Trigonometry Essentials: Learn It All Step-By-Step

A Complete, Visual, and Application-Driven Journey Through Trigonometry

What you'll learn

What is trigonometry and triangles

How angles work—degree and radian measure, angular velocity, and reference angles

The Pythagorean Theorem, distance formula, and how to determine right triangles

Definitions and properties of the six trigonometric ratios, including reciprocal and quotient identities

How to solve right triangles, including real-world applications like measuring heights and distances

The unit circle, trigonometric identities, and graphing sine, cosine, tangent, and their transformations

How to use double angle, half angle, and sum-to-product identities to simplify expressions and solve equations

Techniques for solving trigonometric equations, including those with modified periods and identities

The Law of Sines, Law of Cosines, and how to solve oblique triangles, including the ambiguous case

How to compute area of triangles using trigonometric formulas, including Heron’s formula

Introduction to vectors, including static equilibrium, dot product, and angle between vectors

Deep dive into complex numbers, including polar form, Euler’s identity, and De Moivre’s Theorem

How to work with parametric equations, polar coordinates, and graph polar functions

Requirements

Basic understanding of algebra and geometry

A willingness to engage with visual and analytical problem-solving

Description

Mastering Trigonometry: From Triangles to Polar Graphs and Complex NumbersTrigonometry is often introduced as a tool for solving triangles—but in reality, it’s a gateway to understanding the geometry of motion, waves, rotations, and even complex numbers. This course is designed to redefine how you learn trigonometry, offering a comprehensive, structured, and visually intuitive approach that goes far beyond textbook basics.You’ll begin with the essentials—right triangles, the Pythagorean Theorem, and angle relationships—and quickly progress to unit circle trigonometry, graphing sinusoidal functions, and solving real-world problems involving elevation, distance, and circular motion. From there, you’ll explore trigonometric identities, inverse functions, and equation solving techniques that are foundational for calculus and physics.But what truly sets this course apart is its breadth and depth. You’ll go beyond triangles to study vectors, complex numbers, parametric equations, and polar coordinates—topics that are rarely covered in standard trigonometry courses but are essential for advanced STEM fields. You’ll learn how trigonometry powers engineering, navigation, signal processing, and computer graphics, and how it connects algebra, geometry, and analysis into a unified mathematical language.Why Take This Course?To gain a deep, intuitive understanding of trigonometry—not just memorize formulasTo prepare for calculus, physics, engineering, or any STEM discipline that relies on trigonometric reasoningTo learn real-world applications of trigonometry in motion, measurement, and modelingTo explore advanced topics like complex numbers, polar graphs, and parametric equations in a clear and accessible wayTo build confidence in problem-solving, graph interpretation, and mathematical thinkingWhat Makes This Course UniqueComprehensive Coverage: From basic triangles to advanced polar graphs and complex rootsVisual Learning: Diagrams, animations, and graphing techniques that make abstract ideas concreteReal-World Applications: Physics, engineering, navigation, and moreStep-by-Step Instruction: Clear explanations, guided examples, and problem-solving strategiesAdvanced Topics Included: Vectors, Euler’s identity, De Moivre’s Theorem, and parametric curvesDesigned for All Levels: Whether you're reviewing or learning for the first time, this course adapts to your pace

Overview

Section 1: The Pythagorean Theorem

Lecture 1 Trigonometry and Triangles

Lecture 2 The Pythagorean Theorem

Lecture 3 The Distance Formula

Lecture 4 Determining If a Triangle is Right

Lecture 5 The Midpoint Formula

Lecture 6 Trigonometry and Circles

Section 2: Angles

Lecture 7 Angles

Lecture 8 Degree Measurement

Lecture 9 Complementary and Supplementary Angles

Lecture 10 Angular Velocity

Section 3: Similar Triangles

Lecture 11 The Alternate Interior Angle Theorem (Trigonometry)

Lecture 12 The Angle Sum Triangle Theorem

Lecture 13 Similar Triangles

Section 4: Right Triangle Trigonometry

Lecture 14 Definitions of the Six Trigonometric Ratios

Lecture 15 Reciprocal and Quotient Identities

Lecture 16 Right Triangle Trigonometry

Lecture 17 The Cofunction Theorem

Section 5: Solving Right Triangles

Lecture 18 Solving Right Triangles

Lecture 19 Solving a Right Triangle Involving a Circle

Lecture 20 Solving Two Connected Right Triangles

Lecture 21 Solving an Isosceles Triangle Using Right Triangles

Lecture 22 Angles of Elevation and Depressions

Lecture 23 Measuring the Height of a Pole When the Distance From the Pole Cannot Be Measure

