Definition
SOHCAHTOA is a mnemonic device that is used in mathematics to remember the definitions of the three most common trigonometric functions. Sine, cosine, and tangent are the three main functions in trigonometry. They’re all based on ratios obtained from a right triangle. Before we can discuss what ratios work for which function, we need to label the right triangle. The ratios that allow you to determine the sine, cosine, and tangent of a right triangle are: Opposite is the side opposite the angle in question, adjacent is the side next to the angle in question, and the hypotenuse is the longest side of a right triangle.1
- SOH = Sine is Opposite over Hypotenuse
- CAH = Cosine is Adjacent over Hypotenuse
- TOA = Tangent is Opposite over Adjacent
Who
It seems that a lot of schooling, such as elementary through high school, requires students to memorize. Memorization is one aspect of learning. Here we are going to discuss the meaning of the trigonometric identities. No memorization required – remember by using.
What
As pointed out in the Sine and Cosine – Definition & Meaning – Part 1 – What is Sin(x) & Cos(x)? video below, the points on the circle represent the sine and cosine values of the angles around the unit circle. But what does that really mean? Let’s find out.
Why
This information is critical to understanding the Sine Wave. Also, see Theoretical Knowledge Vs Practical Application.
Understanding how to measure shapes is a critical feature of geometry. Connections are fascinating to scientists and mathematicians. They investigate how a plant or animal interacts with its surroundings, as well as the relationship between numbers and nature, and music. They give it a name and share the knowledge when they find a specific relationship repeatedly. The Pythagorean Theorem, which defines the relationship between the sides of a right triangle, is one such example. Pythagoras was a Greek mathematician who contributed significantly to the field of mathematics.3
Why are cosine, sine, tangent, and cotangent so important? Why should I be graphing them, using them in calculus, etc.?
When I was in high school, someone in my class asked this very question. I didn’t have a very good teacher and she had a rather disappointing answer at the time, an obscure example of trigonometry in calculations for accurate aircraft movement. During my later studies which covered most STEM subjects, I have thought back about the naivete of the question many times. The importance of the harmonic functions, and the related exp and log functions, is almost impossible to overstate.
In mathematics, they are fundamental for a few things like complex analysis and Fourier analysis. Abstract mathematicians like high levels of abstraction so individual functions are too concrete to be seen as important or fundamental in most contexts. On the other hand, they will be ubiquitous as examples, solutions, tricks and exercises in every field of maths including geometry, differential equations, Hilbert spaces, topology, etc.
In physics and engineering, these functions are everywhere. Whenever something is circular or even remotely round, expect to see sin and cos in its description. Whenever something repeats itself in time, expect to see sin and cos in its description. That includes sound waves, electromagnetic waves (including light), any other kind of wave really (like sound or actual waves on the sea), the behavior of elementary particles (both classical and quantum), planetary motion, signal processing, optics, colloidal chemistry (nano science), … Even the temperature profile inside a wall is described by a coscos function in [-pi,pi].
Aside from their use in fundamental theory and theoretical modeling, these functions serve a role as building blocks and inspiration in secondary uses such as (counter-)examples, visualizations (e.g., Andrews plot), empirical formulae and heuristics (e.g., tanh as an activation function in artificial neural networks), etc. In one way or another, they are a central part of the general thought process of almost anyone who works with science-related topics.2
How
Let’s start by examine the Unit Circle 
, and the information in the videos below, to see what we can ascertain about SOHCAHTOA. You may want to open the Unit Circle in another tab or window to assist in your understanding.
- The sine function reduces the hypotenuse of a right triangle to the projection onto the y-axis. Similarly, the cosine function reduces the hypotenuse of a right triangle to the projection onto the x-axis. Thinking of sine and cosine as projection functions like this helps us to visualize exactly what the sin(x) and cos(x) functions do.
- The sides of the triangle can be considered lines (scalars) or vectors. (In mathematics, physics and engineering, a Euclidean vector, or simply a vector, is a geometric object that has magnitude (or length) and direction. Vectors can be added to other vectors according to vector algebra. A Euclidean vector is frequently represented by a ray (a directed line segment), or graphically as an arrow connecting an initial point A with a terminal point B, and denoted by
. )
View the Sine and Cosine – Definition & Meaning – Part 1 – What is Sin(x) & Cos(x) ? video below. - The tangent is also the slope of the hypotenuse
- Using the “A unit circle showing the relationship of the trigonometric functions” image and the Unit circle table, you can use some combination of the sin, cos, tan, cot, sec, excsc, exsec, versin, cvs, Pythagorean theorem, and the slope-intercept form to determine any parts of the right triangles in the image or their (x, y) coordinates.
