Trigonometry 101

This semester I ran a partially flipped open source Trigonometry course using a free textbook and youTube videos. To give back to the OER community, I have recreated my blackboard shell here in a single blog post. I am offering all of the worksheets that I have created below in PDF and source DOCX format under a creative commons license (cc-by-sa).  Please feel free to copy anything here, modify, and/or redistribute.


Open Textbook:  Trigonometry by Michael Corral
Videos: Mathispower4u Trigonometry by James Sousa

End-page Formulas (Trig. cheat sheet): [ Trigonometry Definitions and Formulas | source ]
Free Graphing Utility: Desmos.com
Free CAS: WolframAlfa

Chapter 1: Right Triangle Trigonometry

1.1 Angles: [ Assignment 1a : source ]

   Video notes: Angle Basics
   Video notes: Degrees, Minutes, and Seconds
   Video notes: Angle Relationships and Types of Triangles
   Video notes: The Pythagorean Theorem
   Video notes: Converse of the Pythagorean Theorem

1.2 Trigonometric Functions: [ Assignment 1b : source ]

   Video Notes: Introduction to Trigonometric Functions
   Video Example: Determine Trigonometric Function Values
   Video Example: Trig. Function Values with Hypotenuse Missing

1.3 Applications: [ Assignment 1c : source ] [ Assignment 1d : source ]

   Video Example: Determine the Length of a Side
   Video Example: Perimeter of an Equilateral Triangle

1.4 Trigonometric Functions of any Angle: [ Assignment 1e : source ]

   Video Examples: Determine the Reference Angle
   Video Notes: Introduction to Trig Functions using Angles

1.5 Rotations and Reflections:  [ Assignment 1f : source ]

*** *** *** [ Sample Exam #1 | source ] *** *** ***

 

Chapter 2: General Triangles

2.1 Law of Sines [ Assignment 2a | source ]

   Video: The Law of Sines: The Basics
   Video: The Law of Sines: The Ambiguous Case
   Example: Solve a Triangle Using the Law of Sines

2.2 Law of Cosines [ Assignment 2b | source ]

   Video: The Law of Cosines
   Example: Application of the Law of Cosines
   Example: Determine an Angle given SSS

2.3 Law of Tangents

2.4 Area of a Triangle [ Assignment 2c | source ]

   Example: Determine the Area of a Triangle Using the Sine Function
   Example: Area using Heron's Formula

Chapter 3: Identities

3.1 Basic Identities [ Assignment 3a | source ]
 
   Video: Fundamental Identities
   Video: Verifying Identities (good using a slightly different method.)

3.2 Sum and Difference Formulas [ Assignment 3b | source ]
 
   Video: Sum and Difference Identities for Cosine
   Video: Sum and Difference Identities for Sine
   Video: Sum and Difference Identities for Tangent

3.3 Double and Half-Angle Formulas [ Assignment 3c | source ]
 
   Video: Double Angle Identities
   Example: Double Angle Example Given Information
   Example: Using the Half Angle Formula (use degrees here)
   Example: Using Sine of a Half Angle

*** *** *** [ Sample Exam #2 | source ] *** *** ***

Chapter 4: Radian Measure

4.1 Radian Measure [ Assignment 4a | source ]

   Lecture: Radian Measure (Great video!)

4.2 Arc Length [ Assignment 4b | source ]
 
   Lecture: Arc Length and Area of a Sector

4.3 Area of a Sector [ Assignment 4c | source ]
 
   Lecture: Arc Length and Area of a Sector

4.4 Circular Motion [ Assignment 4d | source ]
 
   Lecture: Linear Velocity and Angular Velocity

Chapter 5: Graphing and Inverse Functions

5.1 Graphing Trigonometric Functions [ Assignment 5a | source ]
 
   Video Lecture: Graphing the Sine and Cosine Function
   Video Lecture: Graphing the Tangent Function
   Video Lecture: Graphing Cosecant and Secant

5.2 Properties of Graphs[ Assignment 5b | source ]

   Video Lecture: Amplitude and period of Sine and Cosine
   Video Lecture: Horizontal and Vertical Translations of Sine and Cosine
   Video Lecture: Graphing Sine and Cosine with Transformations

5.3 Inverse Trigonometric Functions [ Assignment 5c | source ]

   Video Lecture: Inverse Functions
   Video Lecture: Intro. to Inverse Sine, Cosine, and Tangent
   Video Examples: Evaluating Expressions involving the Inverse Trig. Functions

*** *** *** [ Sample Exam #3 | source ] *** *** ***

Chapter 6: Additional Topics

6.1 Solving Trigonometric Equations
[ Assignment 6a | source ] [ Assignment 6b | source ] [ Assignment 6c | source ] 
 
   Video Lecture: Solving Trigonometric Equations I  (Excellent introduction! Linear)
   Video Lecture: Solving Trigonometric Equations II (Quadratic examples)
   Video Lecture: Solving Trigonometric Equations III (Using Identities first, quadratic formula)
   Video Lecture: Solving Trigonometric Equations IV  (Half and multiple angles)

6.2 Numerical Methods in Trigonometry
 
6.3 Complex Numbers [ Assignment 6d | source ]
 
   Video Lecture: Introduction to Complex Numbers
   Video Lecture: Complex Number Operations
   Video Lecture: Trigonometric Form of Complex Numbers
   Video Lecture: De Moivre's Theorem
   Video Lecture: Nth Roots of a Complex Number

6.4 Polar Coordinates [ Assignment 6e | source ]
 
   Video Lecture: Introduction to Polar Coordinates
   Video Lecture: Converting Polar Equations to Rectangular Equations  (sound is bad)
   Video Lecture: Graphing Polar Equations Part I
   Video Lecture: Graphing Polar Equations Part II

*** *** *** [ Sample Exam #4 | source ] *** *** ***

*** Sample Final Exam - coming soon ***

Trigonometry 101 by John Redden is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License. ---

The Graphs of the Six Trigonometric Functions

The objective here is to have the basic graphs of all six trigonometric functions ready to go. Here is a link to a one-page printable reference sheet.

