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Studying for a test? Prepare with these 7 lessons on The unit circle definition of sine, cosine, and tangent.
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What I want to do in this video is revisit a little bit of what we know about pi, and really how we measure angles in radians. And then think about whether pi is necessarily the best number to be paying attention to. So let's think a little bit about what I just said. So pi, we know, is defined-- and I'll write defined as a triple equal sign, I guess you could call it that way-- pi is defined as the ratio of the circumference of a circle to its diameter, which is the same thing as the ratio of the circumference of the circle to two times the radius. And from that, we get all these interesting formulas that you get in geometry class that, hey, if you have the radius and you want to calculate the circumference, multiply both sides of this definition, or this equation, by two times the radius. And you get two times the radius times pi is equal to the circumference, or more familiarly, it would be circumference is equal to 2 pi r. This is one of those fundamental things that you learn early on in your career and you use it to find circumferences, usually, or figure out radiuses if you know circumference. And from that comes how we measure our angles in radians once we get to trigonometry class. And just as a review here, let me draw myself a circle. Let me draw myself a better circle. So there is my-- it'll do the job. And here is the positive x-axis, and let me make some angle here. I'll make the angle kind of obvious just so that it-- so let me make this angle. And the way that we measure angles, when we talk about radians, we're really talking about the angle subtended by something of a certain arc length. And we measure the arc length in-- well, the way I like to think about it, is the angle is in radians and the arc length itself is in radiuses, which isn't really a word. But that's how I think about it. How many radiuses is this arc length that subtends the angle in radians? So let me show you what I'm talking about. So this arc length right here, if the radius is r, what is the length of this arc length? Well, we know from basic geometry the entire circumference over here is going to be 2 pi r, right? This entire circumference, that's really by definition, this entire circumference is going to be 2 pi r. So what is just this arc length here? And I'm assuming this is a fourth of the circle. So it's going to be 2 pi r over 4. So this arc length over here is going to be 2 pi r over 4, which is the same thing as pi over 2 r. Or you could say this is the same thing as pi over 2 radiuses. One of those-- you know, not a real word, but that's how I like to think about it. Or you could say it subtends an angle of pi over 2 radians. So over here, theta is pi over 2 radians. And so really, when you're measuring angles in radians, it's really you're saying, OK, that angle subtended by an arc that has a length of how many radiusi, or I don't even know what the plural of radius is. Actually, I think it's radii but it's fun to try to say radiuses. Radii, actually let me do that just so no one says, Sal, you're teaching people the wrong plural form of radius. Radii. So this arc length is pi over 2 radii and it subtends an angle of pi over 2 radians. We could do another one just for the sake of making the point clear. If you went all the way around the circle and you got back to the positive x-axis here, what is the arc length? Well now, all of a sudden the arc length is the entire circumference of the circle. It would be 2 pi r, which is the same thing as 2 pi radii. And we would say that the angle subtended by this arc length, the angle that we care about going all the way around the circle, is 2 pi radians. And so, out of this comes all of the things that we know about how to graph trigonometric functions or at least how we measure the graph on the x-axis. And I'll also touch on Euler's formula, which is the most beautiful formula, I think, in all of mathematics. And let's visit those right now, just to remind ourselves of how pi fits in to all of that. So if I think about our trigonometric functions. Remember, if this was-- so on trigonometric functions, we assume we have a unit circle here. So in the trig functions, this is the unit circle definition of the trig function. So this is a nice review of all of that. You assume you have a unit circle, a circle of radius 1. And then the trig functions are defined as, for any angle you have here, for any angle, theta, cosine of theta is how far you have to move in-- or the x-coordinate of the point along the arc that subtends this angle. So that's cosine of theta. And then sine