Many of the References and Additional Reading websites and Videos will assist you with understanding SOHCAHTOA.
As some professors say: “It is intuitively obvious to even the most casual observer.”
References
1 “SOHCAHTOA – Definition & Example”. 2021. Study Queries. https://studyqueries.com/sohcahtoa/.
2 “Why Are Cosine, Sine, Tangent, And Cotangent So Important? Why Should I Be Graphing Them, Using Them In Calculus, Etc.?”. 2021. Quora. https://www.quora.com/Why-are-cosine-sine-tangent-and-cotangent-so-important-Why-should-I-be-graphing-them-using-them-in-calculus-etc.
3 Dynamind. “What Is The Pythagoras Theorem And How Does It Work”. 2022. Medium. https://dynamindsolutions.medium.com/what-is-the-pythagoras-theorem-and-how-does-it-work-450e0a3f553d.
The Pythagoras theorem is a fundamental part of mathematics that helps us understand the relationship between the length of certain sides of a right triangle. A crucial theorem that is used in a variety of applications. When applied to real-world examples, it can help us make better decisions and solve problems. By understanding how the theorem works, students can better understand mathematics and physics which will help them in their future education.
Additional Reading
Dynamind. “Tips And Tricks For Learning Trigonometry”. 2022. Medium. https://dynamindsolutions.medium.com/tips-and-tricks-for-learning-trigonometry-21bc4bc11812.
We recognize that math can be a difficult topic to grasp, but one should always do their hardest to grasp it. One of the most difficult subjects in mathematics is trigonometry. Understanding the principles of trigonometry is critical for success in any STEM discipline. You’ll come across this topic again during your academic experience, so it’s critical that you grasp the core concepts. Trigonometry is a subject that most students learn in high school, but there are a number of tips and tactics that might help you grasp the subject. It’s not difficult to learn trigonometry, and it’s well worth the effort. We’d like to share some ideas and tricks for learning trigonometry and its functions with you in this blog on education.
“Sine And Cosine Explained Visually”. 2021. Explained Visually. https://setosa.io/ev/sine-and-cosine/.
Sine and cosine — a.k.a., sin(θ) and cos(θ) — are functions revealing the shape of a right triangle. Looking out from a vertex with angle θ, sin(θ) is the ratio of the opposite side to the hypotenuse, while cos(θ) is the ratio of the adjacent side to the hypotenuse. No matter the size of the triangle, the values of sin(θ) and cos(θ) are the same for a given θ.
“SOHCAHTOA Explained (19 Step-By-Step Examples!)”. 2021. calcworkshop. https://calcworkshop.com/triangle-trig/sohcahtoa/.
“Sohcahtoa: Sine, Cosine, Tangent”. 2021. mathsisfun.com. https://www.mathsisfun.com/algebra/sohcahtoa.html.
Tailor, Pritesh. “A Visual Guide To Trigonometric Functions And Their Reciprocals”. 2022. Medium. https://medium.com/math-simplified/a-visual-guide-to-trigonometric-functions-and-their-reciprocals-92e94dea95ca.
Sine, Cosine and Tangent (AKA sin, cos and tan) are trigonometric functions we learn at school to solve angles and sides of a triangle using some infamous and quirky acronyms “SOHCAHTOA!”. Although it very much helps us interrogate a triangle and sketch out waves, we lose so much understanding of the origin of these functions and they simply remain as a BlackBox of understanding in our minds as to what does sin, cos and tan do when given a number? Furthermore, we lose any explanation as to where they come from? (other than being those exciting buttons on your calculator!)
There are several ways to explain what sin, cos and tan are but the unit circle method provides one to perfectly visually demonstrate where these fundamental and ubiquitous functions come from without having to get bogged down into the details of its math or a mention of pi! So let’s see how they come about all using a simple unit circle of the form x²+y² =1 in an animated and visual fashion. This can be then further extended to understand where the inverse trigonometry functions secant, cosecant and cotangent (AKA sec, cosec and cot) come from.
Videos
A trigonometry introduction, overview and review including trig functions, cartesian quadrants, angle measurement in degrees and radians, the Unit Circle and the Pythagorean Theorem.
In this lesson, we will learn fundamentally what the sine function and cosine function represent. We will learn that the sine function, also written as sin or sin(x), reduces the hypotenuse of a right triangle to the projection onto the y-axis. Similarly, the cosine function, also written as cos or cos(x), reduces the hypotenuse of a right triangle to the projection onto the x-axis. Thinking of sine and cosine as projection functions like this helps us to visualize exactly what the sin(x) and cos(x) functions do. This has applications in math, engineering, physics, and all branches of science.
Worked example evaluating sine and cosine using soh cah toa definition.
Mathematical Mysteries 