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Current time:0:00Total duration:6:08

Fourier coefficients for cosine terms

Video transcript

so we've been spending some time now thinking about the idea of a Fourier series taking a periodic function and representing it as the sum of weighted cosines and sines and some of you might say well how is this constant a weighted cosine or sine well you could view a sub 0 as a sub 0 times cosine of 0 T and of course cosine of 0 T is just 1 so just it'll just end up being a sub 0 but that could be a weighted cosine if you want to view it that way and in the last video we started leveraging some of the integrals definite integrals of sines and cosines to establish a formula for a sub 0 which intuitively ended up being the average value of our function over over 1 of its period or over 0 to 2 pi which is the interval that we are caring about what I want to do now is find a general expression for a sub n where n is not equal to 0 so for n is greater than 0 for n is an integer greater than 0 and I'm going to use a very similar a very similar technique to figure out a sub 0 I just multiplied just took the integral of both sides what I'm going to do now is I'm going to multiply both sides of this equation by cosine of n T so let me do that so on the left hand side I'm going to multiply by cosine of n T and on the right hand side multiplying it by cosine NT if I distribute that cosine of n T I'm going to multiply every term by cosine NT so it's going to be cosine of n T cosine of n T you might see where this is going especially because we took all that trouble to take to figure out the properties of definite integrals cosine of n T cosine of n T cosine of NT are the properties of definite integrals of products of cosines and sines cosine of n T and this is going to be true for all of the terms there's an infinite number of terms here that's what that dot dot dot is represent we're just going to keep on going on and on and on and now let's take the definite integral of both sides from 0 to 2pi so the definite integral from 0 to 2pi DT well that's going to be the integral of this from 0 to 2pi and actually let me just take that coefficient out of the integral so 0 2 pi DT 0 to 2 pi DT a little bit monotonous but it's worth it 0 to 2 pi DT 0 to 2 PI DT we might as well have fun while we do it 0 to 2 pi DT alright 0 my hand is hurting 0 to 2 pi DT 0 to 2 pi DT once again just integration property I'm taking the integral from 0 the definite go from 0 to PI of both sides and I'm just saying hey the integral of this infinite sum is equal to the infinite sum of the different integrals now what is this going to be equal to well we know we know from before that if I just take the definite go from 0 to 2 pi of cosine n T where n is some nonzero integer this is just going to be 0 so this whole thing is going to be 0 we also know and we've established it before if we take so I just used this property we also know if we take cosine NT times sine of M T for any integers M T over the interval from 0 to 2 pi that's going to be 0 and we also know we also know that if we take the integral of cosine M T cosine NT where M does not equal n where m does not equal n that that is going to be 0 and so we know well here we have we're assuming that we're assuming here that n is not equal to 1 the coefficient here so that is going to be equal to 0 this is sines and cosines with different coefficients and frankly even if they had the same coefficient that's going to be 0 we established that in the last few videos this is going to be 0 same argument this is going to be 0 this is going to be 0 everything's going to be 0 except for this thing right over here but what is this thing this is the definite that what we're taking the integral of this is the same thing as cosine squared of n T DT the definitely go from 0 to 2 pi of cosine squared NT DT well what is that well we established that for any M not nonzero M this is going to be equal to PI so all our work is paying off so this whole integral this whole integral right over here is going to evaluate to PI we do that in another video and so now we can start solving for a sub n we know that a sub n times pi a sub n times pi times pi is equal to is equal to this because everything else ended up being 0 which is very nice it's equal to the definite integral from 0 to 2pi of F of T of F of T times cosine and T cosine of n T DT cosine NT DT and so we can now solve for a sub n we just divide both sides by we just divide both sides by PI and we get a sub n is equal to I think we deserve a little bit of a drumroll a sub n is equal to 1 over pi times all of this business times the definite integral from 0 to 2pi f of t f of t cosine NT cosine n T DT D T so if you want to find the nth coefficient for one of our cosines a sub n well you take your function multiply it times cosine of n T and then take that definite integral from 0 to 2pi and then divide by PI so pretty intuitive and what's cool is that the math kind of works out this way that you can actually do this so hopefully you enjoyed that as much as I did