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0:00
My end goal for this lecture and the following lecture is to draw for you a connection between trigonometry and complex numbers. And I really do think when you understand that connection between the two, it strengthens your understanding of both of them. So for this lecture, we're just going to go back to the basics on some 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
0:24
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. But then one day, you know, it comes up, you hear someone talking about physics, and they mention a sinusoidal wave and the cosine function and things like that. So you go to a graphing calculator, something like Desmos,
0:46
and you just poke it in and say, okay, what does this thing look like? Now before seeing its definition, you can understand, oh, yeah, I can see how this might be relevant to waves and wave type dynamics that people might want to study. It's got kind of a cycling phenomenon to it, so maybe anytime people are studying something associated with cycling, you could play around with the graph and do all the things that you can with any normal function.
1:07
If I tweak the input, it kind of squishes it, right? So as I put a bigger constant in front of that input X, it squishes things, gives it a higher frequency. Okay, that's all well and good. 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.
1:33
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 cosine is clearly related to pi and probably circles in some way. 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.
2:00
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 does it say here? 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
2:29
these it might be a little small to read, but the range is going from 0 to about 2 pi. So in A it hits 1 and then it goes down and hits 1 again at 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.
2:50
C looks just like a sinusoidal wave. 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. Okay, so it looks like we've got a lot of answers coming in. A fair bit of consensus towards the top, but it's not, definitely not a unanimous decision on this one where people are going.
3:21
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. Clink. Click it there.
3:45
And, alright, I think that's probably enough time. Let's go ahead and grade this.
3:50
What do you think it will be? Alright, it looks like a majority of you went with C, which is the one that looks like a sinusoidal wave.
3:59
The second most common one was A, which has a similar shape, but it's kind of more flattened out on the bottom. Third most common was B, and then the last most common was D. And I think that's pretty good instincts because A, B, and C, 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, 1 squared should stay 1.
4:23
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. So the idea that no matter what you're going to get something that dips down to 0, then up to 1, down to 0, and up to 1 should make sense. But the specific shape is what we don't know. 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.
4:47
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? But to get another cosine wave that's oscillating more quickly, it's not at all clear that that's what would happen. And using this, even before you know what the cosine of x means, if you're just playing around being kind of childlike with your graphing calculator,
5:11
you can discover a pretty non-trivial fact about trigonometric functions. And then as we start defining them and getting into them, 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. Okay, and I look at this graph hard, okay, and remember 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, okay?
5:39
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?
5:54
Okay, now because this might take a little moment for you guys to work out, 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.
6:14
So which one of these reflects the fact the cosine of x squared is kind of a scaled version of itself?
7:03
So it looks like we still have a fair bit of answers rolling in, 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.
7:14
While you're answering this, it looks like we've got ourselves a question that's rolled in, okay? So Diego Mathemagician, love the name, asks, what is the new music? Great question. So the guy who usually composes the music for the channel, his name is Vince, Vince Rubenetti. I told him that we were doing these streams, and I mentioned I wanted kind of a jeopardy. Here, I'll turn it down so that I can talk over it. Vince, your music is great, but I want to talk about it a little bit more.
7:41
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. Now that is probably enough to noodle this one out, so I'm going to go ahead and grade this. And I'm guessing the majority have gotten it correct. Okay, great job. So to the 3,165 of you that answered B, which is that it's one half cosine of 2x plus 1, congratulations, you're absolutely correct.
8:07
But let's think about how you might do that in a step-by-step way, and this is one of those things that, again, if you're just 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, okay? And then I'm going to start manipulating it. 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
8:29
because that's what the other one is, so maybe I'm going to shift it up by one, okay? And that makes it all positive. But now it's twice as high as the other graph we're looking at, so if we want to make it equal, we might chop it in half. 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.
9:00
And sometimes people find this very counterintuitive that you scale up that input, that you multiply it by 2 in order to squish it. 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. So if we make that input 2x, then the thing being plugged into cosine will go from 0 to 2 pi as x goes from 0 up to pi.