Lecture 24 Measuring the Height of a Building While in a Neighboring Building

Section 6: Reference Angles

Lecture 25 30-60-90 Triangles

Lecture 26 45-45-90 Right Triangles

Lecture 27 Special Angles

Lecture 28 Reference Angles

Lecture 29 Computing Trigonometric Ratios Using Reference Angles

Section 7: Radians

Lecture 30 Radian Measure

Lecture 31 Computing Arc Length for Angle Measures Other Than Radians

Lecture 32 Arc Length

Lecture 33 Approximating Linear Distance with Arc Length

Lecture 34 Area of a Circular Sector

Section 8: Circle Trigonometry

Lecture 35 The Unit Circle

Lecture 36 The Unit Circle Diagram

Lecture 37 Geometric Representations of the Trigonometric Ratios on the Unit Circle

Lecture 38 The Pythagorean Identity

Lecture 39 Angular and Linear Velocities

Section 9: Trigonometric Graphs

Lecture 40 Domain and Range of Trigonometric Graphs

Lecture 41 Periodicity of Trigonometric Graphs

Lecture 42 Symmetry of Trigonometric Graphs

Lecture 43 The Graph of Sine

Lecture 44 The Graph of Cosine

Section 10: Transformations of Sinusoidal Waves

Lecture 45 Amplitude of Sinusoidal Wave

Lecture 46 Period of Sinusoidal Wave

Lecture 47 Vertical and Horizontal Stretches to a Sinusoidal Wave

Lecture 48 Translations to Sine and Cosine

Lecture 49 Sketching Graphs of Sine and Cosine

Lecture 50 Recognizing Graphs of Sine and Cosine

Section 11: Other Trigonometric Graphs

Lecture 51 The Graphs of Secant and Cosecant

Lecture 52 The Graphs of Tangent and Cotangent

Section 12: Simple Harmonic Motion and Review of Inverse Functions

Lecture 53 Simple Harmonic Motion

Lecture 54 Inverse Functions

Lecture 55 The Inverse Function Property

Section 13: Inverse Trigonometry

Lecture 56 The Inverse Trigonometric Functions

Lecture 57 Computing Inverse Trigonometric Functions

Lecture 58 Inverse Trigonometric Expressions and Triangle Diagrams

Section 14: Trigonometric Identities

Lecture 59 The Fundamental Trigonometric Identities

Lecture 60 Converting Algebraic Expressions in Trigonometric Expressions

Lecture 61 Proving Trigonometric Identity Tip: Always Work from the Left-Hand Side to the R

Lecture 62 Proving Trigonometric Identity Tip: When in Doubt, Convert to Sine and Cosine

Lecture 63 Proving Trigonometric Identity Tip: Try using a Pythagorean Identity When Square

Lecture 64 Proving Trigonometric Identity Tip: When Adding Trigonometric Fractions, Find a

