---
title: 'Trigonometry Fundamentals | Ep. 2 Lockdown Live Math'
source: 'https://youtube.com/watch?v=yBw67Fb31Cs'
video_id: 'yBw67Fb31Cs'
date: 2026-08-10
duration_sec: 4397
---

# Trigonometry Fundamentals | Ep. 2 Lockdown Live Math

> Source: [Trigonometry Fundamentals | Ep. 2 Lockdown Live Math](https://youtube.com/watch?v=yBw67Fb31Cs)

## Summary

In this second episode of the Lockdown live math series, Grant Sanderson (3Blue1Brown) explores the fundamentals of trigonometry, emphasizing the connection between trigonometric functions and complex numbers. Through interactive quizzes and playful graphing calculator experiments, he demonstrates how to derive non-trivial identities, understand sine and cosine through unit circles, and compute exact values using geometric symmetry.

### Key Points

- **Discovering Trig Identities with a Graphing Calculator** [00:13] — By squaring the cosine function and observing the graph, one can discover that cos²(x) = (1 + cos(2x))/2, a non-trivial identity that emerges from playful exploration.
- **The Power of Squaring in Trigonometry** [05:26] — Squaring the cosine function results in a scaled-down version of the cosine graph, highlighting the relationship between squaring and frequency doubling.
- **Exponential Functions and Their Properties** [10:57] — The function 2^x satisfies f(2x) = f(x)^2, illustrating that exponents have a property where squaring the output is equivalent to doubling the input, hinting at a deeper connection to trigonometry.
- **Sine and Cosine via the Unit Circle** [13:05] — Sine and cosine can be understood as the y- and x-coordinates of a point moving around a unit circle, with the input representing the distance walked along the circumference.
- **SOH CAH TOA and Right Triangles** [20:20] — The classic triangle-based definitions (SOH CAH TOA) are introduced, showing how sine, cosine, and tangent relate to the sides of a right triangle.
- **Applying Trigonometry to a Leaning Tower** [22:12] — A practical example: a 100-meter tower leaning at 80 degrees casts a shadow of length 100 * cos(80°), demonstrating the use of cosine in real-world scenarios.
- **Degrees vs. Radians** [26:19] — Degrees are human-friendly but unnatural; radians, based on the distance around a unit circle, are the preferred unit in mathematics.
- **Computing Exact Values of Sine and Cosine** [30:42] — Using symmetry of equilateral triangles, we can derive exact values like sin(π/6) = 1/2 and cos(π/6) = √3/2.
- **Pythagorean Identity** [39:07] — The Pythagorean theorem leads to the identity cos²(x) + sin²(x) = 1, which is fundamental in trigonometry.
- **Handling Angles Beyond 90 Degrees** [42:06] — Using the unit circle, we can compute values for angles like -2π/3, where cosine is -1/2, showing that trig functions are not limited to triangles.
- **Using Identities to Compute Complex Values** [47:16] — The identity cos²(x) = (1 + cos(2x))/2 allows us to compute cos(π/12) as √(1 + √3/2)/2, a value that is otherwise difficult to find.
- **Geometric Interpretation of Tangent** [56:23] — Tangent can be visualized as the length of a line segment tangent to the unit circle, explaining its graph and behavior near 90 degrees.
- **Visualizing Cosine Squared** [01:04:04] — By projecting shadows in a unit circle diagram, we can see that cos²(θ) and sin²(θ) appear as segments, leading to a visual proof of the Pythagorean theorem.

### Conclusion

The lecture emphasizes that trigonometry is not just about triangles but is deeply connected to circles and exponentials, with identities like cos²(x) = (1 + cos(2x))/2 being non-trivial yet discoverable through playful exploration. The next session will explore the connection to complex numbers.

## Transcript

for you a connection between trigonometry and complex numbers. it strengthens your understanding of both of them.
of the trigonometric functions, how to think about them, what they mean. And to dive into it, what I want to first do is imagine that you don't actually know any trigonometry yet, but show how just playing around with a graphing
calculator can lead you to come to some pretty non-trivial facts about trig. So for example, let's say you've never heard of the cosine function. and they mention a sinusoidal wave and the cosine function and things like that.
and you just poke it in and say, okay, what does this thing look like? I can see how this might be relevant to waves and wave type dynamics that people
It's got kind of a cycling phenomenon to it, so maybe anytime people you could play around with the graph and do all the things that you can If I tweak the input, it kind of squishes it, right?
it squishes things, gives it a higher frequency. You could stretch it out in the Y direction by messing with the output. Now what I want to do though is point out something very interesting
about the cosine graph that is something you might want to take note of. So what I'm going to do is square it, and I don't want to show you what it looks like yet. I want you to predict what it's going to look like when we take cosine of X squared, okay?
And in particular, I'm just going to have you focus in on the region from 0 up to 2 pi. So again, if you're just playing around, what you would notice is So it takes a dip down to pi, down to negative 1 at pi,
excuse me, and then it comes right back up to 1 at the input 2 pi. So what I want you to do is predict what it looks like when we take cosine of X squared. And let's actually do this as one of our fun live question dynamics, okay?
So getting rid of our Randy, which was a very informative thing for me, let's dump into our very first genuinely mathematical quiz question. What will the graph of f of X equals cosine of X squared look like?
