# Ge-Alge-Trig: Elements of Geometry, Algebra, & Trigonometry

## What you'll learn

- Viewers will learn some foundational elements of geometry, algebra, and trigonometry that have universal practical application.
- Viewers will see the progression of the mathematics from visual descriptions and definitions into key derivations of mathematical formulas
- Viewers will be able to apply the visual descriptions and related mathematical formula to solve some "real world" problems
- Viewers will become empowered, eliminate mathematical anxiety (because they're not being graded, LOL), and learn to enjoy math.

## Requirements

- A willingness to learn mathematics and eliminate any anxiety by discovering the power of these tools in art, drafting, computers, science, and every day.
- Feeling comfortable with fractions, ratios, proportions, and coordinate plotting would be a good background to have but can be learned on the go by seeing it applied!
- The course is presented as a rather continuous derivation such that no prior knowledge is required, except perhaps some pre-requisite mathematical manipulations typically learned at the junior high school level: using additive inverses and multiplicative inverses

## Description

This free course is intended to provide a core of the mathematical skills that are used in college level science, technology, engineering, and mathematics (STEM) courses. It is not completed and will be completed as time permits.

In these days of standardized tests, geometry, algebra, and trigonometry are integrated. There are arguments for this pro-and-con. Formal proofs and derivations tend to be sacrificed for the sake of applied and integrated problems. That's fine as long as the concepts and formulas aren't thrown at a student, "here, use this," without them knowing where it comes from. The course is a guide while learning in class (not a substitute for class) OR a quick review of the math we all need to brush up on from time to time. I have an agenda with it: to refer to these lectures, or even present them in class, when STEM students aren't on board with what's assumed in a college course.

About half of the lectures are completed with about half of the video time consumed: free courses must remain under 2 hours.

Here are the lecture descriptions:

**Parallel Lines Cut by a Transect [completed]**

We start with some elementary definitions of lines, rays, and angles, and then learn the equivalences of the angles around two parallel lines cut by another line. This leads to a proof: the sum of the interior angles of a triangle sum to 180 degrees. These relationships and the proof set the stage for studying the properties of other geometric figures, and are applied in many drawings to describe and solve problems in science, technology, engineering, and mathematics (STEM). If I had a dollar for every time this arrangement is used in a STEM proof, I would be a trillionaire (move over Bezos).

**Some Properties and Formulas for Triangles [pending]**

In this presentation [when available] we look at triangles. In the preceding lecture we've already proved the most important formula for triangles: the sum of their interior angles is 180 degrees. After defining the four main types of triangles, we'll then prove the formula for the area of a triangle. The four types of triangles are:

Scalene: this is your random triangle folks, the one that does not have any particular defining relationship between its sides and angles.

Isosceles: this is a triangle with two equal sides, the third may or may not be equal. When this case occurs, it turns out the two base angles are also equal.

Equilateral: this is a triangle with all three sides equal, thus it is a type of isosceles triangle with two sides equal (and also the third). In this case, all of the interior angles are 60 degrees.

Right triangle: this is a triangle where one of the angles is 90 degrees. The two legs forming the 90 degree angle are called legs, and last side connecting the legs called the hypotenuse. Certainly the right triangle is important because it can be used in corners, but more importantly, it forms the core of trigonometry, used in the definition of the three main trigonometric functions: sine, cosine, and tangent.

**The Equation of a Line [pending, but exercise available]**

Before we cover quadrilaterals in the next lecture, we review the equation of a line. There are many different types of quadrilaterals, and their classification depends upon congruence of various angles, sides, and diagonals on these figures. In addition to being given diagrams of each with their defining properties, the various figures are plotted in the Cartesian space (standard graph paper) and these angles, sides, and diagonals are proven using the slopes of lines.

The lectures on lines and quadrilaterals are not ready at this point. However a review exercise for lines is provided in "external resources." Use the exercise to test your understanding of lines. If you are confident in plotting points, you do not need equation for a line or quadrilaterals to review the lectures on transformations.

**Quadrilaterals [pending]**

There are many different types of quadrilaterals, and their classification depends upon congruence of their various angles, sides, and diagonals when reviewing their diagrams. In addition to being given diagrams of each, and their classification according to each of their defining properties, the various figures are plotted in the Cartesian space (standard graph paper). The defining relationships of the angles, sides, and diagonals are proven for each.

Understanding the preceding lecture on lines will increase the understanding of this presentation. However, if you are confident enough in plotting points according to their ordered pairs, you do NOT need the equation for a line or quadrilaterals to review the next lectures on transformations.