9:25
And what you'll look at is the graphs are right on top of each other, right? Which is very cool, right on top of each other, and that means you've discovered a pretty non-trivial fact about trigonometry. And it's worth writing down because it's one of those things that comes up later in a very hard to memorize way. I always had a little bit of trouble actually remembering what this thing is. So, writing down what you just found, cosine of x squared, okay,
9:52
is equal to 1 plus cosine of 2x, which is basically saying increase its frequency. So we're going to double its frequency and then add 1 to it. 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, it becomes clearer and clearer just how non-obvious this is.
10:21
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.
10:34
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. And this actually takes us to our next question. 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. What does our question say?
10:58
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.
11:10
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.
11:24
Okay, so I'm going to give you a couple minutes, a little bit to answer that. We won't take too long on this one. Oh, it looks like I accidentally graded it right away.
11:34
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. Okay, and we can see this again if we go and just 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. Look at what that graph looks like. So let's not look at our cosines anymore, 2 to the x.
12:03
And indeed, if I take this whole thing and I square it.
12:09
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. And if you know anything about exponents, this kind of makes 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, it's a lot like scaling the input. And of course, the reason I bring that up is that you're
12:33
looking at something very similar when it comes to cosine. 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. It's not at all obvious how it is, but it absolutely is. And that's one of the things that I'd like to get to by the end of the end of the second lecture. We won't get to the end of there today.
13:00
So with all of that, I think it's high time that we actually talk about what sine and cosine actually are. I don't want to come in assuming that you necessarily know that already, so let's have a little moment to go back to the basics.
13:11
I think one of the best summaries of trigonometry that I ever heard, and this was on Reddit, but I cannot for the life of me find it, or remember who originally said this, is you think that it's about triangles, but really it's about circles. 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, which we'll talk about in a moment, to circles.
13:38
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, and often how to get intuitions about where the values will be. So instead of our Desmos graph, I'm going to go ahead and pull up a couple animations here. 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. So I want you to imagine that you've started on the
14:07
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.
14:25
And as you do that, you get this wave that oscillates. And it kind of makes sense how it's just going to oscillate on and on, no matter how far you walk. 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, let's just do the plain vanilla cosine x, you know, 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.
14:49
If instead I type sine of x, we get a similar looking shape. The only difference is that it starts at zero.
14:57
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, it starts at zero before slowly increasing. Okay? Now by contrast, cosine, it's defined very similarly, but it's giving you the x-coordinate as you walk around that circle. So the distance to the vertical line. And in that context, it starts off at one, and then as you walk around the circle,
15:23
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.
15:33
Okay?
15:35
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, but you could think of that as an angle, And we're going to talk in a moment about the difference between degrees and radians. But when you're thinking about walking around the circle, if it has a radius of one, you can think of that input as being the literal distance that you walk.
15:57
So as an example, notice what happens when we get to the distance pi. Okay? 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. At half of that, if you turn to 90 degree angle, which means walking around a distance of pi halves, that's why cosine is zero at that point.
16:25
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.
16:33
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.
16:41
And this time, let's get it up on here.
16:47
What does our question say? 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, which is to say the distance around the unit circle, 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.
17:25
And D, sine of 3 is around negative 0.99 and cosine of 3 is around 0.14.
17:32
Okay, so just to give you a little moment to think that through. it seems like there's not a hundred percent consensus 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.
17:47
I'll let you just think about it. I don't want you to feel rushed here.
18:20
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. Now the specific numbers here don't matter. 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,
18:50
the cosine, which remember is measuring the x-coordinate of your little dot as you walk around the circle.
18:57
What is that going to be when you input 3? 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. This is really the way to think about pi is you've got a unit circle, it takes you halfway around. So to walk 3 units is going to be something a little bit shy of that. Okay, and if you notice, there's very little change in the x direction as you're at that left side of the circle.
19:24
So when it approaches 1, it gets really close 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,
20:04
well, there's still a little bit of height left. And importantly, it's positive because you're still above that x-axis. So when we turn back to our question, the only thing differentiating our answers was that some of them would switch whether sine or cosine was the one that's close to negative 1. And the others would just switch the sine, s-i-g-n, whether it's positive or negative. So congratulations to the 3,152 of you who got that one correct.
