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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 this playlist where if you had an angle like that you might call that a 30 degree 30 degree angle if you have an angle an angle like this you could call that a 90 degree angle and we'd often use this symbol just like that if you were to go 180 degrees you would essentially form you'd essentially form a straight line let make these proper angles if you go 360 degrees if you go 360 degrees you have essentially done one full rotation and if you watch figure skating on the Olympics and someone does rotation they'll say oh they did a 360 or especially in some type of some skateboarding competitions and things like that but the one thing to realize and it might not be obvious right 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 you're said well why do we call a full rotation 360 degrees and there are some possible theories and I encourage you to think about them why why does 360 degrees show up in our culture as a full rotation well there's a couple of theories there one is ancient calendars and even our calendars close to this but ancient calendars were based on 360 days in a year some ancient astronomers observed that things seemed to move one 360th of the sky or per day another theory is 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 they had 60 so in our system we like to divide things into 10 they probably like to divide things into 60 so if you were to if you had a circle and you divided it into about six equilateral triangles six equilateral triangles and each of those equilateral triangles you divide it into 60 sections because you have a base 60 number system then 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 as intuitive to you from the get-go in some ways is much more mathematically purer than degrees it's not based on these cultural artifacts of base 60 number systems or astronomical patterns to some degree a an alien on another planet would not use degrees especially if the degrees are motivated by these astronomical phenomena but they might use what we're going to define as a Radian there's certain a certain degree of purity here radians radians so let's just cut to the chase 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 the center of the circle and now let me draw this radius and let's say that this radius you might already notice the word radius is very close to the word radians and that's not a coincidence so let's say that this radius the circle has a radius of length R now let's construct an angle I'll call that angle theta so let's construct an angle theta so let's call this angle right over here theta and let's just say for the sake of argument that this angle is just the exact right right measure so that if you look at the arc that subtends this angle that seems like a very fancy word let me draw the angle so if you were to draw the angle so if you look at the arc that subtends the angle that's a fancy word that's really just talking about the arc the arc along the circle that intersects those two sides of the angles so this arc right over here sub tens angle this the angle theta so let me write that down sub tens this arc subtends sub tens angle theta let's say theta is the exact right size so that 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 are defining a new 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 in this angle to be well the most obvious one if you kind of view a radiant is another way of saying radius assistances or I guess radii well you say look this this is subtended by an arc of one radius so why don't we call 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 bit useful as we start to interpret more and more types of circles with with when you give a degrees you really you have to do a little bit of math and think about the circumference and all of that to think about how many radiuses are subtending that angle here the angle and radiance tells you exactly how many how many arc length that is subtending the angle so let's do a couple of thought experiments here so given that what would be what would be the angle in radians if we were to go so let me draw let me draw another circle here so let me draw another circle here so let me so that's the center and we'll start right over there so what would have happened if I had an angle what angle if I wanted to measure in radians what angle would this be in radians and you can almost think of it as radius sssss so what would that angle be going one full revolution and degrees that would be 360 degrees what would you well based on this definition what would this be in radians well let's think about the arc that subtends this angle the arc that subtends this angle is the entire circumference of this circle it's the entire circumference it's the entire circumference of this circle well what's the circumference of a circle in terms of radiuses isis's so if this has length R if the radius is length R what's the circumference of the circle in terms of R well we know that that's going to be 2 pi R 2 pi R so going back to this angle how many the what's the the length of the arc that subtends this angle is how many radiuses this is what's two pi radiuses this is it's two pi times R so this angle right over here this angle this I'll call this a different well it's call this angle X X in this case is going to be two pi radians and it is subtended by an arc length of two pi radiuses assistances if the radius was one unit then this would be two pi times one two pi radiuses so given that let's start to think about how we can convert between radians and degrees and vice versa if I were to have and we can just follow up over here if we do one full revolution that is two pi radians 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 is 360 well I could either write out the word degrees or I can use this little degree notation there actually let me write out the word degrees it might make things a little 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 two in which case we would get on the left hand side we would get pi radians would be equal to how many degrees would be equal to 180 degrees 180 degrees and I could write it that way or I could I could write it that way and you see over here this is 180 degrees and you also see if you were to draw a circle around here if you were to draw a circle around here we've gone half way around half way around the circle so the arc length or the arc that subtends the angle is half their circumference half the circumference are it's pi radiuses so we call this pi radians pi radians is 180 degrees and from this we can come up with conversions so one Radian would be how many degrees well to do that we would just have to divide 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 this isn't some voodoo so I'm just dividing both sides by PI here on the left hand side you're left with one and on the right hand side you're left with 180 over PI degrees so one Radian is equal to 180 over PI degrees which is starting to make it 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 let me rewrite this thing over here we said PI radians pi radians is equal to 180 180 degrees 180 degrees so now we want to think about one degree so let's solve for one degree one degree we can divide both sides by 180 we are left with PI over 180 radians is equal to one degree so PI over 180 radians radians is equal to one degree this might seem confusing and daunting and it was for me the first time I was exposed to especially because we're not exposed to this in our everyday life but we're going to 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 that PI radians is equal to 180 degrees which is the two things that I do keep in my mind we can always read arrive these two things you might say hey how do I remember if it's PI over 180 or 180 over PI to convert the two things we'll just remember which is hopefully intuitive that two pi radians is equal to 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