How to memorize the definitions of the trig functions

The following guide is all about how to memorize the definitions of the trig functions in the easiest way possible. That means, without memorizing any more than necessary. This requires just using three of the definitions, and then some common identities that you will need to remember when working problems in trigonometry anyway. So, let’s look at how this works!

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Step 1: Memorize the definitions of three Using SohCahToa

If you can say “SohCahToa” to yourself enough times (sound it out, just like it’s spelled), then you will have the definitions of Sine, Cosine, and Tangent down. Why? Let’s look at this goofy term and see what it is telling you:

Soh

Soh – sine, opposite over hypotenuse

\(\sin(\theta) = \dfrac{\textrm{opposite}}{\textrm{hypotenuse}}\)

Cah

Cah – cosine, adjacent over hypotenuse

\(\cos(\theta) = \dfrac{\textrm{adjacent}}{\textrm{hypotenuse}}\)

Toa

Toa – tangent, opposite over adjacent

\(\tan(\theta) = \dfrac{\textrm{opposite}}{\textrm{adjacent}}\)

Since these three trig functions come up the most often, this will be really useful. But what about the other functions? The best way to remember these is to work with some common identities.

Step 2: Use the Reciprocal Identities

It may seem like you are just memorizing more stuff here, but typically you will need to know these identities anyway, so why not conserve brain space and use them for recalling the rest of the definitions? These identities are:

\(\csc(\theta)=\dfrac{1}{\sin(\theta)}\)

\(\sec(\theta)=\dfrac{1}{\cos(\theta)}\)

\(\cot(\theta)=\dfrac{1}{\tan(\theta)}\)

What these identities tell you, is that you can just flip the definition of sine, cosine, and tangent to get the definition of cosecant, secant, and cotangent. So:

\(\sin(\theta) = \dfrac{\textrm{opposite}}{\textrm{hypotenuse}} \rightarrow \csc(\theta) = \dfrac{\textrm{hypotenuse}}{\textrm{opposite}}\)

\(\cos(\theta) = \dfrac{\textrm{adjacent}}{\textrm{hypotenuse}} \rightarrow \sec(\theta) = \dfrac{\textrm{hypotenuse}}{\textrm{adjacent}}\)

\(\tan(\theta) = \dfrac{\textrm{opposite}}{\textrm{adjacent}} \rightarrow \cot(\theta) = \dfrac{\textrm{adjacent}}{\textrm{opposite}}\)

And now you have the definitions of all 6 trig functions. How nice is that?

Summary

If you continue to study the Mathbootcamp trigonometry guides, you will see that the trig functions are always written in this order. This is due to using these identities to calculate values instead of rote memorization of each definition. In fact, being able to use these identities even helps later with how to graph the functions and being able to easily identify things like the domain of any one of the functions. In other words, they are very useful!

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Additional Reading

Once you have the definitions of these trig functions down, the next thing to review is how to work with them and angles in a right triangle. You can also check out how to memorize their values for the common angles encountered in trig classes.

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