20:20
So that's loosely how you might think about sine and cosine if we're coming at it from the direction of circles. But this is very different from how it's usually taught if we're doing things like 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
20:42
statement that everyone has ringing through their head called SOH CAH TOA.
20:51
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.
21:01
So we know one of the angles is 90 degrees. 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
21:31
angle that we input, is for such a right triangle, it's the opposite over the hypotenuse. That's what this is telling us, opposite over hypotenuse. Similarly, the cosine of that angle is going to equal the adjacent over the hypotenuse. So that would be adjacent over hypotenuse. And then the tangent of that angle is going to be the opposite over the adjacent. So that opposite side O over the adjacent side. So that is the classic thing to remember if students come out of a trig class,
22:08
if they know one thing, they typically know SOH CAH TOA.
22:12
And to give us an example of how you might apply this, 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.
22:24
Clearing out the last one, let's read what our new question has to say.
22:30
Suppose that a tower is 100 meters high. After several years, the tower starts to lean. So instead of making a 90 degree angle with the ground, it makes an 80 degree angle. When the sun is directly overhead, what is the length of the shadow cast by this leaning tower? Okay, the length of the shadow cast by this leaning tower. I would highly encourage you to try drawing this out yourself. I'm absolutely going to draw it for you in a moment,
22:58
but it's more fun if you're engaged and if you have a pencil and paper to noodle it out. You could also try to imagine it in your head if you want. 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.
23:13
So let's give you a little pause and ponder time on this one.
23:42
Looks like we have some very strong consensus on this one, which is very reassuring.
23:51
And I guess one thing that should maybe go without saying, especially given that I've turned the super chat or the live chat into the very aggressive slow mode. 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.
24:16
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 something, rather than raising their hand to ask it or submitting it quietly, just shouting.
24:29
I get it. You're excited. You really want to do it. 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.
24:47
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. So it looks like you guys correctly knew that you were multiplying it by one of these trigonometric functions, but there might have just been a little swap up for which one was which. So let's think this through. Okay, let's go back to our paper. And our question had us imagine a tower which is 100 meters tall.
25:15
And specifically, it said it was 100 meters tall and then it started tilting. So we can think of the length of the towers 100 meters. 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. Okay, so it's going to be casting a shadow perpendicular to the ground. That's what gives us our nice right triangle. And it further specified that the angle here was 80 degrees.
25:42
Okay, so the thing we want to know is this side here, the length of the shadow, which if we look at our SOHCAHTOA, that's the adjacent side. 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. I'll just call it x, or maybe I'll call it s for shadow, try to give our variables readable meanings, divided by the hypotenuse, which is 100.
26:10
So that means that the shadow, once we rearrange, is 100 times the cosine of 80 degrees.
26:19
Now already this feels very different from the thought of trigonometry via circles. Like I said, you start to think about it in terms of triangles, later it comes to 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. We could also talk about angles in terms of radians, like I'll say in a moment, but when humans assign numbers and
26:43
numerical systems to things, we really like whole numbers between 0 and 100. We just love that. Instead of thinking of proportions, we think in terms of percentages, which is the same thing, but it just turns it to whole numbers between 0 and 100. When we think about like sound with decibels, which scales a whole bunch and it's got this crazy exponential pattern to it, we logarithmically change it and we have this very awkward system that's
27:04
really twisting the math behind its back, so that we can think of whole numbers between 0 and 100. 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.
27:26
So how does any of this connect to the unit circle and the idea of the distance around that unit circle that we were walking, like we saw earlier? Well, let me just pull up a pre-printed unit circle, because trust me, you don't want to see me try to draw a circle.
27:40
What's neat here is we can, for each of these triangles, imagine rescaling it so that that hypotenuse is really 1, 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,
28:14
where theta was our angle down here. Okay. And then that bottom part is cosine of theta. Okay.
28:24
And if we want to think about all possible right triangles who have a hypotenuse of 1, what you might think of is taking a unit circle, because that's saying every point is a distance of 1 away, so think of that line as the hypotenuse of a triangle. 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
28:50
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.