Okay, and it looks like A as an option has these sort of valleys where in all of these it might be a little small to read, but the range is going from 0 to about 2 pi. the input pi and kind of goes down with these flat valleys.
B has a similar behavior for when it hits 1 and 0, but it's a little spikier. So it's kind of bouncing right off that x-axis. Okay, it looks like the same kind of wave, but it's all positive and it's,
it goes from 1 down to 0, then up to 1 in the span of pi. And then D kind of gets scrunched together in the x-direction and starts going more rapidly as you move farther to the right.
A fair bit of consensus towards the top, but it's not, definitely not a unanimous decision on this one where people are going. So it seems like there's one most popular answer and then two
that kind of tie closely for the silver and the bronze medals. So I'm going to give you just a couple more seconds here if you want to change your mind or if you want to go to 3b1b.co slash live and answer this as we're going.
The link is also in the description if you just want to clink it there. And, alright, I think that's probably enough time. What do you think it will be?
which is the one that looks like a sinusoidal wave. but it's kind of more flattened out on the bottom.
Third most common was B, and then the last most common was D. which were the top three choices there, if we go back and look at our actual cosine graph itself, what you'll notice is that at the input 0 it's 1, so when you square that,
Similarly, negative 1 when you square that, that should come up to 1 as well. And then positive 1 at the end of its cycle, again that's 1. then up to 1, down to 0, and up to 1 should make sense.
Turns out, as you just saw, the answer is C. So if we go over here and reveal, it looks just like a little cosine wave. Which is weird, if you think about it, because what we're doing when we square something
is affecting the output, it's affecting stuff happening in the y-direction, right? it's not at all clear that that's what would happen. if you're just playing around being kind of childlike with your graphing calculator,
you can discover a pretty non-trivial fact about trigonometric functions. you will appreciate just how non-trivial it is, because it really is not obvious. So for this, I want to take us to a second question.
that it looks like a scaled-down version of the cosine graph. And what I'm going to want you to do is basically come up with an identity based on that,
So question number two, what does it say? Working from the assumption that the graph of cosine of x squared looks like a scaled-down version of the graph of cosine of x, which of the following equations is true?
pencil and paper, highly encourage you to actively work it out. I'm just going to give you maybe 30 to 60 seconds to start penciling through, noodling it out, and let's introduce a little bit of pause and ponder
music just to put us in the problem-solving mood, I think. So which one of these reflects the fact the cosine of x squared is kind of a scaled version of itself?
and I don't want you to feel rushed, so I'm going to give you time to figure it out for yourself if you want to. got ourselves a question that's rolled in, okay?
So Diego Mathemagician, love the name, asks, what is the new music? So the guy who usually composes the music for the channel, I told him that we were doing these streams, and I mentioned I wanted kind of a jeopardy.
Vince, your music is great, but I want to talk about it a little bit more. I wanted just a jeopardy type theme, but mixed with the usual three blue and brown vibe, so a little clock ticking, and I think it really puts you in the problem-solving mood.
so I'm going to go ahead and grade this. And I'm guessing the majority have gotten it correct. So to the 3,165 of you that answered B, which is that it's one half cosine of 2x plus 1,
But let's think about how you might do that in a step-by-step way, noodling around on a graphing calculator, you can come to.
So what I might do is take, here, I'll just go ahead and write a new cosine function, So cosine of x, we won't look at the original. The first thing you'd notice is, okay, we want to make it entirely positive
because that's what the other one is, so maybe I'm going to shift it up by one, okay? But now it's twice as high as the other graph we're looking at, But it still cycles once every 2 pi, right?
You have to go from 0 all the way up to 2 pi before this blue graph that we're looking at goes through the cycle from 1 down to 0 now, back up to 1. So the last step is going to be to squish it in the x direction.
And sometimes people find this very counterintuitive that you scale up that input, But what's really going on, if you think it through, is we want to say instead of taking 2 pi to go through your full cycle,
I want you to go through that cycle as x ranges from 0 to pi. into cosine will go from 0 to 2 pi as x goes from 0 up to pi. And what you'll look at is the graphs are right on top of each other, right?
and that means you've discovered a pretty non-trivial fact about trigonometry. things that comes up later in a very hard to memorize way.
So, writing down what you just found, cosine of x squared, okay, is equal to 1 plus cosine of 2x, which is basically saying increase its frequency.
So it's going to shift above that y-axis, and then we're going to divide that by 2, okay? So this is a fact that we're going to turn back to at the very end of the lecture,
and as you start to get an appreciation for what the cosine means, And in particular, the one thing I want you to take note of is the weird fact that we've got this function that when you square it, it's not equal to,
but it's somewhat related to the idea of scaling up that input. Now if you were clever, there is another function that behaves very similarly to this, where when you square it, it's the same as doing something to its input.
So let's go ahead and move past from this one, okay, and I'm going to ask you something that seems very unrelated to trigonometry, and it'll take us until the next lecture to realize just how related it is.
Which of the following functions satisfies f of 2x is equal to f of x squared? So when you plug in 2x, it should be the same as f of x squared.
Option A, square root of 1 minus x squared, option B, log of x, option C, x squared, and option D, 2 to the x.
Okay, so I'm going to give you a couple minutes, a little bit to answer that. Oh, it looks like I accidentally graded it right away. Okay, well, sorry for everyone who felt a little bit rushed there, my finger slipped.
