Video: Tattoos on Math

Grant Sanderson • 3Blue1Brown • Boclips

Tattoos on Math

06:56

Video Transcript

Hey folks! just a short kind of out-of-the-ordinary video for you today. A friend of mine, Cam, recently got a math tattoo. It’s not something I’d recommend. But he told his team at work that if they reached a certain stretch goal, it’s something that he’d do. And, well, the incentive worked.

Cam’s initials are CSC, which happens to be the shorthand for the cosecant function in trigonometry. So what he decided to do is make his tattoo a certain geometric representation of what that function means. It’s kind of like a wordless signature written in pure math. He got me thinking though about why on earth we teach students about the trigonometric functions: cosecant, secant, and cotangent. And it occurred to me that there’s something kind of poetic about this particular tattoo. Just as tattoos are artificially painted on but become permanent as if they were a core part of the recipient’s flesh, the fact that the cosecant is a named function is kind of an artificial construct on math.

Trigonometry could just as well have existed intact without the cosecant ever being named. But because it was, it has this strange and artificial permanence in our conventions and, to some extent, in our education system. In other words, the cosecant is not just a tattoo on Cam’s chest. It’s a tattoo on math itself, something which seemed reasonable and even worthy of immortality at its inception, but which doesn’t necessarily hold up as time goes on.

Here, let me actually show you all a picture of the tattoo that he chose cause not a lot of people know the geometric representation of the cosecant. Whenever you have an angle, typically represented with the Greek letter 𝜃, it’s common in trigonometry to relate it to a corresponding point on the unit circle, the circle with the radius one centered at the origin in the 𝑥𝑦-plane. Most trigonometry students learned that the distance between this point here on the circle and the 𝑥-axis is the sine of the angle. And the distance between that point and the 𝑦-axis is the cosine of the angle. And these lengths give a really wonderful understanding for what cosine and sine are all about.

People might learn that the tangent of an angle is sine divided by cosine and that the cotangent is the other way around, cosine divided by sine. But relatively few learned that there’s also a nice geometric interpretation for each of those quantities. If you draw a line tangent to the circle at this point, the distance from that point to the 𝑥-axis along that tangent is, well, the tangent of the angle. And the distance along that line to the point where it hits the 𝑦-axis, well that’s the cotangent of the angle. Again, this gives a really intuitive feel for what those quantities mean. You kind of imagine tweaking that 𝜃 and seeing when cotangent gets smaller, when tangent gets larger. And it’s a good gut check for any students working with them.

Likewise, secant, which is defined as one divided by the cosine, and cosecant, which is defined as one divided by the sine of 𝜃, each have their own places on this diagram. If you look at that point where this tangent line crosses the 𝑥-axis, the distance from that point to the origin is the secant of the angle; that is, one divided by the cosine. Likewise, the distance between where this tangent line crosses the 𝑦-axis and the origin is the cosecant of the angle; that is, one divided by the sine. If you’re wondering why on earth that’s true, notice that we have two similar right triangles here, one small one inside the circle and this larger triangle whose hypotenuse is resting on the 𝑦-axis. I’ll leave it to you to check that that interior angle up at the tip there is 𝜃, the angle that we originally started with over inside the circle.

Now, for each one of those triangles, I want you to think about the ratio of the length of the side opposite 𝜃 to the length of the hypotenuse. For the small triangle, the length of the opposite side is sine of 𝜃 and the hypotenuse is that radius, the one that we defined to have length one. So the ratio is just sine of 𝜃 divided by one. Now, when we look at the larger triangle, the side opposite 𝜃 is that radial line of length one. And the hypotenuse is now this length on the 𝑦-axis, the one that I’m claiming is the cosecant. If you take the reciprocal of each side here, you see that this matches up with the fact that the cosecant of 𝜃 is one divided by sine. Kinda cool, right?

It’s also kind of nice that sine, tangent, and secant all correspond to lengths of lines that somehow go to the 𝑥-axis. And then the corresponding cosine, cotangent, and cosecant are all then lengths of lines going to the corresponding spots on the 𝑦-axis. And on a diagram like this, it might be pleasing that all six of these are separately named functions. But in any practical use of trigonometry, you can get by, just using sine, cosine, and tangent. In fact, if you really wanted, you could define all six of these in terms of sine alone. But the sort of things that cosine and tangent correspond to come up frequently enough that it’s more convenient to give them their own names. But cosecant, secant, and cotangent never really come up in problem solving in a way that’s not just as convenient to write in terms of sine, cosine, and tangent.

At that point, it’s really just adding more words for students to learn with not that much added utility. And if anything, if you only introduced secant as one over cosine and cosecant as one over sine, the mismatch of this co prefix is probably just an added point of confusion in a class that’s prone enough to confusion for many of its students. The reason that all six of these functions have separate names, by the way, is that before computers and calculators, if you were doing trigonometry, maybe cause you are a sailor or an astronomer or some kind of engineer, you’d find the values for these functions using large charts that just recorded known input-output pairs. And when you can’t easily plug in something like one divided by the sine of 30 degrees into a calculator, it might actually make sense to have a dedicated column to this value with a dedicated name.

And if you have a diagram like this one in mind when you’re taking measurements with sine, tangent, and secant having nicely mirrored meanings to cosine, cotangent, and cosecant, calling this cosecant, instead of one divided by sine, might actually make some sense. And it might actually make it easier to remember what it means geometrically. But times have changed and most use cases for trig just don’t involve charts of values and diagrams like this. Hence, the cosecant and its brothers are tattoos on math, ideas whose permanence in our conventions is our own doing, not the result of nature itself.

And in general, I actually think this is a good lesson for any student learning a new piece of math, at whatever level. You just got to take a moment and ask yourself whether what you’re learning is core to the flesh of math itself and to nature itself or if what you’re looking at is actually just inked on to the subject and could just as easily have been inked on in some completely other way.

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