29:18
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.
29:46
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.
29:56
Well, that's a third of that. You're walking a third of the half turn around the circle. Everyone who's an enthusiast about tau is sort of yelling right now because it would make the conventions easier, but pi is the standard, so we're working with it. It's pi, thirds, radians.
30:13
Okay? So from this point forward, and really for a lot of your math career, I would encourage you to, every time you want to think in terms of degrees, where that makes the angle intuitive, try to translate it to radians, because that's going to be the more natural way to understand it. And the farther you get down the road when it comes to how sine and cosine play into calculus, how you actually find values of them,
30:33
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.
30:52
And the honest answer is that it's hard. 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. We would just always work in terms of that formula, and we would call it the circle y coordinate formula, and it might get a name, but it wouldn't be a special function. It wouldn't have a special button on your calculator,
31:12
because effectively what you have to do is, before you know fancy techniques associated with calculus, or associated with even fancier things like 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.
31:34
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.
31:49
This looks roughly equilateral, wouldn't you say?
31:54
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. The total is 180 degrees, so you might think of this as 60 degrees. But like I said earlier, I would encourage you to try to get in the habit of thinking of that instead of 60 degrees, say, okay,
32:21
the full angle is pi, so a third of that is going to be pi thirds.
32:26
And then similarly, this one up here is half that angle. 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. All the tau enthusiasts get very mad at this point. But again, hey, it's the convention. This is the thing that you should come out of a trig class 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.
32:54
So that one will be pi sixths. 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? OK, what do you think it is? And in fact, let's do this as another live question, which, as you can see, I'm like a kid in a candy shop.
33:20
I'm just very enthusiastic with my new toy. So let's go ahead and use it. Pull up another question here. And what does our question ask?
33:33
What is the sine of pi sixths, of course? So again, if you want, if you're just joining now, you can hop over to 3b1b.co. Live. You can answer that. I imagine the best way to do this if you're doing it live is you might be answering on your phone, watching on another screen, or if you want to switch between tabs. That's also good.
33:52
Now, while those answers are rolling in, let's take a look to see if we have anything from the audience. And we do.
33:59
When you do maths problems, do you use pen or pencil? For sure I use pencil.
34:04
One, I'm very error prone, so I'm going to make mistakes all the time. The only reason I'm writing in pen now is because it shows up better on the camera. 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. So if that terror doesn't show in the shaking of my hands, then you'll start to notice it from this point forward. Back to our question, though. It looks like we've got pretty strong consensus now,
34:30
around 2,091 people agreeing on one of the answers, and then a pretty even split among the other ones. Oh, no, no, I guess the length of the bar is a little, 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. Maybe we should have some axes on there indicating that. So there's one very unpopular answer, and then one very popular answer.
34:57
So let's go ahead and see what that turns out to be.
35:03
Correct answer is one half. Congratulations to the 2800, no, no, 2915 of you coming into the last minute there to pull in a correct answer. Okay, so why is that the case? 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.
35:35
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.
35:46
Because each side length of the equilateral triangle is one, 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,
36:13
the next question is going to ask you to find the cosine of pi divided by six. There we go. 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? So if you wanted to kind of check back on your own piece of paper, you could take a look through it as you're working out these questions.
36:40
But for right now, I'm just going to give you maybe 20 seconds or so to kind of noodle through it, think of what it might be. Get a little pause and ponder music going.
37:11
Okay.
37:13
Once again, it looks like we have some pretty strong consensus from about the same number of people that we had before. So let's go ahead and grade the answers.
37:23
And the correct one is C, square root of three divided by two. So congratulations to the 3183 of you who knew that it was square root of three divided by two. Now, how do you figure something like that out?
37:37
Well, the trick is basically if you have one of these right triangles, if you know one of their sides, then you know both of them. So let me draw a little copy for ourselves here.
37:47
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 adjacent side. 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.
38:12
A hypotenuse squared, which means that value is equal to the square root of one minus, well what's one half squared? It's one fourth.
