Author Jonathan David
Author Jonathan David
Lessons

Trigonometry Crash Course

by |Published

Before you Start Trigonometry

Course Outline is Based on Michael Sullivan Trigonometry

Learn The Unit Circle

1. Graphs and Functions

1.1 The Distance and Midpoint Formulas

1.2 Graphs of Equations in Two Variables; Circles (Video Lesson)

  • Graph Equations by Plotting Points
  • Find Intercepts from a Graph
  • Find Intercepts from an Equation
  • Test an Equation for Symmetry with Respect to the x-Axis, the y-Axis, and the Origin
  • Know How to Graph Key Equations
  • Write the Standard Form of the Equation of a Circle
  • Graph a Circle
  • Work with the General Form of the Equation of a Circle

1.3 Functions and Their Graphs

  • Describe a Relation
  • Determine Whether a Relation Represents a Function
  • Use Function Notation
  • Find the Value of a Function
  • Find the Difference Quotient of a Function
  • Find the Domain of a Function Defined by an Equation
  • Identify the Graph of a Function
  • Obtain Information from or about the Graph of a Function

1.4 Properties of Functions

  • Identify Even and Odd Functions from a Graph
  • Identify Even and Odd Functions from an Equation
  • Use a Graph to Determine Where a Function is Increasing, Decreasing, or Constant
  • Use a Graph to Locate Local Maxima and Local Minima
  • Use a Graph to Locate the Absolute Maximum and the Absolute Minimum
  • Use a Graphing Utility to Approximate Local Maxima and Local Minima and to Determine Where a Function is Increasing or Decreasing
  • Find the Average Rate of Change of a Function

1.5 Library of Functions; Piecewise-defined Functions

  • Graph the Functions Listed in the Library of Functions
  • Analyze a Piecewise-defined Function

1.6 Graphing Techniques: Transformations

  • Graph Functions Using Vertical and Horizontal Shifts
  • Graph Functions Using Compressions and Stretches
  • Graph Functions Using Reflections about the x-Axis and the y-Axis

1.7 One-to-One Functions; Inverse Functions

  • Determine Whether a Function is One-to-One
  • Obtain the Graph of the Inverse Function from the Graph of a One-to-One Function
  • Verify an Inverse Function
  • Find the Inverse of a Function Defined by an Equation

2. Trigonometric Functions

2.1 Angles, Arc Length, and Circular Motion

  • Angles and Degree Measure
  • Convert between Decimal and Degree, Minute, Second Measures for Angles
  • Find the Length of an Arc of a Circle
  • Convert from Degrees to Radians and from Radians to Degrees
  • Find the Area of a Sector of a Circle
  • Find the Linear Speed of an Object Traveling in Circular Motion

2.2 Trigonometric Functions: Unit Circle Approach

  • Find the Exact Values of the Trigonometric Functions Using a Point on the Unit Circle
  • Find the Exact Values of the Trigonometric Functions of Quadrantal Angles
  • Find the Exact Values of the Trigonometric Functions of 45 degrees (π/4)
  • Find the Exact Values of the Trigonometric Functions of 30 degrees (π/6) and 60 degrees (π/3)
  • Find the Exact Values of the Trigonometric Functions for Integer Multiples of 30 degrees (π/6), 45 degrees (π/4), and 60 degrees (π/3)
  • Use a Calculator to Approximate the Value of a Trigonometric Function
  • Use a Circle of Radius r to Evaluate the Trigonometric Functions

2.3 Properties of the Trigonometric Functions

  • Determine the Domain and the Range of the Trigonometric Functions
  • Determine the Period of the Trigonometric Functions
  • Determine the Signs of the Trigonometric Functions in a Given Quadrant
  • Find the Values of the Trigonometric Functions Using Fundamental Identities
  • Find the Exact Values of the Trigonometric Functions of an Angle Given One of the Functions and the Quadrant of the Angle
  • Use Even-Odd Properties to Find the Exact Values of the Trigonometric Functions

2.4 Graphs of the Sine and Cosine Functions

  • Graph the Sine Function y = sin(x) and Functions of the Form y = A sin(ωx)
  • Graph the Cosine Function y = cos(x) and Functions of the Form y = A cos(ωx)
  • Determine the Amplitude and Period of Sinusoidal Functions
  • Graph Sinusoidal Functions Using Key Points
  • Find an Equation for a Sinusoidal Graph