Well, in either case, it's now been revealed that the answer is D, 2 to the x. play around a little bit in our Desmos graphs. So what we might do is pop over here and let me plug in 2 to the x.
So let's not look at our cosines anymore, 2 to the x. And indeed, if I take this whole thing and I square it.
On the one hand, squaring usually does a kind of stretching motion in the y direction. But what's neat about this shape is that it's exactly the same as taking 2 to the 2x. sense because that 2 up here can get distributed into that x.
But what this means is that exponents have this funny property that when you square them, And of course, the reason I bring that up is that you're Just before, just by playing around with graphs, before you even know what they mean,
you can have this tiny little instinct in your head that says, maybe, just maybe, cosine is somehow related to exponents. And that's one of the things that I'd like to
We won't get to the end of there today. actually talk about what sine and cosine actually are.
so let's have a little moment to go back to the basics. I think one of the best summaries of trigonometry that I ever heard, or remember who originally said this, is you think that it's about triangles,
And this is absolutely true. I think between the time when I very first learned about trigonometry and now, I think my mind shifted from fundamentally thinking of a triangular-based definition,
And let me just show you what I mean with respect to the circles, because that's the one that explains how it comes up in physics, So instead of our Desmos graph, I'm going to go
So in this one, I'll explain what sine of theta is, where theta is the input, and it's going to be a measure of how far you've walked around a circle.
rightmost side of a circle that has a radius of one, okay? And then as you walk around at a constant rate, as a function of how far you've walked, we're going to graph your height, which is to say your y-axis on this grid.
Maybe you'd think of it as the distance between you and the x-axis. And it kind of makes sense how it's just going to oscillate on and on, Which is why, when we go back over to the Desmos graph,
and you think of something like cosine of x, not the squared version, it goes on forever continuing to cycle like this. And I guess here I'm explaining sine of x, but it works the same way.
If instead I type sine of x, we get a similar looking shape. The only difference is that it starts at zero. And we can take a look to explain why here, because if by definition,
if it's giving your y-coordinate when you start off to the right, Okay? but it's giving you the x-coordinate as you walk around that circle.
And in that context, it starts off at one, and then as you walk around the circle, your distance from that y-axis, your x-coordinate, is going to get lower.
It ultimately gets down to negative one before it starts increasing again. So this is how you might think of sine and cosine with respect to circles. The input is a kind of, it's a kind of distance around a unit circle,
And we're going to talk in a moment about the difference between degrees and radians. you can think of that input as being the literal distance that you walk.
So as an example, notice what happens when we get to the distance pi. pi gives you the semi-circumference of one of these circles. If it's got a radius of one, then the distance that it takes to get halfway around is pi,
which is why at that input pi, you see the output go to negative one. which means walking around a distance of pi halves, So now I'm going to ask you another question where we're going to consider
the input in terms of this idea of a distance around the unit circle. And I'm just going to have you guess at some of the values for sine and cosine. So we'll go ahead and pull up our quiz again.
And this time, let's get it up on here. Without looking it up or using a calculator, which of the following is true?
And it's reminding us that that input 3 is going to be considered in radians, if you're just walking along at the arc length. Option A, the sine of 3 is around 0.14 and the cosine of 3 is around 0.99.
B, the sine of 3 is around 0.14 and the cosine of 3 is around negative 0.99. C, sine of 3 is around 0.99, cosine of 3 is around 0.14.
And D, sine of 3 is around negative 0.99 and cosine of 3 is around 0.14. Okay, so just to give you a little moment to think that through.
on this question and answers are still rolling in. So I'm going to go ahead and give you a little bit of pause and ponder music. I don't want you to feel rushed here.
Okay, so I think this is probably a pretty good time to start grading things. So the correct answer is B. And it looks like the majority of you got the correct answer,
which is to say that sine is 0.14 and cosine is negative 0.99. What's actually relevant is just kind of whether it's positive or negative and whether it's very close to 1 or if it's closer to 0.
So if we think through what this might mean in the context of our graphs here, your little dot as you walk around the circle. Well, what does it mean to walk 3 units around the circle?
Well, like I just mentioned, walking pi units around the circle gets you halfway around. it takes you halfway around. So to walk 3 units is going to be something a little bit shy of that.
the x direction as you're at that left side of the circle. to 1 and then it doesn't really change that much. So maybe it shouldn't be too surprising that our cosine of 3 turned out to be around 0.99.
What's relevant is that it's negative, right? It's very close to negative one. So it's close to negative 0.99. Now on the other hand, if we were doing this with sine,
which same game, but it's measuring your y-coordinate, When you walk almost halfway around, which is to say almost pi radians, well, there's still a little bit of height left.
some of them would switch whether sine or cosine was the one that's close to negative 1. So congratulations to the 3,152 of you who got that one correct.
cosine if we're coming at it from the direction of circles. in a typical high school class where you start off thinking about right triangles.
One is not better, they're just very different and worth understanding the connection. So the classic thing when it comes to trig functions is a little statement that everyone has ringing through their head called SOH CAH TOA.
And what you're supposed to remember from this is if we have a right triangle, so let me go ahead and draw a right triangle. And if you know another one of the angles, theta,
then consider one of the sides to be the hypotenuse. And then you ask what of the other two legs is adjacent to that angle, and we label it a for adjacent, and which of them is opposite.
So then SOH CAH TOA, if that's a phrase that you remember running through your head, tells you that the sine of this angle, this is the definition of sine for some kind of angle that we input, is for such a right triangle, it's the opposite over the hypotenuse.
