If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

Main content

# Euler's formula

## Video transcript

so in this video we're going to talk about Euler's formula and one of the things I want to start out with is why why do we want to talk about this rather odd looking formula what's what's the big deal about this and there is there is a big deal and the big deal is E we love E and I'll underline that twice now the reason is because when we take a derivative of e D DT of e to the x equals e to the X and D DT of e to the ax so we're a is anything equals a e to the ax and so the property is that when you take a derivative of the function the same function comes out or if you take a derivative of the function a scaled version of the same function comes out and we love this because why because when we do differential equations e to the X is the solution almost every time whenever we did a circuit e to the X was the answer if you recall from when we word solving circuits simple circuits with differential equations that we always said something like well we're going to guess that V of T is some constant times e to the St that was a proposed solution this turned out to work every time so there's something else we love too and that is sinusoidal cosines ok we love these and that gets two lines now why do we love these is because they happen in nature if you whistle the air pressure looks like a sine wave if you ring a bell the bell moves in a sine wave any kind of music if you look at the notes and music the sound they make the pressure waves look like sine waves and circuits make sine waves remember if we analyze this circuit in great detail was the l-c circuit we looked at the natural response of this and was a sine-wave okay so electric circuits make sideways all these things make sine waves they occur in nature and we want to be able to analyze things that happen when sine waves are present so we have two things we love and we want to relate these two things and these are going to be related through that Euler's formula that's how we connect these two separate ideas and let me let's go do that so Oilers formula says that e to the J x equals cosine X plus J times sine X though Sal has a really nice video where he actually proves that this is true and he does it by taking the Maclaurin egg series expansions of e and cosine and sine and showing that this expression is true by look by comparing those those series expansions now I'm not going to repeat that here we're just going to state that as fact and now we're going to look at this equation a little bit more so this is this is the expression that relates Exponential's that we love to sines and cosines that we love and part of the price of doing that is we introduce complex numbers into our world here's two complex numbers okay this is where complex numbers come in to electrical engineering so I have to mention the other form of this formula which is e to the I put a minus sign in here e to the minus J X and that equals cosine X minus J sine X so these two these two expressions together are Euler's formula or Euler's formulas and we're going to exploit this by taking we'll be able to take the cosines and sines that we find in nature we're going to be able to fashion them into Exponential's and these Exponential's then go into our differential equations and give us solutions and we're going to come back and pull out the cosines and sines that's the rhythm of how we're going to use this equation to help us solve solve circuits one small point I want to share notice that in both these equations the cosine comes first and the sine is over here on the imaginary side so the cosine is on the real side this is the reason that we have a preference in the future we're going to have a preference for talking about our real-world signals in terms of the cosine function it's because in this Euler's formula the cosine comes first in both cases so what I want to do now is take a take a second and I want to see if we had our our signal expressed in these Exponential's how do we recover the cosine and the sine term how do we how do we flip these equations around so we can solve for the cosine and the sine it's a simple bit of algebra here it's good to see all right so if I want to isolate the cosine term if I want to isolate the cosine term let me let me get rid of these guys here so now to isolate the cosine term what I'm going to do is add these two equations together and that plus and minus are going to cancel out this second term here that's what I'm going for so if I add I'll get e to the J X plus e to the minus J x equals cosine doubles two cosine X and the two sine terms cancel out right so then I can write I'll write it over here I can write cosine of X equals e to the plus J X plus e to the minus J x over 2 all right so that's the expression that's the expression for cosine in terms of complex Exponential's ok let's go back and see if we can get sine so what I'm going to do with sign is I'm going to subtract and that gives me what that'll do is it'll get the cosine terms to fall away and then I get e to the J X subtracting right minus e to the minus J x equals cosine terms I'm going they subtract out and I get two times J times sine X all right and that means I can write sine x equals e to the plus J X minus e to the minus J x over two J and it's real easy don't that J term is down there that's easy to forget sometimes so they put a square around these guys because that's important so there's the two expressions that's the two expressions for if you have complex Exponential's and you want to extract the cosine this is how you do it if you want to extract the sine this is how you do it okay and you can either you know you can put these in your head or probably easier for me if you just remember Oilers formula this is a pretty straightforward quick derivation so the other thing we put a square around is this guy here so we have Euler's formula and basically the cosine and sine extracted from Euler's formula