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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 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 actually show you all a picture of the tattoo that 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 related 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 theta
and seeing when cotangent get smaller, 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, which is defined as 1 divided by the 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 interior angle up at the tip there is theta,
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 theta to the length of the hypotenuse.
For the small triangle, the length of the opposite side is sine of theta,
and the hypotenuse is that radius, the one that we define 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 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 theta is 1 divided by sine.
Kinda cool, right?
It's also kind of nice that
sine, tangent and secant all correspond length of lines
that somehow go to the x-axis,
and then the corresponding cosine, cotangent and cosecant are all the 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 is 1 over cosine
and cosecant is 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 are 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 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 onto the subject,
and could just as easily have been inked on in some completely other way.
Before you go, I've got a book recommendation for you,
meant primarily for those of you who don't already listen to audio books.
You see, this video was supported by audible.com,
which, as many of you know, provides audio books and other audio materials.
And if you go to audibletrial.com/3blue1brown, you'll get a one-month free trial,
meaning you can listen to pretty much anything you want for free!
Today, I want to recommend Zen and the Art of Motorcycle Maintenance.
This is actually what I listen to right after the Art of Learning,
which is what I recommend it to you guys at the end of the last video on the zeta function,
since the author of the Art of Learning was actually a big fan of Zen and the Art of Motorcycle Maintenance.
Actually, I ended up listening to it twice,
since it's just that kind of book.
It's very unusual,
in a way that feels neither like fiction nor nonfiction.
It's really philosophical and very thought-provoking,
but at the same time, it's a powerful lesson in empathy and storytelling.
And what's more, the version that you can hear on audible.com is beautifully read.
Again, if you go to audibletrial.com/3blue1brown,
you can listen to it for free, or any other book you might want.
So I hope you enjoy it, and I'll see you all next video.