2.5 Graphs of the Tangent, Cotangent, Cosecant, and Secant Functions

  • Graph the Tangent Function y = tan(x) and the Cotangent Function y = cot(x)
  • Graph Functions of the Form y = A tan(ωx) + B and y = A cot(ωx) + B
  • Graph the Cosecant Function y = csc(x) and the Secant Function y = sec(x)
  • Graph Functions of the Form y = A csc(ωx) + B and y = A sec(ωx) + B

2.6 Phase Shift; Sinusoidal Curve Fitting

  • Graph Sinusoidal Functions of the Form y = A sin(ωx – φ) + B
  • Build Sinusoidal Models from Data

3. Analytic Trigonometry

3.1 The Inverse Sine, Cosine, and Tangent Functions

  • Define the Inverse Sine Function
  • Find the Value of an Inverse Sine Function
  • Define the Inverse Cosine Function
  • Find the Value of an Inverse Cosine Function
  • Define the Inverse Tangent Function
  • Find the Value of an Inverse Tangent Function
  • Use Properties of Inverse Functions to Find Exact Values of Certain Composite Functions
  • Find the Inverse Function of a Trigonometric Function
  • Solve Equations Involving Inverse Trigonometric Functions

3.2 The Inverse Trigonometric Functions (Continued)

  • Define the Inverse Secant, Cosecant, and Cotangent Functions
  • Find the Value of Inverse Secant, Cosecant, and Cotangent Functions
  • Find the Exact Value of Composite Functions Involving the Inverse Trigonometric Functions
  • Write a Trigonometric Expression as an Algebraic Expression

3.3 Trigonometric Equations

  • Solve Equations Involving a Single Trigonometric Function
  • Solve Trigonometric Equations Using a Calculator
  • Solve Trigonometric Equations Quadratic in Form
  • Solve Trigonometric Equations Using Fundamental Identities
  • Solve Trigonometric Equations Using a Graphing Utility

3.4 Trigonometric Identities

  • Use Algebra to Simplify Trigonometric Expressions
  • Establish Identities

3.5 Sum and Difference Formulas

  • Use Sum and Difference Formulas to Find Exact Values
  • Use Sum and Difference Formulas to Establish Identities
  • Use Sum and Difference Formulas Involving Inverse Trigonometric Functions
  • Solve Trigonometric Equations Linear in Sine and Cosine

3.6 Double-angle and Half-angle Formulas

  • Use Double-angle Formulas to Find Exact Values
  • Use Double-angle Formulas to Establish Identities
  • Use Half-angle Formulas to Find Exact Values

3.7 Product-to-Sum and Sum-to-Product Formulas

  • Express Products as Sums
  • Express Sums as Products

4. Applications of Trigonometric Functions

4.1 Right Triangle Trigonometry; Applications

  • Find the Value of Trigonometric Functions of Acute Angles Using Right Triangles
  • Use the Complementary Angle Theorem
  • Solve Right Triangles
  • Solve Applied Problems

4.2 The Law of Sines

  • Solve SAA or ASA Triangles
  • Solve SSA Triangles
  • Solve Applied Problems

4.3 The Law of Cosines

  • Solve SAS Triangles
  • Solve SSS Triangles
  • Solve Applied Problems

4.4 Area of a Triangle

  • Find the Area of SAS Triangles
  • Find the Area of SSS Triangles

4.5 Simple Harmonic Motion; Damped Motion; Combining Waves

  • Build a Model for an Object in Simple Harmonic Motion
  • Analyze Simple Harmonic Motion
  • Analyze an Object in Damped Motion
  • Graph the Sum of Two Functions

5. Polar Coordinates; Vectors

5.1 Polar Coordinates

  • Plot Points Using Polar Coordinates
  • Convert from Polar Coordinates to Rectangular Coordinates
  • Convert from Rectangular Coordinates to Polar Coordinates
  • Transform Equations between Polar and Rectangular Forms

5.2 Polar Equations and Graphs

  • Identify and Graph Polar Equations by Converting to Rectangular Equations
  • Test Polar Equations for Symmetry
  • Graph Polar Equations by Plotting Points

5.3 The Complex Plane; De Moivre’s Theorem

  • Plot Points in the Complex Plane
  • Convert a Complex Number between Rectangular Form and Polar Form or Exponential Form
  • Find Products and Quotients of Complex Numbers
  • Use De Moivre’s Theorem
  • Find Complex Roots

