The magic (unit) circle

This is the story of an index card, a trigonometry book, and a magic circle.

Once upon a time* I was helping my friend, John, prepare for a test in his trigonometry class. His teacher told him that he could take one 3×5 index card into the test. He sharpened a pencil to a fine point and started writing on the card. There are three tables in his textbook that he copied onto his card. One shows the values of trig functions for “special angles” (30°, 45°, and 60°). The book recommends memorizing that table. Another table showed values of trig functions for quadrantal angles (0°, 90°, 180°, and 270°). The third showed signs of trig functions in different quadrants: cosine and sine both positive in Quadrant I, sine positive and cosine negative in Quadrant II, etc.

John copied those tables onto his index card along with some definitions (sin = opp/hyp, etc.) Then he asked me for suggestions for other things to write on the card. I looked at the three tables and suggested that he replace the three tables with a drawing of the unit circle.

The trig book John had does talk about the unit circle a little bit, and John knew what I was talking about when I referred to the unit circle, but I had to look at the book pretty closely to see any mention of it. As far as I could tell, the place where it’s explained is in the section about wrapping functions. Wrapping functions? Never heard of them. Why bury the unit circle in a section about some kind of mathematical formalism that no one ever uses unless they have to learn it for a test? To me that’s heresy.

When I learned trig, back in the days when dinosaurs roamed the earth (~1975), the unit circle was the key to learning trigonometry. At least it was in Uncle Bill’s class. I remember sitting in a classroom at Skyline High in Salt Lake City, Utah and watching the teacher, Bill Earl (“Uncle Bill”), draw a unit circle on the board at the beginning of class. He would stand by the board and swing his arm around to draw the circle. He was just the right height to get a circle that fit nicely on the board if he had his arm fully extended when he swung it around.

I used the unit circle a lot in Uncle Bill’s class and before long had it memorized, kind of like the authors of John’s math book suggested that students memorize Table 2.2, Trigonometric Functions of Special Angles. But it wasn’t just that I had it memorized. It was that I understood how it worked so that I could draw it any time I wanted to. You could say that I could re-derive it any time I wanted to. Even forty-plus years later, which is what I did when I drew it for John.

Here’s my picture, scribbles and all:
drawing of the unit circle

As you can tell by the scribbles, I didn’t get it right the first time. And I cheated a little bit. But given a little more time I could have reproduced it without using a book or a calculator.

That picture, along with some of the basic definitions (tan = sin/cos) includes all of the information in Tables 2.2 (Trigonometric Functions of Special Angles), 2.4 (Values of Trigonometric Functions of Quadrantal Angles), and 2.5 (Signs of the Trigonometric Functions) in John’s textbook. So John could replace three tables on his index card with one diagram of the unit circle.

Memorizing the circle is easier than memorizing the tables in the book. If you’re familiar with the four quadrants, it’s easy to see why the sign of π/2 is positive and the sine of 3π/2 is negative. Instead of memorizing a table (2.4 or 2.5) filled with arbitray numbers or words you’re looking at quadrants on a graph. It’s true that the textbook’s explanation gives some context for Table 2.5, and there are some diagrams, so they’re not completely arbitrary numbers and words, but why not put everything together in one diagram, and then memorize, or even better learn, that one diagram?

Yes, there might be a few things that you need to memorize, like the values for sine and cosine in Quadrant I. But really the main thing you need to memorize is the sequence 1/2, √2/2, and √3/2. They all have 2 in the denominator, and you can think of the 1 in 1/2 as √1 so you get a nice sequence of √1/2, √2/2, and √3/2. If you understand the overall idea, it’s pretty easy to take care of a few memorization details like that.

If you learn the unit circle and understand it well you won’t need to draw it on an index card to take it into the test. You’ll understand the relationships between different parts of the circle and see how they all fit together, and that makes a big difference.

So why does the title of this post refer to a “magic circle”? Am I trying to convince you that the unit circle magic? No, I’m not. The unit circle is definitely useful, but it’s not magic.

I’m referring to the magic circle of games and game design. The idea is that when players play a game they agree to play by the rules. They pretend that the things in the game are important, even if (or especially if) those things are not at all important in the real world. They get into the game. If they do those things they are entering the magic circle.

If they don’t enter the magic circle, players will probably not enjoy the game. Even worse, they might prevent others from enjoying the game by arguing about the rules or not paying attention and taking their turn when it’s time, or saying out-of-character things.

I have another friend who was a year or two behind me in high school. At first he didn’t like math, he didn’t get good grades in his math classes, and he more or less considered it a waste of time. In game design terms he wasn’t in the magic circle and wasn’t even close to it. Then, for some reason, he decided that math was interesting, or even fun. I don’t know why. Maybe it was something a teacher said or did. Maybe it was an interesting application or just an interesting problem. Maybe he solved a particularly tricky problem and got a confidence boost. But something happened and he entered the magic circle.

Math wasn’t easy for him just because he entered the magic circle. But it was easier. Maybe a better way to put it is that it became easier for him to put in the time and effort needed to learn it. Because he put more time and effort into it, he made better progress, which in turn helped with motivation so that he could continue putting more time and effort into it. Instead of being stuck in a negative feedback loop, he was in a positive feedback loop. Or, as I like to put it, he was “spiraling up” inside the magic circle.

And that’s a good place to be.

Extra Credits video about the magic circle

Compensated links to
• The trig book used in John’s class: College Trigonometry
• The game design book where I first read about the magic circle:
Andrew Rollings and Ernest Adams on Game Design

* I’ve taken the liberty of changing some aspects of this story.