1. One period graphed for the sine function. ( y = sin x )

2. One period graphed for the cosine function. ( y = cos x )

Notice that cosine function is a horizontal translation of sine by π/2 units. To see this click on the link below for an interactive graph.

Desmos Interactive Graph

3. One period graphed for the cosecant function. ( y = csc x = 1 / sin x )

4. One period graphed for the secant function. ( y = sec x = 1 / cos x )

Notice that the zeros of sine and cosine define vertical asymptotes for the cosecant and secant functions. To see this click on the link below for an interactive graph.

Desmos Interactive Graph

5. One period graphed for the tangent function.  ( y = tan x = sin x / cos x )

6. One period graphed for the cotangent function. ( y = cot x = cos x / sin x )

Notice that for the tangent function, the zeros of cosine determine the vertical asymptotes. And for the cotangent function, the zeros of sine determine the vertical asymptotes.

Desmos Interactive Graph

Enjoy.

Visually Searchable Algebra Video Compendium

[ OpenAlgebra.com Visually Searchable Video Compendium ]

I have been working with beginning Algebra students for a good many years now. In that time, I have found myself answering many similar questions where students often tell me that when they are alone they get stuck and do not know where to look for help.  This is certainly the case when problems are posed out of sectional context. After some interaction, it becomes clear that folks simply do not have the algebraic vocabulary needed when searching the index or internet for help and quickly become overwhelmed.

"I do not know what to type in when searching for help!"
 

To help, my idea was to create a visually searchable database of commonly asked algebra questions. This compendium of algebra videos is intended to be more of a "how to" type of approach serving students looking to solve a particular problem in a short amount of time.  More instructional or "lecture" type of videos are valuable and quite abundant. For these I would like to recommend James Sousa's MathIsPower4u website.

After authoring two online algebra textbooks published by Flatworld Knowledge, I created a similar visually searchable list of videos for all of the "Thy this!" problems found in the textbook.

[ Elementary Algebra (Algebra 1): Try This Videos ]

[ Intermediate Algebra (Algebra 2): Try this Videos ]

This approach is not the silver bullet for success in Algebra, however, it does review well is appreciated by students who are more inclined toward a visual learning style.  Everything on the openAlgebra.com site is offered free of charge and is openly licensed with a creative commons license.  This means that you are free to and encouraged to link to it, copy-and-paste into your CMS, build upon, improve and share the material.
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This semester I posted mobile phone friendly solutions to exams in my Elementary Algebra and Intermediate Algebra courses.  Both are one-semester courses. This was easy to do using a basic snipping tool and Blogger.

[ Elementary Algebra Exam #1 Solutions ]

[ Elementary Algebra Exam #2 Solutions ]

[ Elementary Algebra Exam #3 Solutions ]

[ Elementary Algebra Exam #4 Solutions ]

--- EA Sample Final Exam Ch. 1-7 ---

[ Intermediate Algebra Exam #1 Solutions ]

[ Intermediate Algebra Exam #2 Solutions ]

[ Intermediate Algebra Exam #3 Solutions ]

[ Intermediate Algebra Exam #4 Solutions ]

--- IA Sample Final Exam Ch. 1- 7 ---

In doing this, I was able to save some valuable class time. We are not quite finished, I still need to cover conic sections (chapter 8) in Intermediate Algebra, but students definitely said they appreciated this approach. Please feel free to comment or share.

App Inventor Tutorial

Here I have created a mathematical programming tutorial using the MIT App Inventor platform.

[ Algebra App: Evaluating a Linear Expression ]

The tutorial guides students through the process of making the following android application.

The almost completed program follows:

Please feel free to try it, distribute, and remix.  This is a "work in progress," so any suggestions would be greatly appreciated.

MathJax Equation Builder

Here I'm testing a web app using what I learned in Dart flight school. I built the app and then uploaded the build folder to Google docs... embedded below in an iframe. Very easy using angular dart. Some LaTeX to try follows:

new line = \\  fraction = \frac{x}{y}  square root = \sqrt{x}

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Over 100,000 Views!

My algebra YouTube videos broke the 100,000 view mark! I know it is not a lot compared to many others out there but it is a milestone for me nonetheless.

I have received a great many "thank you" type comments and emails. It still amazes me that I can help so many so easily.  And I am getting better at tagging and SEO so hopefully this run will continue to accelerate.

Free Mathematics eBooks

Hundreds of E-Books for free online viewing and/or download.

http://www.e-booksdirectory.com/listing.php?category=3

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Open Algebra Study Guide Website

Do you have a student that missed a day? Well... send him a link to that days topic!

Our free algebra study guide makes for a rather useful website. It is coming together quite nicely using Blogger - have a look.

http://www.openAlgebra.com/

Feel free to share a link, copy-and-paste anything you find there.  Also, notice that there are links to videos at the bottom of each page. Hope this helps somebody out there!