5.4 Vectors

  • Graph Vectors
  • Find a Position Vector
  • Add and Subtract Vectors Algebraically
  • Find a Scalar Multiple and the Magnitude of a Vector
  • Find a Unit Vector
  • Find a Vector from Its Direction and Magnitude
  • Model with Vectors

5.5 The Dot Product

  • Find the Dot Product of Two Vectors
  • Find the Angle between Two Vectors
  • Determine Whether Two Vectors Are Parallel
  • Determine Whether Two Vectors Are Orthogonal
  • Decompose a Vector into Two Orthogonal Vectors
  • Compute Work

5.6 Vectors in Space

  • Find the Distance between Two Points in Space
  • Find Position Vectors in Space
  • Perform Operations on Vectors
  • Find the Dot Product
  • Find the Angle between Two Vectors
  • Find the Direction Angles of a Vector

5.7 The Cross Product

  • Find the Cross Product of Two Vectors
  • Know Algebraic Properties of the Cross Product
  • Know Geometric Properties of the Cross Product
  • Find a Vector Orthogonal to Two Given Vectors
  • Find the Area of a Parallelogram

6. Analytic Geometry

6.1 Conics

  • Know the Names of the Conics

6.2 The Parabola

  • Analyze Parabolas with Vertex at the Origin
  • Analyze Parabolas with Vertex at (h, k)
  • Solve Applied Problems Involving Parabolas

6.3 The Ellipse

  • Analyze Ellipses with Center at the Origin
  • Analyze Ellipses with Center at (h, k)
  • Solve Applied Problems Involving Ellipses

6.4 The Hyperbola

  • Analyze Hyperbolas with Center at the Origin
  • Find the Asymptotes of a Hyperbola
  • Analyze Hyperbolas with Center at (h, k)
  • Solve Applied Problems Involving Hyperbolas

6.5 Rotation of Axes; General Form of a Conic

  • Identify a Conic
  • Use a Rotation of Axes to Transform Equations
  • Analyze an Equation Using a Rotation of Axes
  • Identify Conics without Rotating the Axes

6.6 Polar Equations of Conics

  • Analyze and Graph Polar Equations of Conics
  • Convert the Polar Equation of a Conic to a Rectangular Equation

6.7 Plane Curves and Parametric Equations

  • Graph Parametric Equations
  • Find a Rectangular Equation for a Plane Curve Defined Parametrically
  • Use Time as a Parameter in Parametric Equations
  • Find Parametric Equations for Plane Curves Defined by Rectangular Equations

7. Exponential and Logarithmic Functions

7.1 Exponential Functions

  • Evaluate Exponential Functions
  • Graph Exponential Functions
  • Define the Number e
  • Solve Exponential Equations

7.2 Logarithmic Functions

  • Change Exponential Statements to Logarithmic Statements and Logarithmic Statements to Exponential Statements
  • Evaluate Logarithmic Expressions
  • Determine the Domain of a Logarithmic Function
  • Graph Logarithmic Functions
  • Solve Logarithmic Equations

7.3 Properties of Logarithms

  • Work with the Properties of Logarithms
  • Write a Logarithmic Expression as a Sum or Difference of Logarithms
  • Write a Logarithmic Expression as a Single Logarithm
  • Evaluate Logarithms Whose Base Is Neither 10 Nor e

7.4 Logarithmic and Exponential Equations

  • Solve Logarithmic Equations
  • Solve Exponential Equations
  • Solve Logarithmic and Exponential Equations Using a Graphing Utility

7.5 Financial Models

  • Determine the Future Value of a Lump Sum of Money
  • Calculate Effective Rates of Return
  • Determine the Present Value of a Lump Sum of Money
  • Determine the Rate of Interest or the Time Required to Double a Lump Sum of Money

7.6 Exponential Growth and Decay Models; Newton’s Law; Logistic Growth and Decay Models

  • Model Populations That Obey the Law of Uninhibited Growth
  • Model Populations That Obey the Law of Uninhibited Decay
  • Use Newton’s Law of Cooling
  • Use Logistic Models

7.7 Building Exponential, Logarithmic, and Logistic Models from Data

  • Build an Exponential Model from Data
  • Build a Logarithmic Model from Data
  • Build a Logistic Model from Data

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