Similarly, the cosine of that angle is going to equal the adjacent over the hypotenuse.
And then the tangent of that angle is going to be the opposite over the adjacent.
So that is the classic thing to remember if students come out of a trig class, if they know one thing, they typically know SOH CAH TOA.
I'm going to go ahead and pull up another question here for you, which is going to ask us about a leaning tower, a leaning tower of Pisa.
Clearing out the last one, let's read what our new question has to say. Suppose that a tower is 100 meters high. So instead of making a 90 degree angle with the ground, it makes an 80 degree angle.
length of the shadow cast by this leaning tower? Okay, the length of the shadow cast by this leaning tower. I'm absolutely going to draw it for you in a moment,
but it's more fun if you're engaged and if you have a pencil and paper to noodle it out. But for me, at least, I always think much more clearly once I have pencil and paper, no matter how much I think I can think things through in my head.
So let's give you a little pause and ponder time on this one.
Looks like we have some very strong consensus on this one, which is very reassuring. especially given that I've turned the super chat or the live chat into the very
This is most fun if you don't know what the majority of people are already thinking. The whole idea of the statistics is right now you can see there's a bandwagon. There's 2600 people who are voting for one of the answers who think that's it,
but we don't know what one of those answers is. So if you have the answer, try not to just blurted out in the live chat. That's kind of the equivalent of a student in a classroom where when the teacher asks
just shouting. But in this case, it's a little bit more fun if we have the slow reveal. All right, so that's probably enough time for people to have rolled their answers in.
Now let's go ahead and see what they see what the majority has submitted. And the correct answer is B, which is 100 times the cosine of 80.
And it looks like the second most common answer was A, which is 100 times the sine of 80. multiplying it by one of these trigonometric functions, but there might have just been a little swap up for which one was which.
And our question had us imagine a tower which is 100 meters tall. And specifically, it said it was 100 meters tall and then it started tilting.
That's not necessarily how far its new top is off of the ground. And then it has us imagine that there's the sun directly overhead.
That's what gives us our nice right triangle. And it further specified that the angle here was 80 degrees. Okay, so the thing we want to know is this side here, the length of the shadow,
So we might write for ourselves cosine of this angle, which is 80 degrees, is equal to that adjacent side, the thing that we want to know.
try to give our variables readable meanings, divided by the hypotenuse, which is 100. So that means that the shadow, once we rearrange, is 100 times the cosine of 80 degrees.
Now already this feels very different from the thought of trigonometry via circles. One of the main differences here is that we're dealing with a different unit, right?
Here I was saying degrees, and that's very natural when we're in the context of triangles. like I'll say in a moment, but when humans assign numbers and numerical systems to things, we really like whole numbers between 0 and 100.
Instead of thinking of proportions, we think in terms of percentages, When we think about like sound with decibels, which scales a whole bunch we logarithmically change it and we have this very awkward system that's
really twisting the math behind its back, so that we can think of whole Same deal with what's going on with degrees, but mathematicians don't really like that. They don't really like to think in terms of degrees, because it's so unnatural.
Instead, they want some sort of natural unit, something where you imagine if you talk to an alien civilization about math, they would have the same conventions. distance around that unit circle that we were walking, like we saw earlier?
because trust me, you don't want to see me try to draw a circle. What's neat here is we can, for each of these triangles, because all we care about are ratios of lengths of sides.
So we might as well say, okay, let's scale this so that the hypotenuse is 1, meaning we divide everything by h. And what that's going to get us is the opposite
side is now o over h, whatever that was, and then the adjacent side is a over h, meaning that the y, or the vertical component here, is sine of theta,
where theta was our angle down here. Okay. And then that bottom part is cosine of theta. And if we want to think about all possible right triangles who have a hypotenuse of 1,
because that's saying every point is a distance of 1 away, If you have a triangle with a bigger angle, it shows up there.
If you have a triangle with a smaller angle, it shows up there. And it's as if you take all possible right triangles and you fix them so that their points are all in common, and such that their hypotenuses are all 1, you scale them down,
and the other tip of the triangle is going to trace out a unit circle like this, and each one of these points for the corresponding angle is going to have coordinates, cosine of theta as the x-coordinate, and then sine of theta for the y-coordinate.
And I'm just writing it vertically so I can fit it all on here. Now typically, like I said, mathematicians like to think in terms of radians, which is really a way of saying if we know that the radius of our circle is 1,
what is the distance that you've walked along the outside here? So for example, if you were to say 180 degrees, excuse me, degrees, which is walking halfway around the circle, that's the same as pi radians.
And if you were to take something like 60 degrees, okay, which would maybe end up, oh, I don't know, around here, that might be 60 degrees. You're walking a third of the half turn around the circle.
would make the conventions easier, but pi is the standard, so we're working with it. It's pi, thirds, radians.
So from this point forward, and really for a lot of your math career, where that makes the angle intuitive, try to translate it to radians, And the farther you get down the road when it comes to how sine and
almost all of the time it's easier to think of that input in terms of radians. So I think it's high time I ask you a couple questions about these,
and you might wonder, you know, how do we actually compute some of the values, right? How to compute. If there was an easy answer to say, oh, well, this is the formula that shows
you what sine of theta is, we wouldn't have given sine of theta a fancy name. and we would call it the circle y coordinate formula, and it might get a name, It wouldn't have a special button on your calculator,
because effectively what you have to do is, before you know fancy techniques what's known as the Kordak algorithm, you just have to have a handful of values that you do know that come from geometric settings where there's enough symmetry.