**Transformations [completed: 4 Lectures]**

Transformations are mathematical operations that "transform" (change) a figure (2D) or object (3D) in the "space" in which it is plotted. These transformations can be "isometric," preserving the distances and angles of the figure or object, or "non-isometric," where the figure or object can be scaled (changed in size) or deformed (changed in shape such as skewed). We will cover the most common isometric examples 1) translation: moving a figure or object in "space," 2) rotation, rotating a figure about a point, and 3) reflection, where the figure or object is reflected by a line or plane. We then look at "composites," combinations of transformations that are performed sequentially, with the result depending upon the sequential order of the transformations.

We restrict this discussion to two-dimensional (2D) figures.

We look at the "rules" of each operation and what they do to the ordered pairs (x,y) of the points that define the figure.

**An Interesting Proof [exercises and their answers complete the proof; a video following the exercises/ proof is pending].**

We prove the Pythagorean theorem using translations from the previous section on transformations! The proof is centered on two quadrilaterals, but they are the easy one everyone knows: the square. We'll prove the two figures are squares first, but it is straightforward from inspection. The Pythagorean theorem, restated as "the distance formula," is used to calculate distances given sides of right triangles. It is thus usually used in math classes long before it is proven. The Pythagorean theorem is also used to define a circle. In the next lecture, we then look at the circle that forms the core of trigonometry, the "unit" (radius = 1) circle.

Though the presentation is not yet available, two "Google Forms" present the proof as an exercise. The presentation will follow this proof.

**The Unit Circle: the Core of Trigonometry [pending]**

In this presentation, we use the Pythagorean theorem along with the definition of sine and cosine to describe a "unit" (radius = 1) circle. We plot the points of some special commonly used angles.

**An Equation for a Wave [pending]**

Let the unit circle roll!!!!!!!! We let the unit circle roll while plotting sine (or cosine) and we get a wave. HOW COOL IS THAT???. We look at the resultant formula that defines that wave, then tweak it a bit.

**Triangulation [pending]**

We use the law of sines to calculate the distance to a mountain when we know a baseline between two points sighting on the mountain. We then calculate its height using the tangent function and an angle measured between the summit and the horizontal.

## Who this course is for:

- High school students (any K12) and early college students that need to shore up the mathematics that is required for most college "STEM" courses.
- Algebra and trigonometry are quite visual, and the "transformations" (expressed by either "rules" or matrices in linear algebra) have practical application in computer assisted design (CAD), science, programming, and graphical arts.

## Instructor

John R. Hoaglund, III, Ph.D.

Dr. Hoaglund is a geologist with more than 35 years of experience in environmental research, teaching, and consulting in the private sector, government, and academia. He received his BS (1985) and MS (1987) degrees in geology from the University of Wisconsin, worked in research and consulting in Wichita, Kansas on projects related to ground water supply and contamination, then returned to academics in 1991, receiving his doctoral degree in geological sciences from Michigan State University in 1996. As part of his dissertation, he completed the US Geological Survey (USGS) Regional Aquifer Systems Analysis (RASA) groundwater model of the Michigan Basin, a model used to calculate modern and Pleistocene groundwater and brine discharge to the Great Lakes and rivers in Michigan. He taught hydrogeology, groundwater modeling, environmental geology, and glacial / climate geology at the University of Michigan before joining the Pennsylvania State University research on regional climate-hydrologic models, sponsored by the National Science Foundation, and groundwater nitrate studies, funded by the US Department of Agriculture. In 2007, Dr. Hoaglund resumed groundwater consulting, focusing primarily on sites involving perchlorate groundwater contamination in southern California. While reviewing the reactions involved in the manufacture of perchlorate, he recognized the potential for the electrolysis reaction to consume salt waste while producing hydrogen. Later, reviewing DOW documents describing the reasons and methods for the air-tight conditions required for the storage of sodium hydroxide byproducts, he recognized the potential for the aeration reaction to sequester carbon into bicarbonate.

Dr. Hoaglund founded Carbon Negative Water Solutions, LLC in 2010 to pursue the trifecta of desalination, water resource development, hydrogen production, and CO2 sequestration. In addition to continued groundwater consulting, he wrote extensively about the potential for coupling ocean desalination with carbon sequestration, and approached several water and energy companies with the idea to promote mutually beneficial cooperation. He discovered these companies operate in separate universes on projects that are planned over a decade or more, and are reluctant to adopt new technologies over the established and state-approved method for greenhouse gas (GHG) mitigation: offsetting. In 2015, Dr. Hoaglund relocated to Las Vegas to accept contract work with Navarro Research and Engineering for the Department of Energy, assisting with groundwater characterization and modeling of the Nevada National Security Site (formerly Nevada Test Site), work related to the legacy groundwater contamination associated with historic nuclear testing. In 2019 he transitioned the Carbon Negative Water Solutions LLC to the non-profit Carbon Negative Water and Energy. In addition to the non-profit, he maintains a private research consulting and e-learning service, Provenance Geosciences.