Fundamentals of Trigonometry
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# Fundamentals of Trigonometry

A primer on the theory and beauty of angles, triangles, and their relationships
4.7 (3 ratings)
80 students enrolled
Created by Joseph Gerth
Last updated 6/2015
English
Price: \$20
30-Day Money-Back Guarantee
Includes:
• 4 hours on-demand video
• 4 Supplemental Resources
• Full lifetime access
• Access on mobile and TV
• Certificate of Completion
What Will I Learn?
• Complete right triangles given two sides
• Complete 45-45-90 and 30-60-90 triangles given one side
• Evaluate trig functions (e.g. sin(90)=1)
• Convert between radians and degrees and vice versa
• Use the Unit Circle to evaluate Trig functions
• Use the arc functions to determine a missing angle measure
• Prove identities
• Complete non-right triangles using the Law of Sines, the Law of Cosines, and the Law of Tangents
View Curriculum
Requirements
• Algebra 1 or equivalent
• A little bit of Geometry knowledge
Description

This course is a rigorous approach to basic Trigonometry. It sets out to show the foundations of the subject in a theoretical way. By reading a formal textbook, watching PowerPoints on each topic, and taking actual tests, students will gain a deep and thorough understanding of the necessities of the subject.

This course will teach concepts such as the sine, cosine, and tangent function, radian measure, the use of the Unit Circle, identities, and more. There are 15 lectures with almost 4 hours of content. Students will receive a full textbook (with selected answers) and have access to three tests is .pdf format.

This course has been designed to help introduce students to Trigonometry and its basic skills. High school students seeking to prepare themselves for a Pre-Calculus or year-long Trigonometry course are part of the main audience, but this course certainly can benefit others looking to review this material, or simply learn it from a more theoretical standpoint.

Who is the target audience?
• Those looking for an introduction to Trigonometry
• Those looking to relearn Trigonometry
• Those looking to learn Trigonometry properly and theoretically
• Not for those needing an entire semester-long Trigonometry course
• Not for those requiring advanced level trigonometry (concepts such as DeMoivre's Theorem, for instance)
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Curriculum For This Course
22 Lectures
06:07:20
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Introduction
3 Lectures 25:30

The purpose of this intro video is to overview what the course is all about. You'll see the content that you'll learn, how you will learn it, who the intended audience is, and more.

Preview 15:42

Please see the video titled "How to use the textbook" for some basic advice on how to get the most from it. It's a bit different from most modern math textbooks, and, like most things, requires some practice and awareness to be comfortable with.

If you notice any spelling, grammar, format, or other errors, please let me know.

As the license mentions, this book is free to all. You may copy, print, share etc. this textbook as much as you like.

Preview 110 pages

Unfortunately, most students are used to math books that are essentially cookbooks. This means that most people have been conditioned to expect that the textbook is nothing more than a reference for the various steps to get to an answer.

This text is much different, and for very good reason. It's main purpose is to help foster intuition and thought in the reader. It is also designed for self-study, so all of the explanations and meanings are there in full.

However, different also presents a bit of a learning curve for most people. As such, I felt it necessary to provide some tips and pointers on how to get the most from this textbook. The video here offers a fair bit of advice so that you can succeed.

Preview 09:48
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Part I: Introduction to Trigonometry
7 Lectures 01:16:55

We begin with a concept from Geometry, and one well-known for millenia. It gives us a method by which we can solve a right triangle, if we have the right information. We will also use this formula to set up the next section.

1.1 Pythagoras' Theorem
08:49

This section shows us the two ever-important special right triangles: The 45-45-90 and the 30-60-90 triangles. They follow a sort-of-formula that allows us to complete them knowing only one side. We will use these relationships in this section and very, very often in the rest of the course.

1.2 Special right triangles
10:27

We finally begin our first steps into the world of Trigonometry. We meet our first Trig function, the Sine function. We learn that it takes an angle as an input, and outputs a side ratio.

1.3 The sine function
13:20

We next meet the cosine function. It is very closely related to the sine function, in both its name and its overall process.

1.4 The cosine function
13:00

To close out Part I we meet the Tangent function, which we derive from the sine and cosine function. It operates just like the sine and cosine function, but it outputs a different side ratio.

