## Sohcahtoa!

Mathematics can ask you to remember things that have no obvious connection to common sense. Either because it’s arbitrary (the name of a function in respect to what it computes) or because you haven’t quite figured all the details out. In either cases, a few mnemonics are always useful!

The first series of mnemonics you’ll learn is to remember constants. Constants like $\pi$ and $e$ have special importance in modern mathematics—beyond plain creepy piphilia.

The notation for $\pi$ is due to William Jones who first used it in print in 1707. He chose $\pi$ probably because it had some mnemonic connection to perimeter, which make sense given the context. The notation was adopted and popularized by Euler, which in turn introduced the notation $e$ for the base of the natural logarithm. It appeared in print in 1736.[1]

The constants $\pi$ and $e$ are both transcendental irrational numbers.

The discipline of remembering digits of either—and of $\pi$ in particular—has risen to Olympic levels. For every day use knowing the values to 6 or 16 digits is probably quite sufficient1. To remember $\pi$ to 20 digits, the following English poem does the trick:

3      Pie
141592 I wish I could determine Pi
65358  Eureka, cried the great inventor
9793   Christmas pudding, Christmas pie
23846  Is the problem’s very center

And, indeed, $\pi \approx 3.14159265358979323846264338327950288\ldots$. For $e$, we have the same kind of poem:

2718281 We require a mnemonic to remember $e$

And, indeed, quite so, $e \approx 2.71828182845904523536\ldots$. If you really want a whole lot of digits, you can have a look at Simon Plouffe‘s page of world records.

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Trigonometric functions such as sine, cosine, and tangent are somewhat named arbitrarily. Maybe not tangent, because, more than the two others, it suggest something of a straight line, a slope. Fortunately, there’s a rather simple—if Japanese-sounding—mnemonic for those. Take a deep breath and say it out loud: SOHCAHTOA!

Indeed, sine is opposite over hypotenuse, cosine is adjacent over hypotenuse, and tangent is opposite over adjacent:

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Keeping up with a Japanese theme, a friend of mine, understanding nothing of the right-hand rule, came up with a simple mnemonic for remembering the vector cross product rules:

$\mathbf{ij} = \mathbf{k}$

$\mathbf{jk} = \mathbf{i}$

$\mathbf{ki} = \mathbf{j}$

$\mathbf{ji} = -\mathbf{k}$

$\mathbf{kj} = -\mathbf{i}$

$\mathbf{ik} = -\mathbf{j}$

There’s also:

$\mathbf{ii} = \mathbf{jj} = \mathbf{kk} = \mathbf{0}$

Let us look at a mitsudomoe crest:

You can imagine it spinning clockwise:

or counter-clockwise:

If the mitsudomoe is spinning clockwise, pen down $\mathbf{i}$, $\mathbf{j}$, and $\mathbf{k}$ in a clockwise manner:

If you read the products clockwise, the answer is the next vector clockwise. Indeed, $\mathbf{ij}=\mathbf{k}$.

The same works if you use it counter-clockwise: the answer is the next vector in counter-clockwise order: $\mathbf{ji} = -\mathbf{k}$.

You can recompute the identities using the definition of cross-product based on the matrix determinant:

$\mathbf{a} \times \mathbf{b} = \det \left[~\begin{array}{ccc}\mathbf{i} & \mathbf{j} & \mathbf{k}\\a_1 & a_2 &a_3\\b_1 &b_2 & b_3\end{array}~\right]$

But that’s a lot of work.

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Mnemonics are always useful, if you manage to remember them. I mean it seriously. I think it’s easier to remember $\pi$ to 6 digits than to remember a 20-word poem. The clockwise/counter-clockwise analogy is a lot easier to remember if you’re a visual like I am.

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The Hidari Mitsudomoe images are based on a work released under the creative commons. The derived works complies with the license. The hiragana ‘sohcahtoa’ calligraphy is mine, I retain full rights on this image.

References

[1] Florian Cajori — A History of Mathematical Notations — Dover, 1993 (also available in two volumes: V.1 and V.2).

1 6–7 decimal digits corresponds to the safe precision of IEEE 754 float, 16 corresponds to double.

### 5 Responses to Sohcahtoa!

1. wheels says:

And, if you can remember e to 6 places, you’ve got it to 10 places.

2. Steven Pigeon says:

True!

3. […] Link via Harder, Better, Faster, Stronger. […]

4. MsD says:

This neat SOHCAHTOA video is great for inspiring your trig students.

• Steven Pigeon says:

ah ah that’s cute :p