38:24
Which means it is the square root of one minus a fourth is three fourths. So we can go ahead and take the square root of that bottom and just leave the radical in the top. And what we get is the square root of three all divided by two. Now this is a value that anybody coming out of a trig class becomes very familiar with. They're like, okay, if the answer isn't one half, it's going to be square root of three over two. Or root two over two also comes up in another circumstance.
38:50
But this is one of those that kind of becomes burned in your memory. 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? Because if we have a right triangle whose hypotenuse is one, and we know that one of the side lengths, let me switch to thinking with respect to this lower left angle again.
39:16
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. Okay?
39:38
Now this is a very important identity. It basically tells you if you know cosine, you know sine. If you know sine, you know cosine. And it also is giving you a little reminder that it's all really about the Pythagorean theorem. In fact, this is equivalent to the Pythagorean theorem. Because if you think of how cosine and sine are defined, it's just defined in terms of the leg lengths of a right triangle, and then everything scaled down as needed.
40:00
Now there's one very silly convention with how we write these things. So we've already started to make friends with the cosine of x squared function, and the fact that it's a little more interesting than you might think at first sight. But you never see it written down as open parenthesis 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. It's all very nice.
40:30
How lovely. We don't have to write too many symbols. What a wonderful convention. 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. That's almost never what it means. 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. That's what that 2 usually refers to. Sometimes later on it refers to the second derivative,
41:02
especially if it has some parenthesis around it. But the idea is that it's very different from the convention in trigonometry. Trigonometry is just flying by its own rules and saying, yeah, yeah, we've got our own conventions for what it means when you put a little 2 up there. I understand some students will be confused, but I'm trigonometry. I don't care. I'm moving forward with my conventions. And that's fine. You get used to it. This is the standard way that you'll end up seeing it.
41:26
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 of x.
41:37
Now, very interesting is that this is not just a consequence of the Pythagorean theorem that gives us a relation between our trig functions. At the end of the lecture, I want to show you how you can use trig functions to prove the Pythagorean theorem in a sneaky way. And it's pretty sneaky. Like it's not one of the common proofs that you'll see that's associated with, you know, rearranging triangles or anything like that. But it is very visually satisfying.
42:02
So that's something that we'll get to by the end of the lecture.
42:06
But in the meantime, I want to think through a couple more of our of our trig function questions. OK, so we kind of come back up here. We just found cosine of pi six, which is square root of three over two.
42:19
But let's answer it for a couple more complicated ones. So far, I'm dealing only with angles that were between zero and 90 degrees, or I should say between zero and pi halves. But let's try a question that's a little bit different from that.
42:33
And what if I instead said cosine of negative two pi over three?
42:42
OK, now that's one that is hard to think of in terms of triangles, because what does it mean to have a negative angle? 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
43:09
chopping the triangle into thirds and then using that to help guide the intuitions. But again, I kind of want to do this as a live quiz. So let's go ahead and pull up the next question.
43:24
And have you guys answer this for me. What do you think the cosine of two pi divided by three? Let's go ahead and pull it up. Cosine of negative two pi divided by three.
43:37
So again, if you're keeping notes, that can be very helpful. You reference your notes now.
43:41
But just to make sure that you have enough time to think this through, I'm going to go ahead and give you a little bit more of that pause and ponder music.
43:59
It looks like one of you figured out how to submit an answer that is not one of the four multiple choices.
44:08
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.
44:17
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. I'm very curious about. Oh, now we have a sixth. OK, so two of you have now gone outside the realm of the typical multiple choice universe and have entered your own things. Two of you sit there and say, I don't want one of your choices. I figured out my own answer and you just didn't put it in your form.
44:43
So I'm going to figure out a way to submit it differently. If only you could do that on the SAT. What is the answer? A through E. Sir, I think the answer is J.
44:55
1729. We got Ramanujan's constant. Oh, if I pause it now, though, it's not going to stay there for a brief, glorious moment. We had a wonderful number. Maybe I can pause it around the year, though, because it's 1999.
