Remembering Trig Identities: An Elegant Mnemonic

Hello! Today I’ll tell you of an elegant mnemonic to remember the main trigonometric identities. You may have heard of it, but it’s not widespread at all.

First, let’s make sure we’re on the same wavelength. There are 6 main trigonometric functions – sine, cosine, tangent, secant, cosecant, cotangent. They represent various ratios of various sides of a right triangle. If you don’t understand what they stand for, I can’t explain it now, it’s beyond the scope of this blog post; but you can look them up in Wikipedia right here.

It turns out that certain trigonometric ratios are related to other ratios in very interesting ways. Many students have a hard time remembering them (me too!) Only recently have I found out that there is a very elegant way to represent all the major trig identities concisely, and I’ll share this technique with you now.

It’s called “the hexagon” of trigonometric identities. Since there are 6 trig functions, it just makes sense to use a hexagon, right? 🙂

On the edges of the hexagon you can see the trig functions, in the center – you can see the number “1”. The key to this hexagon is that you don’t need to remember the trig identities, only the structure of the hexagon. It’s relatively easy to recreate the hexagon, or you may keep a draft of it and carry it with you at all times. So, that said, let’s begin exploring!

The product of any two opposing edges of the hexagon equals 1. So, we get:

sin(x) * csc(x) = 1
cos(x) * sec(x) = 1
tan(x) * cot(x) = 1

That shouldn’t be very hard to remember. Just multiply two opposing edges, and you’ll get 1 as a result. Now, for the next portion:

The two top vertices of any inner triangle (initially squared), when added, give the bottom vertex (squared).

sin2 x + cos2 x = 1
1 + cot2 x = csc2 x
tan2 x + 1 = sec2 x

These are what we call the “standard” identities. They might be a bit confusing, but they aren’t. Just take the top two vertices (or corners) of an inner triangle, square them, and add them. As a result, you’ll get the bottom vertex of the inner triangle, also squared. It takes some practice to get these results quickly, but you’ll get the hang of it pretty soon. 🙂

A trigonometric function on the hexagon equals the product of the two adjacent trigonometric functions.

tan(x) = sin(x) * sec(x)
sin(x) = cos(x) * tan(x)
cos(x) = sin(x) * cot(x)
cot(x) = cos(x) * csc(x)
csc(x) = cot(x) * sec(x)
sec(x) = csc(x) * tan(x)

This one should have been a bit easier than the last one. You just take any trig function. It equals the product of the two adjacent trig functions (on both sides of the initial function).

This whole hexagon thing may seem difficult at first, but actually, we derived 12 (!) trig identities from it! I’m sure it will help you a lot, especially if you’re in high school. Trigonometry is really an important topic, and these trigonometric identities play a crucial role in your success in this subject.

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