a Conic
  • 7.4.2 Name That Conic
  • 7.4.3 Rotation of Axes
  • 7.4.4 Rotating Conics