of theta is the y value of that point. Let me make that clear. Cosine of theta is the x value, sine of theta is the y value. And so if you were to graph one of these functions, and I'll just do sine of theta for convenience. But you could try it with cosine of theta. So let's graph sine of theta. Let's do one revolution of sine of theta. And we tend to label it-- so let's do sine. When the angle is 0, sine of theta is 0. Let me draw the x- and y-axis just so you remember. This is the y-axis and this is the x-axis. So when the angle is 0, we're right here on the unit circle. The y value there is 0. So sine of theta is going to be right like that. Let me draw it like this. So this is our theta, and this is-- I'm going to graph sine of theta along the y-axis. So we'll say y is equal to sine of theta in this graph that I'm drawing right over here. And then we could do-- well, I'll just do the simple points here. And then if we make the angle go-- if we did it in degrees, 90 degrees, or if we do it in radians, pi over 2 radians. What is sine of theta? Well, now it is 1. This is the unit circle. It has a radius 1. So when theta is equal to pi over 2, then sine of theta is equal to 1. So if this is 1 right here, sine of theta is equal to 1. If theta-- and then if we go 180 degrees or halfway around the circle-- theta is now equal to pi. Let me do this in a color, orange. I have already used orange. Theta is now equal to pi. When theta is equal to pi, the y value of this point right here is once again 0. So we go back to 0. Remember, we're talking about sine of theta. And then we can go all the way down here, where you can see there's 270 degrees. Or you could view this as 3 pi over 2 radians. So this is in radians, this axis. So 3 pi over 2 radians, sine of theta is the y-coordinate on the unit circle right over here. So it's going to be negative 1. So this is negative 1. And then finally, when you go all the way around the circle, you've gone 2 pi radians, and you're back where you began. And the sine of theta, or the y-coordinate, is now 0 once again. And if you connect the dots, or if you'd plotted more points, you would see a sine curve over just the part that we've graphed right over here. So that's another application. You say hey, Sal, where is this going? Well I'm showing you-- I'm reminding you of all of these things because we're going to revisit it with a different number other than pi. And so I want to do one last visit with pi. You say, look, pi is powerful because-- or one of the reasons why pi seems to have some type of mystical power, and we've shown this in the calculus playlist-- is Euler's formula, that e to the i theta is equal to cosine of theta plus i sine of theta. This, by itself, is just a crazy-- it's just one of those mind boggling formulas. But it sometimes looks even more mind boggling when you put pi in for theta because then, from Euler's formula, you would get e to the i pi is equal to-- well, what's cosine of pi? Cosine of pi is negative 1. And then sine of pi is 0. So 0 times i. So you get this formula, which is pretty profound, and then you say, OK, if I want to put all of the fundamental numbers together in one formula, I can add 1 to both sides of this. And you get e to the i pi plus 1 is equal to 0. Sometimes this is called Euler's Identity, the most beautiful formula or equation in all of mathematics. And it is pretty profound. You have all of the fundamental numbers in one equation. e, i, pi, 1, 0, although for my aesthetic taste it would have been even more powerful if this was a 1 right over here. Because then this would have said, look, e to the i pi, this bizarre thing, would have equaled unity. That would have been super duper profound to me. It seems a little bit of a hack to add 1 to both sides and say, oh, look, now I have 0 here. But this is pretty darn good. But with that, I'm going to make-- well, I'm not going to argue for it. I'm going to show an argument for another number, a number different than pi. And I want to make it clear that these ideas are not my own. It comes from-- well, it's inspired by-- many people are on this movement now, the Tau Movement, but these are kind of the people that gave me the thinking on this. And the first is Robert Palais on "Pi is Wrong." And he doesn't argue that pi is calculated wrong. He still agrees that it is the ratio of the circumference to the diameter of a circle that is 3.14159. But what he's saying is that we're paying attention to the wrong number. And also, you have Michael Hartl, "The Tau Manifesto." All of this is available online. And what they argue for is a number called tau, or what they call tau. And they define tau, and it's a very simple change from pi. They define tau not as the ratio of the circumference the diameter, or