So for example, let's say I try to draw an equilateral triangle. An equilateral triangle will be nice and symmetric, and we're going to be able to leverage that symmetry to figure out a concrete
value of sine and cosine. This looks roughly equilateral, wouldn't you say? And let's say each of those side lengths is one because,
you know, if we get to choose our side lengths, why not one? Because remember, sine and cosine have everything to do with just ratios. So on an equilateral triangle, this angle here is a third of the total.
But like I said earlier, I would encourage you to try to get in the full angle is pi, so a third of that is going to be pi thirds.
Here's where we're leveraging that symmetry to some extent. That one is 30 degrees, which in terms of radians is now a sixth of a half turn.
But again, hey, it's the convention. knowing in your head how to think in terms of radians measured in pi, because that's how we often write it down in tests and such.
So one thing I might ask you now is if you can leverage the symmetry of the setup, the fact that equilateral triangles are going to be very nice and that the right triangle coming about from it is very nice,
to answer something like a very concrete calculation, what is the sine of pi sixths? And in fact, let's do this as another live question, I'm just very enthusiastic with my new toy.
Pull up another question here. And what does our question ask? So again, if you want, if you're just joining now, you can hop over to 3b1b.co.
I imagine the best way to do this if you're doing it live is you might be answering That's also good.
a look to see if we have anything from the audience. When you do maths problems, do you use pen or pencil? One, I'm very error prone, so I'm going to make mistakes all the time.
And let me just tell you, with every stroke that I write, I'm deeply terrified that there's about to be an error, because I know there will be. then you'll start to notice it from this point forward.
It looks like we've got pretty strong consensus now, and then a pretty even split among the other ones.
whoever worked on the scaling, maybe a little bit misleading. It looks like one of the answers is much less popular than the other two. But I guess it's considering them to start from somewhere like that.
So there's one very unpopular answer, and then one very popular answer. So let's go ahead and see what that turns out to be.
Congratulations to the 2800, no, no, 2915 of you coming into the last minute there to pull in a correct answer. Why is that necessarily going to be one half?
Well, remember, sine of pi six, we're asking SOHCAHTOA. So kind of remembering your head, sine is opposite over adjacent, opposite over hypotenuse, excuse me.
So with respect to this smaller angle, maybe I'll change colors. With respect to this smaller angle, the opposite side is this segment here.
And because of the symmetry of our triangle, that's going to be one half. so half of it should be one half. And divided by the hypotenuse, but by definition the hypotenuse was one here.
So one half divided by one, it's just the same as one half. Now, slightly trickier is if I'd asked you the cosine of pi six. And let's do this one again as a live quiz, but maybe a little bit more quickly this time.
I'm going to hop over, get to our next question, and as you might expect, the next question is going to ask you to find the cosine of pi divided by six. And your options are A, one half, B, square root of two divided by two, C,
square root of three divided by two, or D, square root of five divided by two. If you can, I would highly encourage you to be keeping your own notes, right? you could take a look through it as you're working out these questions.
But for right now, I'm just going to give you maybe 20 seconds Get a little pause and ponder music going.
Once again, it looks like we have some pretty strong consensus from about the same number of people that we had before. And the correct one is C, square root of three divided by two.
that it was square root of three divided by two. Well, the trick is basically if you have one of these right triangles,
So let me draw a little copy for ourselves here. We know that the hypotenuse is one, one of the side lengths is one half, and the only thing that's missing if we're thinking the cosine of 30 degrees is this
So because we were doing it with the angle up there, adjacent side is here. Now the Pythagorean theorem tells us that that adjacent value squared plus the other leg length squared is going to equal one squared.
A hypotenuse squared, which means that value is equal to the square root of one minus, It's one fourth.
Which means it is the square root of one minus a fourth is three fourths. that bottom and just leave the radical in the top. Now this is a value that anybody coming out of a trig class becomes very familiar with.
it's going to be square root of three over two. Or root two over two also comes up in another circumstance. Square root of three divided by two.
And one very important fact that's come about here that's worth highlighting is that, hmm, how does the Pythagorean theorem play into our trigonometry?
and we know that one of the side lengths, let me switch to thinking with respect to If one of the side lengths is cosine of theta, and one of them is sine of theta,
well the Pythagorean theorem tells us that cosine of x squared plus sine of x squared is always equal to one squared.
It basically tells you if you know cosine, you know sine. And it also is giving you a little reminder that In fact, this is equivalent to the Pythagorean theorem.
it's just defined in terms of the leg lengths of a right triangle, Now there's one very silly convention with how we write these things.
and the fact that it's a little more interesting than you might think at first sight. cosine of x closed parenthesis squared.
Because people don't want to write too many parenthesis, instead they write cosine squared of x plus sine squared of x. We don't have to write too many symbols.
Except for the fact that literally every other time in math you see something written as f squared of x, what that means is not f of x squared.
Instead, what it's supposed to mean for most of math is taking a function and applying it to itself, like layering it over itself.
Sometimes later on it refers to the second derivative, But the idea is that it's very different from the convention in trigonometry.
we've got our own conventions for what it means when you put a little 2 up there. I don't care. You get used to it.
