Welcome to Trig!

Welcome from She Loves Math!   I know it’s been awhile, but I wanted to welcome everyone back to school and talk a little about Trigonometry, which is usually part of a Pre-Calc class these days.

I’ve completed the Right Triangle Trigonometry, Graphs of Trig Functions, Transformations of Trig Functions sections, and the beginning of the Inverses of the Trigonometric Functions section.

Here is an excerpt from my Right Triangle Trigonometry section to get you started with Trig:

You may have been introduced to Trigonometry in Geometry, when you had to find either a side length or angle measurement of a triangle.  Trigonometry is basically the study of triangles, and was first used to help in the computations of astronomy.  Today it is used in engineering, architecture, medicine, physics, among other disciplines.

The 6 basic trigonometric functions that you’ll be working with are sine, cosine, tangent, cosecant, secant, andcotangent.  (Don’t let the fancy names scare you; they really aren’t that bad).


With Right Triangle Trigonometry, we use the trig functions on angles, and get a number back that we can use to get a side measurement, as an example.    Sometimes we have to work backwards to get the angle measurement back so we have to use what a call an inverse trig function But basically remember that we need the trig functions so we can figure out triangles’ sides and angles that we don’t  otherwise know.  

Later, we’ll see how to use trig to find areas of triangles, too, among other things.

You may  have been taught SOH – CAH – TOA  (SOHCAHTOA)  to remember these.  Back in the old days when I was in high school, we didn’t have SOHCAHTOA, nor did we have fancy calculators to get the values; we  had to look up trigonometric values in tables.

Remember that the definitions below assume that the triangles are right triangles, meaning that they all have one right angle.  Also note that in the following examples, our angle measurements are in degrees; later we’ll learn about another angle measurement unit, radians, which we’ll discuss here in the Angles and Unit Circle section.

Basic Trigonometric Functions (SOH – CAH – TOA)

Here are the 6 trigonometric functions, shown with both the SOHCAHTOA and Coordinate System Methods.   Note that the second set of three trig functions are just the reciprocals of the first three; this makes it a little easier!


Note that the cosecant (csc), secant (sec), and cotangent (cot) functions are called reciprocal functions, or reciprocal trig functions, since they are the reciprocals of sin, cos, and tan, respectively.

For the coordinate system method, assume that the vertex of the angle in the triangle is at the origin (0, 0):

Right Triangle


Coordinate System Method

\(\displaystyle \begin{align}\text{SOH: Sine}\left( A \right)=\sin \left( A \right)=\frac{{\text{Opposite}}}{{\text{Hypotenuse}}}\\\text{CAH: Cosine}\left( A \right)=\cos \left( A \right)=\frac{{\text{Adjacent}}}{{\text{Hypotenuse}}}\\\text{TOA: Tangent}\left( A \right)=\tan \left( A \right)=\frac{{\text{Opposite}}}{{\text{Adjacent}}}\end{align}\)


\(\displaystyle \begin{align}\text{cosecant}\left( A \right)=\csc \left( A \right)=\frac{1}{{\sin \left( A \right)}}=\frac{{\text{ Hypotenuse}}}{{\text{Opposite}}}\\\text{secant}\left( A \right)=\sec \left( A \right)=\frac{1}{{\cos \left( A \right)}}=\frac{{\text{ Hypotenuse}}}{{\text{Adjacent}}}\\\text{cotangent}\left( A \right)=\cot \left( A \right)=\frac{1}{{\tan \left( A \right)}}=\frac{{\text{ Adjacent}}}{{\text{Opposite}}}\end{align}\)

 \(\displaystyle \begin{align}\sin \left( A \right)=\frac{y}{h}\\\cos \left( A \right)=\frac{x}{h}\\\tan \left( A \right)=\frac{y}{x}\end{align}\)


\(\displaystyle \begin{align}\csc \left( A \right)=\frac{1}{{\sin \left( A \right)}}=\frac{h}{y}\\\sec \left( A \right)=\frac{1}{{\cos \left( A \right)}}=\frac{h}{x}\\\cot \left( A \right)=\frac{1}{{\tan \left( A \right)}}=\frac{x}{y}\end{align}\)

Keep checking back, as I’m trying to write every day, and please let me know if you have any questions or comments.    Keep working hard!   Lisa 🙂

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