45:08
2031. OK, well, I've paused it in the future.
45:11
So the correct answer in this context is negative one half. OK. And it looks like the second most common answer was D, which is negative root three over two. OK, so that's very good because often it's kind of this back and forth 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 degree angles and things that feel like 60 degree angles.
45:36
So let's take a look back at our drawing and see how we might think about this.
45:41
So, like I said, negative two pi divided by three. That's taking us a negative distance over here. And this is why cosine will be a negative number, because the x coordinate is pointing in the negative direction.
45:56
And if you want to think of this in terms of triangles, 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. So if it takes 120 degrees to get around here,
46:26
then that minus the 90 degree angle between the x and y axis gives us 30 degrees.
46:33
OK, so then the question is, what is the cosine of this value? What is the x coordinate there? 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. Because if you imagine taking the equilateral triangle and chopping it in half,
47:00
then the shorter side has length one half and then the longer side comes from the Pythagorean theorem.
47:08
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, associated with cosine squared and cosine of two thetas, and show how that extends our capability for making some of these computations and figuring out certain values. 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,
47:36
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. And I think that when you're able to make concrete computations,
48:00
it's often a reflection of the fact that you have a deeper conceptual understanding of what's going on. So even if it's always going to be your calculator computing this, you're not sitting here working out cosine values by hand, the fact that you can is an illustrate that you understand that cosine function a little bit better. So over on our paper, let me write down again what that identity was. Let's go back to black. Good old friendly black.
48:24
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.
48:38
OK, so now I'm going to ask you a trickier question, which is going to be what is the cosine of pi twelfths?
48:50
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.
49:04
It's looking like this.
49:07
That's 30 degrees. I'm now going to be taking half of that.
49:12
Half of that. And I want to know the cosine of this value, which is going to be the x coordinate of that value. And right away, you can probably make a guess that it's very close to one, right, because it's a small angle.
49:25
And the small angle means that we haven't walked that far around a circle, 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. All right, so cosine of pi twelfths, 15 degrees. You know what's about to happen. You know that this is a live quiz dynamic. So let's pull it up.
49:50
All right.
49:53
What is the cosine of pi divided by 12?
49:58
OK, lots of answers rolling in. Very fun, very exciting. Again, if you just happen to be tuning in now, you haven't been following along before, go to 3b1b.co slash live. Your answers will become part of what you're seeing on screen. It affects the overall distribution. This one, unlike what we had before, much less overall consensus. And it definitely seems to be taking a bit more time. So I'm going to just stop talking and give you some some pause and ponder moments here,
50:27
because this one is tricky. This one takes time to think through.
50:44
So.
51:15
Now it looks like there's a total of five of you who have extended beyond the realm of usual multiple choice.
51:24
By the way, as you guys are entering answers here, let me just say a huge thanks to Ben Eater and Cam Christensen. They're the ones who actually worked on the item pool website that is making all of this possible.
51:39
Eater, many of you might recognize from his own YouTube channel, named Ben Eater about computer engineering. If you haven't checked it out, you absolutely should.
51:47
And then item pool is a pretty cool thing that will be unfolding over the next coming months and year. So you might want to keep your eye on that.
51:57
It looks like some more consensus is starting to form. So I'm going to go ahead and grade this question.
52:05
All right, it looks like I'm getting a phone call.
52:10
Awesome. 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. Two thousand two hundred forty three of you. How are the answers still changing? How are the answers still changing? Going to have to talk to Cam and Eater about that. My hands are not on this. Who got that correct?
52:36
Now it looks like the second most common answer was A, which is of a very similar form. There's just a minus sign in there. So it's probably the case that while doing the arithmetic, something got flipped. We're going to walk through it in a moment so you can see. And a special congratulations to the two of you who figured out how to submit an at sign on this. Oh, the three of you who submitted an at sign. That's not the correct answer, but well done. Well done, four of you. Very nice.
53:02
All right. So clearly this is a complicated answer, but where does it come from?
53:08
Well, if we look at our identity here, the key is that it's relating two times an angle to the angle itself. 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.
53:26
OK, if we enter that as X is equal to one plus the cosine of two times that value, which is pi six.
