# Tattoos on Math ## Метаданные - **Канал:** 3Blue1Brown - **YouTube:** https://www.youtube.com/watch?v=IxNb1WG_Ido - **Дата:** 06.01.2017 - **Длительность:** 6:57 - **Просмотры:** 761,872 ## Описание After a friend of mine got a tattoo with a representation of the cosecant function, it got me thinking about how there's another sense in which this function is a tattoo on math. Thanks to these viewers for their contributions to translations Spanish: Yago Iglesias ------------------ 3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with YouTube, if you want to stay posted on new videos, subscribe, and click the bell to receive notifications (if you're into that). If you are new to this channel and want to see more, a good place to start is this playlist: http://3b1b.co/recommended Various social media stuffs: Website: https://www.3blue1brown.com Twitter: https://twitter.com/3Blue1Brown Patreon: https://patreon.com/3blue1brown Facebook: https://www.facebook.com/3blue1brown Reddit: https://www.reddit.com/r/3Blue1Brown ## Содержание ### [0:00](https://www.youtube.com/watch?v=IxNb1WG_Ido) Segment 1 (00:00 - 05:00) Hey folks, just a short, 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 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. It 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 show you a picture of the tattoo he chose, because not a lot of people know the geometric representation of the cosecant. Whenever you have an angle, typically represented with the Greek letter theta, it's common in trigonometry to relate it to a corresponding point on the unit circle, the circle with the radius 1 centered at the origin in the xy plane. Most trigonometry students learn 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 links 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 relatively few learn 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 the tangent of the angle. And the distance along that line to the point where it hits the y-axis is the cotangent of the angle. Again, this gives a really intuitive feel for what those quantities mean. You can imagine tweaking that theta and seeing when cotangent gets smaller and when tangent gets larger, and it's a good gut check for any students working with them. Likewise, secant, which is defined as 1 divided by the cosine, and cosecant, sine of theta, 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 1 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 1 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 the interior angle up at the tip there is theta, the angle that we originally started with over inside the circle. For each one of those triangles, I want you to think about the ratio of the length of the side opposite theta to hypotenuse. For the small triangle, the length of the opposite side is sine of theta, and the hypotenuse is that radius, the one we defined to have length 1, so the ratio is just sine of theta divided by 1. Now, when we look at the larger triangle, the side opposite theta is that radial line of length 1, and the hypotenuse is now this length on the y-axis, the one 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 theta is 1 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. ### [5:00](https://www.youtube.com/watch?v=IxNb1WG_Ido&t=300s) Segment 2 (05:00 - 06:00) 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 introduce secant as 1 over cosine, and cosecant as 1 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 because you're 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 1 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 1 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 gotta 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. --- *Источник: https://ekstraktznaniy.ru/video/16309*