1.5 The tangent function
13:12

Although not in the textbook, my experience has told me that distinguishing which Trig function to use is a challenge for students. I attempt to show some of the intuition involved in determining whether one should use the sine, cosine, or tangent function in order to complete a right triangle.

The three Trig functions
18:07

You are free to take this test however you like, however, to do this "properly" you should not use any notes, your textbook, and you should not receive help from anyone else. Once you start this test, you must finish it (i.e. you can't do a problem, run an errand, do a few more, then run another errand...). You may use a calculator and you may take as long as you wish.

This test was worth 30 points.

Answer keys are available by request (i.e. email me).

Preview 3 pages
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Part II
6 Lectures 01:18:55

To begin Part II, we take a look at a different way to measure angles. Radians are very natural to use with Trig functions since they are also a ratio. We'll work a little bit with the definition of a radian, but ultimately, the most important thing in this section is to get comfortable with radians, and be able to quickly recognize what pi/6, pi/4, and other fundamental measurements represent.

18:22

We introduce perhaps the most important part of Trig: The unit circle. We will build some basic knowledge of the unit circle, and show how it can be used to evaluate Trig functions. To do this, we'll practice finding the coordinates of a point on the unit circle.

2.2 The unit circle
13:06

We reveal the true usefulness of the unit circle: It allows us to evaluate Trig functions that we don't have memorized, but are awfully similar to ones we already have memorized. This section is challenging, but mastering this content is well worth it.

We combine a host of lessons in this section, so make sure you are comfortable with 2.2, 1.3, 1.4, and 1.5

2.3 Angles greater than pi/2
18:50

We consider negative angles in this section. This is neat in and of itself, but it also provides a nice little shortcut for evaluating some Trig functions, particularly those in Quadrant IV.

2.4 Angles less than 0
14:38

Up to this point, we've used angles to figure out side lengths. Finally we go in the opposite direction. You'll know a side length ratio, and from that information derive an angle. Thus, we'll learn about the inverse of the Trig functions.

2.5 The arc functions
13:59

You are free to take this test however you like, however, to do this "properly" you should not use any notes, your textbook, and you should not receive help from anyone else. Once you start this test, you must finish it (i.e. you can't do a problem, run an errand, do a few more, then run another errand...). You may use a calculator and you may take as long as you wish.

This test was worth 30 points.

Answer keys are available by request (i.e. email me).

Preview 3 pages
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Part III
6 Lectures 01:08:00

The goal in this section is to show you identities in the friendliest way possible. This is always a very challenging concept, so we've split it up into two parts, the first of which being very elementary. This section requires intuition and patience, so dig in and be as diligent as possible.

3.1 Introduction to identities
11:27

This is a more formal showing of identities, but it's still not at the full scale difficulty you'll encounter in more challenging courses. We spend time distinguishing between equations and identities, which can be very confusing for students first working with identities.

The hope is that these two sections will make you much better prepared for the challenge to come.

3.2 Proving identities
12:00

Up until now, we've only been using right triangles when working with Trig functions. In the last three sections, we'll work with non-right triangles. The first method we'll use is probably the easiest: The Law of Sines.

3.3 The Law of Sines
13:17

The Law of Cosines is able to solve most non-right triangles that the Law of Sines cannot. It's a bit more complicated than the Law of Sines, but it's not that bad if you recognize some of the relationships in the formula.

3.4 The Law of Cosines
15:30

Perhaps as expected, there is also a Law of Tangents. This Law doesn't give us new functionality, as it is only capable of solving non-right triangles that either the Law of Sines or Law of Cosines can also solve. But the approach may be preferable to some.

This topic is almost never covered or even discussed in any math classes. So while not "required," we hope that you will find it interesting, if not useful.

3.5 The Law of Tangents
15:46

You are free to take this test however you like, however, to do this "properly" you should not use any notes, your textbook, and you should not receive help from anyone else. Once you start this test, you must finish it (i.e. you can't do a problem, run an errand, do a few more, then run another errand...). You may use a calculator and you may take as long as you wish.

This test was worth 30 points.

Answer keys are available by request (i.e. email me).

Preview 2 pages