53:37
Pi six.
53:40
All divided by two. Now, the cosine of pi six. s, we know, right?
53:45
That's the same as our 30 degree angle here. 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.
53:59
And then the longer side is square root of three over two, assuming that the hypotenuse is one, which in the context of the unit circle, we always do assume. 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.
54:19
When we just want to say cosine of pi twelfth, no squared about it is going to be the square root of this value. OK, very interesting. 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, but easier to compute values like cosine of pi, where you would look at your unit circle and say, oh, pi. That means I've walked halfway around.
54:49
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? You like draw out with a protractor and you try to carefully see 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.
55:17
And just to check ourselves, we can actually go and plug this into a calculator if we wanted to, something like Desmos and just see, one, 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.
55:34
Cosine of pi over 12.
55:38
Point nine six five. 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. By two. Right. That looks correct.
55:55
But then all of that was divided by two. All that stuff on the inside 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.
56:06
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, not at all a coincidence. I think it's very fun.
56:30
I'm excited to show you guys all about that. Now, what I wanted to do just to finish things off here, though, 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. Where is tan?
56:47
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?
57:03
So pulling up another one of my infinite supply of unit circles here.
57:09
Actually, you know what? 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. I'm sort of terrible at using compasses because it always slips out for me. We're just going to do this. I'm so nervous that this is going to somehow slip out. Oh, yeah, that actually didn't turn out too badly. Hopefully the pencil shows up OK on screen.
57:32
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.
57:44
You could just ask about the opposite divided by the adjacent, which is where that SOHCAHTOA TOA comes from. 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, is supposed to be the opposite over the adjacent, I want a triangle where that adjacent is equal to one.
58:15
OK, so a right triangle such that this one is the adjacent side to our angle theta. Well, in order to do that, let's just draw a right angled perpendicular line off of our radial line.
58:30
And some of you might know that if you draw a perpendicular 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.
58:41
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. Well, indeed, this length also is getting very close to zero.
59:09
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. But the idea is that when you draw that tangent line, tangent is this length that I just pointed out to you. And in that context, you could also think about cotangent.
59:38
There's this whole other triad of trigonometric functions that aren't worth teaching, but are kind of historical quirks. That's actually what that whole video was about. But from here, you can just play around with moving the angle theta and seeing what that does to this tangent value, to that tangent length.
59:55
And using that intuition, actually, I want you to place a guess on what the graph of tangent of theta is going to look like. So once again, let's go over to our live question dynamic. 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, over below the values we have available here. The final question, I just want you to guess what
1:00:23
the tangent of theta looks like when we graph it.
1:00:27
And it should be pulled up now.
1:00:31
And oh, there we go. It is rather tall, though. Okay. All right
1:00:35
So let me go ahead and pull up a separate image for you here so that you can see all of the options. Let's pop over here and take a look. 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?
1:01:00
Unfortunately, we're going to have to kind of switch back and forth to see the statistics. It looks like there's pretty strong consensus in one direction, but I'm going to give you I'm going to give you a little bit more time to think about it if you want to. Don't feel rushed. Don't feel like you have to jump right into it.
1:01:38
I want a little ambient pause and ponder music while we're at it.
1:01:43
I'm mildly addicted to it.
1:02:05
All right. So while you're answering.
1:02:09
So as we can see, some people are clearly messing with the website and having fun, or there's a bug in the website to answer various other things. Other fun thing going on. 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. I really enjoyed this class. Thank you. I appreciate that.
1:02:40
And I appreciate your your hacky instincts. You're going to go far in life, my friend. You're going to go very, very far.
1:02:47
So let's go ahead and grade this question here. It looks like we have very strong consensus, hopefully around the correct answer.
1:02:55
And the correct answer is a again, maybe a little bit hard to see on our on our indication of the question over here. But a is the graph that looks like this. 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.
1:03:27
The angle does a very modest thing of simply going to pi halves a fine number. But the tangent of that angle is going to just get bigger and bigger. And ultimately, it's it's not defined. It's the same as dividing by zero once you get to pi halves. And then after that, if you play the same game, 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
1:03:50
triangle are going to be considered to have separate signs.