Lecture 65 Guidelines For Proving Trigonometric Identities

Lecture 66 Proving Trigonometric Identities Involving Pythagorean Identities

Lecture 67 Proving Trigonometric Identities Involving R

Section 15: Double Angle Identities

Lecture 68 Double Angle Identity for Sine

Lecture 69 Double Angle Identity for Cosine

Lecture 70 Graphing a Trigonometric Function Using Double Angle Identities

Lecture 71 Double Angle Identity for Tangent

Lecture 72 A Trigonometric Substitution Using Double Angle Identities

Section 16: Half Angle Identities

Lecture 73 Half Angle Identities for Sine and Cosine

Lecture 74 Half Angle Identity for Tangent

Lecture 75 Graphing a Trigonometric Function Using Half Angle Identities

Lecture 76 Proving a Trigonometric Identity Using Half Angle Identities

Section 17: Additional Trigonometric Identities

Lecture 77 Inverse Trigonometric Expressions and Trigonometric Identities

Lecture 78 Sum to Product Identities

Section 18: Solving Trigonometric Equations

Lecture 79 Solving Linear Trigonometric Equations

Lecture 80 Solving Quadratic Trigonometric Equations

Section 19: Solving Trigonometric Equations Using Trigonometric Identities

Lecture 81 Solving Trigonometric Equations Using Ratio Identities

Lecture 82 Solving Trigonometric Equations Using Double Angle Identities

Lecture 83 Solving Trigonometric Equations Using Pythagorean Identities

Lecture 84 Solving Trigonometric Equations By Squaring

Section 20: Solving Trigonometric Equations with Modified Periods

Lecture 85 Solving Trigonometric Equations With Modified Periods

Lecture 86 Solving Trigonometric Equations Using Angle Sum Identities

Lecture 87 Solving Quadratic Trigonometric Equations With Modified Periods

Lecture 88 Solving Trigonometric Equations By Squaring (Reprise)

Section 21: Law of Sines

Lecture 89 Solving Oblique Triangles

Lecture 90 The Law of Sines

Lecture 91 Solving Oblique Triangles (AAS) Using the Law of Sines

Lecture 92 Solving Oblique Triangles (ASA) Using the Law of Sines

Lecture 93 Using the Law of Sines to Find the Altitude of a Satellite

Section 22: Law of Cosines

Lecture 94 Law of Cosines

Lecture 95 Solving Oblique Triangles (SAS) Using the Law of Cosines

Lecture 96 Solving Oblique Triangles (SSS) Using the Law of Cosines

Lecture 97 Using the Law of Cosines to Find the Dimensions of a Parallelogram

Section 23: The Ambiguous Case

Lecture 98 The Ambiguous Case (SSA)

Lecture 99 Solving the Ambiguous Case (SSA): No Solution

Lecture 100 Solving the Ambiguous Case (SSA): Two Solutions

Lecture 101 Solving the Ambiguous Case (SSA): One Solution

Section 24: Triangle Area

Lecture 102 Finding the Area of an Oblique Triangle (SAS)

Lecture 103 Finding the Area of an Oblique Triangle (AAS and ASA)

Lecture 104 Finding the Area of an Oblique Triangle (SSS) and Heron's Formula

Section 25: Geometric Vectors

Lecture 105 Geometric Vectors (Trigonometry)

Lecture 106 Static Equilibrium (Right Triangle)

Section 26: Vector Applications

Lecture 107 Static Equilibrium (Oblique Triangle)

Lecture 108 Headings and True Course

Lecture 109 Headings and True Course (The Ambiguous Case)

Section 27: Algebraic Vectors

Lecture 110 Algebraic Vectors (Trigonometry)

Lecture 111 Unit Vectors (Trigonometry)

Lecture 112 Static Equilibrium (Reprise)

Section 28: The Dot Product

Lecture 113 The Dot Product

Lecture 114 The Angle Between Two Vectors and Perpendicular Vectors

Section 29: Complex Numbers

Lecture 115 Dot Products and Work

Lecture 116 Multiplication of Complex Numbers

Lecture 117 Division of Complex Numbers

Lecture 118 The Complex Plane

Lecture 119 Polar Form of Complex Numbers

Lecture 120 Computing the Polar Form of a Complex Number

Lecture 121 Euler's Identity

Section 30: Products and Quotients of Complex Numbers

Lecture 122 Multiplication of Complex Numbers in Trigonometric Form

Lecture 123 De Moivre’s Theorem and Complex Exponents

Lecture 124 Division of Complex Numbers in Trigonometric Form

Section 31: Roots of Complex Numbers

Lecture 125 Roots of Complex Numbers

Lecture 126 Solving a Cubic Equation Using Complex Roots

Lecture 127 Finding All Complex Roots of a Biquadratic Polynomial

Section 32: Parametric Equations

Lecture 128 Parametric Equations

Lecture 129 Parametric Equations and Conic Sections

Lecture 130 The Cycloid

Section 33: Polar Coordinates

Lecture 131 Polar Coordinates

Lecture 132 Conversion Between Polar and Cartesian Coordinates

Lecture 133 Polar Equations

Section 34: Polar Graphs

Lecture 134 Graphs of Polar Functions

Lecture 135 Graphs of Cardioids and Limaçons

High school and college students studying geometry, trigonometry, or precalculus,STEM learners in physics, engineering, or computer science,Educators seeking a structured and visual approach to teaching trigonometry,Anyone who wants to understand the math behind motion, waves, and circular systems