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Current time:0:00Total duration:7:15

Video transcript

we've been doing several videos now to establish a bunch of truths of definite integrals of various combinations of trigonometric functions so that we will have a really strong mathematical basis for actually finding the Fourier coefficients and I think we only have one more video to go in the last video we said hey if you take any combinations of sines where m and n are integers that either don't equal each other I don't equal the negative each other you're going to get that integral is going to be equal to 0 and that if they did equal to each other well it's just going to be the same thing as sine squared of some multiple of T and that that actually over the interval from 0 to 2pi is going to be equal to pi and just to be clear I wasn't as clear as I should have been in the last video this is going to be true where m is a non-zero integer if M was 0 then the inside of this integral would just would just simplify to 0 and then the integral would be 0 so M has to be a non-zero integer for this right over here to be true now what we want to do in this video is contain thing we did in the last video but now do it for cosines that the product of two cosines where m and n are different integers or they're not the negative of each other that that's going to be 0 but if they are the same integer and they are not 0 so that will boil down to cosine squared of M T that that is going to be equal this definite integral is going to be equal to pi now we're going to do it the same way that we did it with the sines we are going to use some of our trigonometric some of our trigonometric identities and so let's rewrite let's rewrite this right over here what we're take trying to take the integral of and so this is going to be the integral from 0 to 2pi so cosine M T times cosine n T using a product to some trig identity and if this is unfamiliar you can review it on Khan Academy that is going to be 1/2 times cosine of the difference of M t minus NT so I could write that as M minus n T M minus n t plus ko sine plus cosine of MT plus NT which I could write is M plus n T DT D DT so let's think about two situations let's think about the first situation when let me go do the DT in blue so DT so when actually let me let me now use some integration properties to expand this out a little bit this is going to be equal to so I'm gonna write this is two different integrals so one integral from 0 to 2pi and I'm going to put the DT right over here and then have an utter integral from 0 to 2pi and I'm going to throw that DT out here and so just using some integration property is going to be 1/2 times this integral of cosine of M minus n T DT and then plus I'm just distributing the 1/2 and using some integration properties one half and now this integral is going to be cosine of M plus n m plus and T DT now let's think about it when when M and n are integers that don't equal each other don't equal they're a negatives so let's think about M not equaling N or M not equaling negative end and we're always assuming that these things are going to be integers m and n well in that situation this right over here is going to be a non zero integer and this right over here is going to be a non zero integer and we've already established we've already established that if you have a non zero coefficient here that this definite integral is going to be equal to zero the definitely go from zero to 2pi of cosine of some nonzero integer times te DT well that's exactly that's exactly what both of these integrals are this is the integral from 0 to PI of cosine times some nonzero integer T or nonzero integer times T DT so in this case where m and n are integers that don't equal each other don't equal the negatives of each other both of these integrals are going to be 0 and then you multiply that times 1/2 1/2 times 0 0 1/2 times 0 0 it's all going to end up being a zero so that should so that should hopefully make you pretty feel pretty good about the case this first case and now let's think about the second case where m is a nonzero integer or we could say where m is equal to n so in that situation N and M are the same and they are not equal to 0 so let's just take that situation especially because when we're looking at 48 coefficients we care about the non-negative coefficients at least the way that we've defined it so let's just assume that M is equal to n and that there and that M is not equal to 0 well and that would just resolve that would take this integral and turn it into that integral well in that situation what's going to happen well this first integral right over here if M is equal to n and M is not equal to 0 well it's going to be M minus M you're going to get this is going to be 0 T so this whole thing is going to simplify to 1 and then this right over here going to have M plus M that's going to simplify to 2 m and so let's rewrite the integrals here this is going to be equal to 1/2 times the definite integral from 0 to 2pi 0 to 2 pi of 1 times I'll write that one here 1 DT 1 DT plus 1/2 plus 1/2 that's a new color times the integral from 0 to 2pi of cosine let's let me do it in that same of cosine of 2m T DT to MT D DT DT so once again we're assuming M is not equal to 0 this is the definite integral from 0 to 2pi of cosine times some non-zero coefficient times T well once again we've established multiple times that this is going to be 0 so this whole second term is going to be 0 and this first one is going to be equal to let's go to neutral color it's going to be 1/2 times the antiderivative of 1 the well that's just T evaluated from 0 to 2 pi so that's going to be equal to 1/2 times 2 pi minus 0 2 pi minus 0 well that's just 1/2 times 2 pi which is equal to pi and so we have now established this one as well and now we have a full toolkit we now have a full toolkit for evaluating the Fourier coefficients which we will now do in the next video which is very exciting