1:03:56
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.
1:04:08
And you might think, oh, if we can see that in our image somewhere, 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. I'm just going to draw the square based on that.
1:04:29
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.
1:04:41
It certainly shows us what we already saw before, which is that the square of that value looks like another cosine wave.
1:04:49
But this is something where just by staring at the image, it actually doesn't yield much insight. I think oftentimes in geometry or math, when you simply write down the definition of things, the answer pops out. This is a sign that again, it's not trivial. It's very unclear why the way that that square, that literal square would oscillate is going to be related to cosine, much less cosine of two theta. So it's going to come from somewhere else inside our image.
1:05:16
And for that, let me go ahead and draw a much bigger triangle for ourselves.
1:05:24
Let's say we've got one of these guys that was living inside the unit circle.
1:05:33
This, by the way, I think is the most exciting part of the lecture. So if there was just one thing that you were going to stick around for, I would say that this is the thing that you should. Because it gives a very sneaky proof of the Pythagorean theorem. It might feel like a circular argument at first, but I assure you that it's not. 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.
1:05:56
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. Let me ask you, if I draw, well, okay, so, you know, the example we had earlier was this leaning tower thing where we kind of imagined casting a shadow from a leaning tower onto the ground. This is sort of going to flip it on its head and say, well,
1:06:15
this angle theta between them, who's to say who's the tower and who's the ground? What if I wanted to have a light source over here and I wanted to cast a shadow from the ground onto the tower? A nice perpendicular line in this direction.
1:06:28
That way my angle theta is going to do double duty for me. It's describing the angle between these two lines, but in one context, this ground line was the adjacent side. It was the short 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?
1:06:55
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. I'm going to write down what it equals.
1:07:12
It's going to be the adjacent side divided by the hypotenuse. But for this little triangle, the hypotenuse is cosine. 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.
1:07:29
Let me just write it down in a different color to really emphasize.
1:07:33
Cosine squared of theta. 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 that this sits on a circle where x squared plus y squared equals one. We're just using the definition of cosine twice to cast one shadow and then to cast another. 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.
1:08:00
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. Or you might think up here, I could just give it a different name. If I called this other angle alpha, then we know that alpha plus theta, alpha plus theta plus 90 degrees, or pi halves, if you want to get used to the better conventions, has to equal pi.
1:08:32
It's got to equal 180 degrees. This is maybe confusing. Just think alpha plus theta plus pi halves. In the back of your mind, you can go to 90 degrees if that's something you want for comfort.
1:08:44
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, so that something's got to be theta. 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,
1:09:11
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, so that's sine of theta. Sine of theta. Which means sine squared of theta is our other shadow. This here is sine squared of theta. So that's kind of cool. The side length, which was by definition one, because we've always been rescaling our triangles such that the hypotenuse is one,
1:09:44
can be broken down into one part that's cosine squared and another part that's sine squared. 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,
1:10:14
the x coordinate is going to be cosine, and our hypotenuse is always one.
1:10:21
So that hypotenuse is one, this is theta. And you ask yourself, where is cosine squared in this diagram? 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.
1:10:48
And what we'll do in a future lecture, actually, I want to talk about geometry and geometry proofs, and in particular, one that comes up all the time, I've used it in many past videos, it's high time we just sit down and say, how would you prove it, is the inscribed angle theorem. 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
1:11:10
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 value divided by two. 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 alpha halves for some angle, and then 2x would be alpha.
1:11:37
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.
1:11:53
I, for one, I have such trouble memorizing the trig identities, 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, which is that squaring cosine gets you a smaller cosine graph. And the next time I'll show you how complex numbers help you kind of re-derive a bunch of other identities on the fly.
1:12:15
But this one in particular, I just always had trouble remembering. It's called the half-angle identity. 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.
1:12:41
Thanks for sticking around this long. Thank you once again to Cam and Ben Eater, who have been making the item pool magic behind the scenes to get us our live statistics. And I'll see you go at the very end. I had almost everything down until the very end. I will see you all on Friday.
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