What I was writing earlier when we were thinking about the cosine squared function, which again is very interesting, you would almost always see it written as cos squared
Now, very interesting is that this is not just a consequence of the Pythagorean theorem that gives us a relation between our trig functions. trig functions to prove the Pythagorean theorem in a sneaky way.
Like it's not one of the common proofs that you'll see that's associated with, But it is very visually satisfying. So that's something that we'll get to by the end of the lecture.
couple more of our of our trig function questions. We just found cosine of pi six, which is square root of three over two.
So far, I'm dealing only with angles that were between zero and 90 degrees, But let's try a question that's a little bit different from that.
And what if I instead said cosine of negative two pi over three? OK, now that's one that is hard to think of in terms of triangles,
So this is a good reminder that cosine and sine, even if you think they're about triangles, are really about circles. And remember, if you walk all the way around the circle, and if it has a radius of one,
walking all the way around the circle is going to take you a distance of two pi. So if you want to think about two pi thirds, what you might start by doing is chopping the triangle into thirds and then using that to help guide the intuitions.
So let's go ahead and pull up the next question. What do you think the cosine of two pi divided by three?
Cosine of negative two pi divided by three. You reference your notes now. I'm going to go ahead and give you a little bit more of that pause and ponder music.
It looks like one of you figured out how to submit an answer that is not one of the four multiple choices. So other than that, we've got a very strong lead with 1152 of you,
as I say this, agreeing on one of the answers. And then second place by about a factor of four is going to sit on another one. I'm curious about actually what what all the answer distribution here is.
OK, so two of you have now gone outside the realm of the typical multiple choice universe and have entered your own things. I figured out my own answer and you just didn't put it in your form.
If only you could do that on the SAT. Sir, I think the answer is J.
Oh, if I pause it now, though, it's not going to stay there for a brief, glorious moment. Maybe I can pause it around the year, though, because it's 1999.
So the correct answer in this context is negative one half. which is negative root three over two.
between whether you think it's a half or whether you think it's root three over two, because it's sort of the difference between things that feel like 30
So let's take a look back at our drawing and see how we might think about this. That's taking us a negative distance over here.
because the x coordinate is pointing in the negative direction. I might draw this triangle here and ask, what is this angle?
OK, we know that it's 90 degrees plus something or pi halves plus something and negative two pi over thirds. If we wanted to think in terms of degrees, if that's something we're more comfortable
with, that's negative 360 divided by three degrees, which is negative 120 degrees. then that minus the 90 degree angle between the x and y axis gives us 30 degrees.
OK, so then the question is, what is the cosine of this value? With your 30, 60, 90 triangles, it's just kind of a nice thing to remember that the shorter side is one half and the longer side is root three over two.
And if you forget why, it's exactly the diagram that we just drew before associated with equilateral triangles. then the shorter side has length one half and then the longer
side comes from the Pythagorean theorem. So congratulations to those of you who correctly found that. Now, I want to bring back that identity that you guys found at the very beginning,
and show how that extends our capability for making some of these computations and So let's remind ourselves what that actually was.
I'm going to go back here and I'm going to pull up again the Desmos graph. And remember how what we said is that cosine squared, which looks like this little scaled down version of cosine,
oscillating more quickly, is the same as cosine of two x, that quicker oscillation. But we had to kind of shift it and rescale it to make them equal. So now we're going to use that fact that you can discover just by playing around
with graphs to make a computation that otherwise would be pretty tricky to make. it's often a reflection of the fact that you have a deeper conceptual understanding So even if it's always going to be your calculator computing this,
the fact that you can is an illustrate that you understand that cosine So over on our paper, let me write down again what that identity was.
Good old friendly black. Cosine squared of x, using the dumb dumb convention, is one plus cosine of two x.
So it's kind of oscillating more quickly, all divided by two. OK, so now I'm going to ask you a trickier question, which is going to be what is the cosine of pi twelfths?
And for those of you keeping track, pi twelfths is the same as 15 degrees. So if we were doing this with a unit circle, what you might think is, OK, a 30 degree angle.
That's 30 degrees. I'm now going to be taking half of that. And I want to know the cosine of this value, which
And right away, you can probably make a guess that it's very close to one, right, because it's a small angle. so we haven't gotten that far away from one, especially given the fact that
as you start to walk, almost all of your velocity is in the up and down direction. You know that this is a live quiz dynamic. So let's pull it up.
What is the cosine of pi divided by 12? Very fun, very exciting.
you haven't been following along before, go to 3b1b.co slash live. It affects the overall distribution.
This one, unlike what we had before, much less overall consensus. So I'm going to just stop talking and give you some some pause and ponder moments here,
This one takes time to think through.
So.
have extended beyond the realm of usual multiple choice. let me just say a huge thanks to Ben Eater and Cam Christensen.
pool website that is making all of this possible. named Ben Eater about computer engineering.
And then item pool is a pretty cool thing that will So you might want to keep your eye on that.
So I'm going to go ahead and grade this question. All right, it looks like I'm getting a phone call.
So the correct answer is B, which is square root of it's a very complicated thing, right? Square root of one plus the square root of three over two divided by two. So congratulations to the two to two nine of you who two to two for two to four three.
Going to have to talk to Cam and Eater about that. Who got that correct? Now it looks like the second most common answer was A, which is of a very similar form.
