Daniel Kopsas

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Pre-Calculus Tutorials

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ALGEBRA


Adding, Subtracting, Multiplying, and Dividing Functions (YouTube)
Understanding Composition of Functions with Numeric Examples (YouTube)
Determining the Domain of a Composition of Functions (YouTube)

Definition of an Injective (One-to-One) Function (YouTube)
Definition of an Inverse Function with Basic Examples (YouTube)
Finding the Equation of an Inverse Function (YouTube)
Using Function Composition to Verify that Two Functions are Inverses (YouTube)
Graphing an Inverse Function Using the Original Function Graph (YouTube)

Exponential and Logarithmic Functions

Exponent Review: Zero, Negative, and Fractional Exponents (YouTube)
What is a Logarithm? (YouTube)
The Number e, Common Logarithms, & Natural Logarithms (YouTube)

Basic Exponential Functions (YouTube)
Graphing an Exponential Function Using Transformations (YouTube)

Restrictions on Logarithms (YouTube)
Finding the Domain of Functions with Logarithms (YouTube)
Advanced Example: Finding the Domain of Functions with Logarithms (YouTube)


Factoring Polynomials

Synthetic Division of Polynomials (YouTube)
Using Synthetic Division to Evaluate a Polynomial (YouTube)
How Zeros and Factors of a Polynomial are Related

General Method of Completing the Square (YouTube)
Solving a Quadratic Equation by Using a Square Root (Completing the Square) (YouTube)
Factoring Quadratic Polynomials (YouTube)

Rational Zeros of a Polynomial (YouTube)
Factoring a Polynomial by First Finding a Rational Zero (YouTube)

Polynomial & Rational Functions

Determining What Values Make a Polynomial Zero, Positive, or Negative (YouTube)
Using the Leading Term of a Polynomial to Graph its "Ends" (YouTube)

When a Rational Function is Zero, Not Defined, Positive, or Negative
Finding the x-Intercepts and Vertical Asymptotes of a Rational Function
The "Ends" of a Rational Function - The Horizontal Asymptote
Graphing a Factored Rational Function
Graphing a Non-Factored Rational Function
Graphing a Rational Function Where the Ends are not Horizontal (YouTube)

TRIGONOMETRY

Angles

Angles Measured in Degrees
Angles Measured in Radians
Relationship Between Radians and Degrees
Converting Between Radians and Degrees
Radian Measures Not in Terms of Pi

Unit Conversion

Converting Units Using Conversion Factors

Arc Length and Sector Area

Using an Angle or an Arc Length to Calculate a Percentage of a Circle
The Area of a Circular Sector (Want to redo; too formulaic.)
Calculating Arc Length (Want to redo; too formulaic.)

Circular Motion

Circular Motion: Angular Speed
Circular Motion: Linear Speed

Right Triangles

The Pythagorean Theorem
Right-Triangle Trigonometry
Using Trigonometric Functions to Find the Side of a Right Triangle
Using Inverse Trigonometric Functions to Find an Angle of a Right Triangle
Special Right Triangle: 30, 60, 90
Special Right Triangle: 45, 45, 90

Understanding a Trigonometric Function of a Non-Acute Angle
Understanding a Trigonometric Function of a Multiple of 900
Examples: Evaluating Trigonometric Functions of any Angle

Determining the Sign of a Trigonometric Function
Finding the Value of Any Trig. Function Given the Value of One

The Pythagorean Identities

Understanding the Pythagorean Identities
Using a Pythagorean Identity to Algebraically Write One Trig. Function in Terms of Another

The Area of a Triangle

The Area of a Triangle
Finding the Area of a Triangle, Given Two Sides and the Included Angle

Oblique Triangles: Law of Sines, Law of Cosines

Basic Properties of Triangles
The Law of Sines
The Difficulty with Using Sine Inverse when Solving a Triangle
Using Law of Sines to Solve a Triangle with "No Solution"
Using Law of Sines to Solve a Triangle with One Solution
Using Law of Sines to Solve a Triangle with Two Solutions

The Law of Cosines
Using the Law of Cosines to Solve for a Side of a Triangle
Using the Law of Cosines to Solve for an Angle

Navigation and Direction

Navigation: Basics of Bearings

The Unit Circle

Arc Lengths and Angles with the Unit Circle
Cutting the Unit Circle into Common Angles or Arc Lengths
Terminal Points on the Unit Circle
Determining if a Point is on The Unit Circle

Even & Odd Functions

What are Even and Odd Functions?
Even and Odd Trigonometric Functions
Determining Whether a Function is Even, Odd, or Neither

Graphing Trigonometric Functions

The Basic Sine Function Graph
The Basic Cosine Function Graph
Period, Amplitude, and Average Value of a Sine or Cosine Function
Graphing a Transformed Sine or Cosine Function
Determining the Equation for a Transformed Sine or Cosine Function Given its Graph

The Basic Tangent Function Graph
Graphing a Transformed Tangent Function

Graphing Functions that are Reciprocals
Graphing a Secant Function
Graphing a Cosecant Function

Simple Harmonic Motion

Simple Harmonic Motion

Proving Trigonometric Identities

Proving or Disproving an Identity (YouTube)
Basic Strategies for Proving Trigonometric Identities (YouTube)

A Trigonometric Function of a Sum or Difference of Two Angles (Addition/Subtraction Formulas)

A Trigonometric Function of a Sum or Difference (YouTube)
Some Ways to Use the Addition and Subtraction Formulas (YouTube)
A Proof Using Addition & Subtraction Formulas (YouTube)

A Trigonometric Function of Double an Angle (Double-Angle Formulas)

Deriving the Double-Angle Formulas from the Addition Formulas (YouTube)
Basic Examples Using the Double-Angle Formulas (YouTube)
A Proof Using Double-Angle Formulas (and other techniques) (YouTube)

Writing the Square of a Trigonometric Function in Terms of the First Power of Cosine (Power-Lowering Formulas)

Deriving Power-Lowering Formulas (YouTube)
Example Using Power-Lowering Formula (YouTube)

A Trigonometric Function of Half an Angle (Half-Angle Formulas)

Deriving Half-Angle Formulas (YouTube)
Examples Using Half-Angle Formulas (YouTube)
Inverse Trigonometric Functions

The Meaning of an Inverse Trig. Function (YouTube)
The Inverse Sine Function (YouTube)
The Inverse Cosine Function (YouTube)
The Inverse Tangent Function (YouTube)


Complex Numbers in Polar Form and in "a+bi" Form

Graphing Complex Numbers in the Complex Plane (YouTube)
Converting a Complex Number from "a+bi" Form to Polar Form (YouTube)
How to Multiply and Divide Complex Numbers in Polar Form (YouTube)
Examples of Multiplying and Dividing Complex Numbers in Polar Form (YouTube)
DeMoivre's Theorem: Raising a Complex Number in Polar Form to a Power (YouTube)

The nth Roots of a Complex Number (YouTube)
Examples of Finding Roots of a Complex Number (YouTube)

Vectors

What is a Vector? (YouTube)
Operations on Vectors (YouTube)
Unit Vectors (YouTube)
Writing Vectors in Terms of i and j (YouTube)

An Application of Vectors: True Velocity (YouTube)

Dot Product (YouTube)
Calculating the Angle Between Vectors (YouTube)
I will eventually put a video here about orthogonal vectors

What is a Vector Projection? (YouTube)
Finding a Unit Vector in the Same Direction as a Given Vector (YouTube)
Calculating the Magnitude of a Projection Vector (YouTube)
The Strategy for Calculating a Projection Vector (YouTube)