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Sal explains the definition and motivation for radians and the relationship between radians and degrees. Created by Sal Khan.
Video transcript
You are by now probably used to the idea of measuring angles in degrees. We use it in everyday language, we've done some examples on........ where if you had an angle like that, you might call that a 30 degree 30 degree angle if you had an angle like this call that a 90 degree angle you'd often use a symbol just like that. If you were to go a 180 degrees, you'd essentially form a straight line. You go 360 degrees, you've essentially done one full rotation and if you watch figure skating on the olympics and someone does a rotation they say "Oh! they did a 360" or espescially in some type of some skateboarding competitions and things like that. But the one thing to realize is that it might not be obvious from the get-go but this whole notion of degrees, this is a human constructed system. this is not the only way that you can measure angles. And if you think about it yourselves, Why do we call a full rotation 360 degrees? And there's some possible theories and I encourage you to think about them, Why does 360 degrees show up in our culture as a full rotation? Well, there's a couple of theories there... One is ancient calenders and even our calenders close to this, but ancient calenders were based on 360 days in a year Some ancient astronomers observed that things seemed to move one 360th of the sky per day. Another theory is that the ancient babylonians liked equilateral triangles a lot and they had a base 60 number system. So they had 60 symbols, we only have 10. We have a base 10 and they had 60. So in our system we like to divide things under ten, they liked to divide under 60. So if you were to, if you had a circle and you divided it into 6 equilateral triangles. And each of those equilateral triangles you divided into 60 sections cause you have a base 60 number system. And you might end up with 360 degrees. What I want to think about in this video is an alternate way of measuring angles. And that alternate way, even though it might not seem intuitive to you from the get-go in some ways is much more mathematically pure, than degrees. It's not based on the cultural artifacts of base 60 number system or, astronomical patterns. To some degree an alien on another planet would not use degrees, especially if degrees are motivated by astronomical phenomena But they might use what we are going to define as a radian. Tere's a certain degree of purity here, Radians. So, lets just cut to the cheese and define what a radian is, so let me draw a circle here, my best attempt at drawing a circle. Not bad, and let me draw, centre of the circle and now let me draw this radius, and lets say that this radius, you might already notice that this word radius is very close to the word radian. And that's not a coincidence. So let's say that this radius, the circle has a radius of length r. Now lets construct an angle, I'll call that angle theta. So lets construct an angle theta. So let's call this this angle right over here theta and let's just say for the sake of argument that this angle is just the exact right measure, so that if you look at the arc that subtends this angle, That seems like a very fancy word. So let me draw the angle, so if you look at the arc that subtends the angle that seems like a fancy word and that's just talking about the arc along the circle that intersects the sides of theses two angles So this arc right over here subtends angle, this is the angle theta. So let me write this down, subtends this arc, subtends angle theta. Lets say theta's exactly the right size, so this arc is also the same length as the radius of the circle so this arc is also of length r. So given that if you were to find a new type of angle measurement and, you wanted to call it a radian which is very close to a radius. How many radians would you define this angle to be ? Well, the most obvious one, if you kind of view a radian as another way of saying radiusss or I guess a radii... Well, you say look, this is subtended by an arc of one radius. So why don't we this right over here one radian? Why don't we call this one radian? Which is exactly how a radian is defined, When you have a circle and you have an angle of one radian, the arc that subtends it is exactly one radius long. Which you can imagine might be a little useful as we start to interpret more and more types of circles, When you give in degrees, you have to do math and you have to think about the circumference and all of that To think about how many radiuses are subtending an angle. Here the angle in radians tells you exactly how many arc length is subtending the angle. So let's do a couple of broad experiments here. So, given that what would be the angle in radians if we were to go, so let me draw another circle here..... that's the center, start right over there. So what would happen if I had an angle, what angle if I were to measure in radian, what angle would this be in radian? You can almost think of it as radiussssss..... So what would that angle be? Going one full revolution, In degrees that would be 360 degrees. What would you, based on this definition, What would this be in radians? Well, lets think about the arc that subtends this angle is the entire circumference of this circle. Its the entire circumference of this circle. Well what's the circumference of a circle in terms of radiussss? So if this is length r, what's the circumference of the circle in terms of r? Well we know that it's gonna be 2 pi r. So going back to this angle, how many... whats the length of the arc that subtends this angle? How many radiusss? Well, it's two pi radiussss.... Its two pi times r. So this angle right over here , this angle, I'll call this a different.. lets call this angle x. x in this case is going to be two pi radians. And it is subtended by an arc of length two pi radiussss.... If the radius was one unit then this would be two pi times one, two pi radiusss... So, given that, lets start to think about how we can convert radians and degrees and vice versa. If I were to have, and we can just follow up over here. To do one full revolution, that is two pi radians, how many degrees is this going to be equal to? Well we already know this. A full revolution in degrees is 360 degrees. Well I could either write degrees or I could use this degree notation there, Actually let me write the word degrees. It might make thing a bit clearer that we're kind of using units in both cases. Now if we wanted to simplify this a little bit, we could divide both sides by 2, in which case we would get on the left hand side, we would get pi radians would be equal to how many degrees? Welll, it'd be equal to 180 degrees. 180 degrees and I could write it that way or I could write it that way. And you see over here, this is a 180 degrees and you also see if you were to draw a circle around here you've got halfway around the circle so the arc length or the arc that subtends the angle is half the circumference, half the circumference or pi radiusss... So we call this pi radians. pi radians is a 180 degrees. And from this we can come up with conversions. So one radian would be how many degrees? Well to do that, we just have to divide both sides by pi and on the left hand side you'd be left with one, I'll just write it singular now, one radian is equal to, I'm just dividing both sides, let me make it clear what I'm doing here just to show you it not voodoo. So, I'm just dividing both sides by pi here. On the left hand side we're left with one and on the right hand side we're left with 180 over pi degrees. So 1 radian = 180 over pi degrees. Which is starting to make an interesting way to convert them. Let's think about it the other way, if I were to have one degree, how many radians is that? Well, let's start off with, let me rewrite this thing over here. We said, pi radians is equal to 180 degrees. So now we want to think about one degree. So let's solve for one degree, we could divide both sides by 180. We are left with pi over 180 radians is equal to 1 degree. So pi over 180 radians is equal to one degree. This might seem confusing and daunting and it was for me the first time ...... Specially because we're not exposed to this in our everyday life. What we're gonna see over the next few examples is that as long as we keep in mind, this whole idea that two pi radians is equal to 360 degrees or pi radians is equal to 180 degrees, which is the two things that I do keep in my mind. We can always re derive these two things. You might say, "Hey how do I remember if its pi over 180 or 180 over pi?" To convert the two things, well just remember and which is hopefully intuitive that two pi radians = 360 degrees. And we'll work through a bunch of examples in the next video, to just make sure that we're used to converting one way or the other. ENDSs.