the ratio of the circumference to 2 times the radius. They say, hey, wouldn't it be natural to define some number, the ratio of the circumference to the radius? And as you see here, this pi is just one half times this over here, right? Circumference over 2 r, this the same thing as one half times circumference over r. So pi is just half of tau. Or another way to think about it is that tau is just 2 times pi. Or, and I'm sure you probably don't have this memorized, because you're like, wait, I spent all my life memorizing pi, but it's 6.283185 and keeps going on and on and on, never repeating just like pi. It's 2 times pi. And so you're saying, hey Sal, pi has been around for millennia, really. Why mess with such a fundamental number, especially when you just spent all this time showing how profound it is? And the argument that they'd make, and it seems like a pretty good argument, is that actually things seem a little bit more elegant when you pay attention to this number instead of half of this number, when you pay attention to tau. And to see that, let's revisit everything that we did here. Now, all of a sudden, if you pay attention to 2 pi, as opposed to pi. Or we should call it-- if you pay attention to tau instead of tau over 2. What is this angle that we did in magenta? Well first of all, let's think about this formula right over here. What is the circumference in terms of the radius? Well, now we could say the circumference is equal to tau times the radius because tau is the same thing as 2 pi. So it makes that formula a little bit neater, although it does make the pi r squared a little bit messier. So you could argue both sides of that. But it makes the measure of radians much more intuitive because you could say that this is pi over 2 radians, or you could say that this is pi over 2 radians is the same thing as to tau over 4 radians. And where did I get that from? Remember, if you go all the way around the circle, that is the circumference. The arc length would be the circumference. It would be tau radii, or it would be to tau radians would be the angle subtended by that arc length. It would be tau radians. All the way around is tau radians. So that by itself is intuitive. One revolution is one tau radians. If you go only one fourth of that, it's going to be tau over four radians. So the reason why tau is more intuitive here is because it immediately-- you don't have to do this weird conversion where you saying, oh, divide by 2, multiply by 2, all that. You're just like, look, however many radians in terms of tau, that's really how many revolutions you've gone around the circle. And so, if you've gone one fourth around, that's tau over four radians. If you've gone halfway around, that'd be tau over two radians. If you go 3/4 around that'd be three tau over four radians. If you go all the way around that would be tau radians. If someone tells you that they have an angle of 10 tau radians, you'd go around exactly 10 times. It would be much more intuitive. You wouldn't have to do this little mental math, converting, saying, do I multiply or divide by 2 when I convert to radians in terms of pi? No, when you do it in terms of tau radians, it's just natural. One revolution is one tau radians. So that makes-- and it makes a sine function over here. Instead of writing pi over 2-- well, when you look at a graph like this you're like, where was this on the unit circle? Was this one fourth around the circle? Was this one half? And this is actually one fourth around the circle, right over here? But now it becomes obvious if you write it in tau. Pi over 2 is the same thing as to tau over 4. Pi is the same thing as to tau over 2. 3 pi over 2 is 3 pi-- oh, sorry, 3 tau over 4, 3/4 tau. And then one revolution is tau. And then immediately, now when you look at it this way, you know exactly where you are in the unit circle. You're one fourth around the unit circle, you're halfway around the unit circle, you're 3/4 of the way around the unit circle. And then you're all the way around the unit circle. And so the last thing that I think the strong pi defenders would say is well, look Sal, you just pointed out one of the most beautiful identities or formulas in mathematics. How does tau hold up to this? Well, let's just try it out and see what happens. So if we take e to the i tau, that will give us cosine of tau plus i sine of tau. And once again, let's just think about what this is. Tau radians means we've gone all the way around the unit circle. So cosine of tau-- remember, we're back at the beginning of the unit circle right over here-- so cosine of tau is going to be equal to 1. And then sine of tau is equal to 0. So e to the i tau is equal to 1. And I'll leave it up to you to decide which one seems to be more aesthetically profound.