So it's probably the case that while doing the arithmetic, something got flipped. And a special congratulations to the two of you Oh, the three of you who submitted an at sign.
Well done, four of you. So clearly this is a complicated answer, but where does it come from? Well, if we look at our identity here, the key is that
So if we just plug this in for 15 degrees or for pi twelfth, what we would get is that the cosine squared of pi twelfth.
OK, if we enter that as X is equal to one plus the cosine of two times that value, Pi six.
s, we know, right? Again, when you have a 30, 60, 90 triangle, the shorter side of that,
which is going to be the sign in this context, the shorter side is one half. assuming that the hypotenuse is one, which in the context of the unit circle,
So that means that we now have one plus the square root of three over two. That's what the cosine of pi six was all divided by two, which means our final answer.
When we just want to say cosine of pi twelfth, no squared about it is going to be the square root of this value. So this is showing us that cosine and sine can get complicated.
We've got a handful of values like pi six and pi thirds where we can figure them out. We can have even more, I won't call them trivial values, where you would look at your unit circle and say, oh, pi.
So my x coordinate is negative one. But outside of just a handful of values like that, they're very hard to compute by hand. and in the olden days, the only way that people could compute them by hand,
before something like the existence of calculus, would be measuring it, right? what the coordinates are or layering on usage of identities like this over and over in ways that you can get approximate values like this.
And just to check ourselves, we can actually go and plug this into a calculator how big the value is and kind of confirm for ourselves that they turn out to be the same.
So if we pop over here and I just type something like. Cosine of pi over 12. OK, so like I said, very close to one, and that's kind of what we would expect.
And the answer we just got was the square root of one plus the square root of three. Got to make sure I do the divided by two correctly. But then all of that was divided by two.
And indeed, they're the same value. So even just something like pi 12, we end up getting a very convoluted image. Now, like I said, this is a reflection of the fact that our cosine identity that we found
at the very beginning, just by playing through with graphs, it's definitely not obvious. And what we'll find next time is that this is actually a shadow of the fact that trigonometric functions are related to complex numbers.
And the idea of doubling the angle here corresponding to multiplying by something twice, I'm excited to show you guys all about that. was to show two things that are not commonly shown about the unit circle.
OK, so we see where sine and cosine are, but there might be a couple of questions. OK, where in our picture do we see tangent of theta?
And then also, where is cosine of x or cosine of theta squared or with the dumb convention, cosine squared of theta?
So pulling up another one of my infinite supply of unit circles here. This one's not going to have enough y axis, so I'm going to do something that I really shouldn't do, which is to try to use a compass live on camera.
We're just going to do this. Oh, yeah, that actually didn't turn out too badly. The reason I'm doing this is the tangent is going to demand that we think outside of
the box, that we think outside of the circle in order to geometrically interpret it. You could just ask about the opposite divided by the adjacent, But what I really want is a single length in our diagram that corresponds to
the tangent so we can kind of get that nice geometric intuitive feel for it. And the idea is we take that radius, which is one, and if we think of the fact that tangent of theta, SOHCAHTOA,
I want a triangle where that adjacent is equal to one. OK, so a right triangle such that this one is the adjacent side to our angle theta.
angled perpendicular line off of our radial line. to the radius of a circle, it actually lies tangent to the circle.
So you see tangent, this is where the word comes from. Then what we have is a triangle where the adjacent side to theta is one and then the opposite side is this length here.
So that length there is the tangent of theta. And so right now you can kind of intuitively understand that as theta is very small, when the opposite over adjacent is very close to zero,
because that opposite is close to zero and the adjacent is close to one. And then you can also think through what happens as that angle approaches 90 degrees.
This is something I actually illustrated once before in a previous video, a lesser known video entitled Tattoos on Math. That, fun fact, includes someone who is behind the scenes as we say all of this right now.
tangent is this length that I just pointed out to you. There's this whole other triad of trigonometric functions that aren't worth teaching,
That's actually what that whole video was about. and seeing what that does to this tangent value, to that tangent length.
on what the graph of tangent of theta is going to look like. Again, congratulations to those of you who got the last one right.
And as our final question before the end of this particular stream, which is a rather tall question, so it might actually pop up over, The final question, I just want you to guess what
the tangent of theta looks like when we graph it. And it should be pulled up now. Okay. All right
for you here so that you can see all of the options. Great. So again, if you go to 3b1b.co slash live, where the link is in the description,
you can submit your answers to this. Basically, which one of these graphs shows you the tangent of theta? Unfortunately, we're going to have to kind of switch back and forth to see the statistics.
but I'm going to give you I'm going to give you a little bit more time to think about Don't feel like you have to jump right into it.
I want a little ambient pause and ponder music while we're at it. I'm mildly addicted to it.
So while you're answering. or there's a bug in the website to answer various other things. It seems like we've got a couple comments in the in the Json associated with the answers.
So to the person who decided to skip over my very draconian limits on the live chat dynamic and instead entered in Json comment through the answer forum forum.
And I appreciate your your hacky instincts. You're going to go very, very far. It looks like we have very strong consensus, hopefully around the correct answer.
And the correct answer is a again, maybe a little bit hard to see on our on our indication of the question over here. And what you'll notice is it starts at zero.
And then as you approach pi halves, it blows up to infinity. OK, now, in terms of the animations and the geometric interpretation of tangent, that should actually make sense, because if we go back to what we were
looking at as the angle blows up, the angle doesn't blow up. The angle does a very modest thing of simply going to pi halves a fine number. And ultimately, it's it's not defined.
