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Current time:0:00Total duration:16:22

Video transcript

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 and radians and then think about whether PI is necessarily the best number to be paying attention to so let's think of a little bit about what I just said so pi we know is defined and I'll write defined as an as a triple equal sign I guess you could call it that way pi is defined as the ratio of a 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 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 really this definition or this equation by 2 times the radius and you get 2 times the radius times pi is equal to the circumference or more familiarly familiar li it would be it would be circumference is equal to 2 pi 2 pi r this is one of those fundamental things that you learn early on in your career and and you use it to find find circumference is usually or figure out radiuses if you know circumference and from that comes how we measure how we measure our angles in radians once we get to trigonometry class and just as a review here so let me draw myself a circle let me draw myself a better circle so there is my well it's 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 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 rate well the way I like to think about it is the angle is in radians and the arc length itself is in radiuses which will you know which isn't really a word but that's how I think about how many how many radiuses is this arc length that subtends the angle in radians so let me tell you what 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 sub 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 that this is 1/4 of the circle so it's going to be 2 pi R over 4 so this arc length over here this arc length 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 two radiuses radius this is this is one of those you know not not a real word but that's how I like to think about it or you could say it sub 10 it subtends an angle of PI over 2 radians so over here paid 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 is subtended by an arc of that has a length of how many radius I or I don't even know the plural of radius is actually I think it's radii but it's um it's fun to try to say radiuses radii actually me do that just those no one says that Sal you're teaching people the wrong plural form of radius R a radii so this arc length is PI over 2 radii and it sub it subtends an angle of PI over 2 radians we could do another one just just for the sake of making the point clear if you went all the way around the circle so 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 the arc length is the entire circumference of the circle it would be 2 PI R which is the same thing as 2 pi 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 two pi radians and so out of this comes out of comes all of the things that we know about how to how to graph trigonometric functions or at least how we measure the graph on the x-axis and 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 into all of that so if I think about our trigonometric functions remember if this was so on the trigonometric functions we assume we have a unit circle here so on 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 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 sine of theta is the y-value of that point sine of theta let me make that clear cosine of theta is the x value is the x value sine of theta is the Y value and so if you were to graph 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 so and we tend to label so let's do sine when the angle is zero sine of theta is zero let me draw the x and y axis just so you remember this this is the y axis y axis and this is the x axis this right here is the x axis so when the angle is zero we're right here on the unit circle the Y value there is zero so sine of theta is going to be right like that so let me 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 Y is equal to sine of theta in this graph that I'm drawing over here and then we could do well I'll just do the simple points here then if we make the angle goat if we did 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 a unit circle has a radius 1 so when we get to PI over 2 so when theta is equal to PI over 2 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 theta is let me do this in any color I'll do an orange I've 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 could do this either it's 270 degrees or you could view this as as 3 PI over 2 3 PI over 2 radians 3 PI over 2 so this is in radians this right of this this this this axis so 3 PI over 2 radians sine of theta is it's the y coordinate on the unit circle right over here so it's going to be negative 1 so this is negative 1 negative 1 and then finally when you go all the way around the circle you've gone 2 pi radians you've run 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 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 we 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 there's 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 to create its you know it's just one of those mind-boggling forms but it sometimes looks even more mind-boggling when you put pie 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 I could add 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 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 aí pi 1 0 although for my aesthetic tastes 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 this bizarre thing would have equaled unity that would have been super duper profound to me it seems a little bit a little bit of a hack to add 1 to both sides 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 it well it's inspired by many people are on this movement now the Tao movement but these are kind of the the people that that 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 circuit the ratio of the circumference to the diameter of the 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 härtel 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 by they define tau naught as the ratio as of the circumference the diameter the ratio the circumference to two times the radius they say hey wouldn't it be natural to define some number that's the ratio of the circumference to the radius and as you see here this pi is just 1/2 times this over here right circumference over 2r that's the same thing as 1/2 times circumference over r so pi is just half of tau or another way to think about it is that pot tau is just 2 times pi or if you I'm sure you probably don't have this memorized because you're just like wait I've spent all my life memorizing PI but it's six point two eight three one eight five and keeps going on and on and on never repeating just like PI it's two times pi and so you're saying hey Sal you know this is you know pi has been around for you know for millennia really you know why mess with such a fundamental number especially when you just spend all of this time showing how profound it is and the argument that they 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 if you pay attention to two pi as opposed to pi or if you we should call it we pay if you pay attention to tau instead of tau over 2 what is this angle that we did in magenta what is this angle we did in magenta well first of all let's think about it 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 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 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 tau raid 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 towel is more intuitive here is because it immediately you don't have to do this weird conversion where you're saying oh you know divided by two multiplied by two all that yours 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 half way around that be tau over two radians if you go three fourths around that be three tau over four radians if you go all the way around that would be tau radians if someone tell you went if someone tells you that they have an angle of ten tau radians you'd go around the egg you go around exactly ten times so it'd be much more intuitive you wouldn't have to do this little mental math converting you know the multiple just saying do I multiply or divide by two when I convert to radians in terms of pi know when you do it in terms of tau radians it's just natural one revolution is one tau Radian so that makes and it makes a sine function over here instead of writing PI over two well PI over two you know when you look at a graph like this like where was this on the unit circle was this one fourth around the circle was this was this 1/2 and this is actually one fourth around the circle you're right over here but now it becomes obviously if you write it in tau PI pie nut pie top PI over two is the same thing as tau over four pi is the same thing as tau over 2 3 PI over 2 is 3 PI I'm sorry 3 tau over 4 3 fourths tau 3 fourths 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 three fourths the way around the unit circle and then you're all the way around the unit circle and so the last thing that the I think the strong PI defenders would say is well look Sal you just pointed out one of the most beautiful 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 II 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 we tau is the Tau radians means we've gone all the way around the unit circle so cosine of tau remember we're just 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 sine of tau is equal to 0 so e to the I tau e to the I tau is equal to 1 and I'll leave it up to you to decide which one is seems to be more aesthetically profound