It's the same as dividing by zero once you get to pi halves. it's going to be it's going to be considered a negative value, which basically comes down to the fact that the opposite and adjacent sides of our
triangle are going to be considered to have separate signs. So the very last thing that I want to talk about, like I said, is the question of where does cosine squared of theta show up in our image?
Because this is the very thing that we opened with is trying to understand cosine squared. maybe it really reveals why it looks like a miniature version of the cosine wave. Now, one of the naive things, not naive, but one of the first things you might
try to do is pull up the original cosine animation that we had or something like it and say, oh, well, I'm going to take that cosine length and literally square it. And if I were to do this, where I'm going to plot the length of that horizontal line
and I'm also going to plot the area of that square, it's a little bit complicated. which is that the square of that value looks like another cosine wave.
it actually doesn't yield much insight. write down the definition of things, the answer pops out. It's very unclear why the way that that square,
that literal square would oscillate is going to be related to cosine, So it's going to come from somewhere else inside our image. And for that, let me go ahead and draw a much bigger triangle for ourselves.
Let's say we've got one of these guys that was living inside the unit circle. So if there was just one thing that you were going to stick around for,
Because it gives a very sneaky proof of the Pythagorean theorem. And I also assure you that the word circular was not meant as a pun there.
So remember, if that's our angle theta, then one of our side lengths is sine theta. And that's the angle theta. And that's the angle theta. And that's the angle theta. Then one of our side lengths is sine of theta. And the other one is cosine of theta.
we kind of imagined casting a shadow from a leaning tower onto the ground. this angle theta between them, who's to say who's the tower and who's the ground? I wanted to cast a shadow from the ground onto the tower?
A nice perpendicular line in this direction. That way my angle theta is going to do double duty for me. but in one context, this ground line was the adjacent side.
But now, if I ask you a question like, what is this side length? The shadow that the ground lays down on the tower, if that's not more confusing than I want it to be, what are we going to get?
Well, in terms of this smaller right triangle that we've just created, by definition, the cosine of that angle is going to be the adjacent side, the thing we want to know, maybe I'll call it s for shadow again.
It's going to be the adjacent side divided by the hypotenuse. So it's divided by the cosine of theta, which means cosine squared of theta,
dumb convention, is equal to the answer that we want. Let me just write it down in a different color to really emphasize. And let me emphasize, this is not using the Pythagorean theorem in any way.
This is not having the cosine squared arrive because we were assuming We're just using the definition of cosine twice Now, you can probably see where this is going, do the same thing for the sine of theta.
So another place where theta shows up in our diagram is right here. And a way that you can see that is that this angle plus something must equal 90 degrees. But this other something plus r theta had to be the
remaining angles of the little right triangle that we had. If I called this other angle alpha, then we know that alpha plus theta,
if you want to get used to the better conventions, has to equal pi. This is maybe confusing.
In the back of your mind, you can go to 90 degrees But that also implies that when we have a new right triangle sitting here,
this little angle here has got to be theta because it's alpha plus 90 plus something, Now, the reason I have stated that so much is because we're going to play our same shadow casting game to figure out what this side length is.
Now, with respect to that, we have sine of theta of this theta in particular is the opposite, the new shadow we don't know, I'll call it s prime, is equal to s prime divided by the hypotenuse.
Well, on this smaller right triangle, what was the long side has become the hypotenuse, Which means sine squared of theta is our other shadow.
This here is sine squared of theta. The side length, which was by definition one, because we've always been can be broken down into one part that's cosine squared and another part
And the way to break that down is to just draw a single line, the perpendicular projection from our corner point onto that hypotenuse.
So what that means is every time you see a unit circle out in the wild, which I'm sure you do pretty regularly, and you draw the standard triangle to understand sine and cosine, so that height is going to be sine,
the x coordinate is going to be cosine, and our hypotenuse is always one. So that hypotenuse is one, this is theta. Where is sine squared?
You can just draw a little perpendicular line here, and the part on the bottom is cosine squared. So, cos squared of theta is this length, this length here.
I want to talk about geometry and geometry proofs, and in particular, it's high time we just sit down and say, how would you prove it,
So, after showing you how complex numbers play into things next time, I want to turn back to this particular diagram to give another very intuitive understanding for the identity that you found simply by playing around with graphs
at the very beginning here, which, again, is a very non-trivial identity, that the cosine squared of some value is one plus the cosine of two times that The more standard way you see it, by the way, if you were to look this up in a textbook,
instead of writing this as x, this is commonly written as something like And the whole thing ends up looking like cosine of alpha divided by two
is equal to the square root of one plus cosine of alpha divided by two. and I always would just sort of re-derive them on the fly.
And this is one where I think the easiest way to re-derive is to practice kind of smooshing around graphs in your head, when you have the one key insight, And the next time I'll show you how complex numbers help
But this one in particular, I just always had trouble remembering. And I just think it's delightful that before you even know any trigonometry,
simply hopping over to something like Desmos and being a little bit playful, that, I mean, that lands you on a highly non-trivial fact. And once again, it's one that indicates that cosine is related to exponentials.
That's not obvious, and hopefully next time we can start to get a glimpse of why. Thank you once again to Cam and Ben Eater, who have been making And I'll see you go at the very end.
I will see you all on Friday.