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# Maclaurin series of sin(x)

AP.CALC:
LIM‑8 (EU)
,
LIM‑8.E (LO)
,
LIM‑8.E.1 (EK)
,
LIM‑8.F (LO)
,
LIM‑8.F.2 (EK)

## Video transcript

in the last video we took the Maclaurin series of cosine of X we approximated it using this polynomial and we saw this pretty interesting pattern let's see if we can find a similar pattern if we try to approximate sine of X using a Maclaurin series and once again I Maclaurin series is really the same thing as a Taylor series where we are centering our approximation around X is equal to 0 so it's just a special case of a Taylor series so let's take f of X in this situation to be equal to sine of X f of X is now equal to sine of X and let's do the same thing that we did with cosine of X let's just take the different derivatives of sine of X really fast so if you have the first derivative of a sine of X is just cosine of X the second derivative of sine of X is derivative of cosine of X which is negative sine of X the third derivative is going to be the derivative of this so I'll just write a 3 in parenthesis there instead of doing all the prime prime prime so third derivative is derivative of this which is negative cosine of X the fourth derivative the fourth derivative is the derivative of this which is positive sine of X again so you see just like coasts of X it kind of cycles after you take the derivative enough time and we care in order to do the Maclaurin series we care about evaluating the function and each of these derivatives at X is equal to 0 so let's do that so for this let me do this in a different color not that same blue so I'll do it in this purple color so f that's hard to see I think so let's do this other blue color so f of 0 in this situation is 0 and F the first derivative evaluated at 0 is 1 cosine of 0 is 1 negative sine of of 0 is going to be 0 so f prime prime the second derivative evaluated 0 is 0 the third derivative evaluated at 0 is a negative 1 cosine of 0 is 1 you have a negative out there it is negative 1 and then the fourth derivative evaluated at zero is going to be zero again we could keep going but once again it seems like there's a pattern zero 1 zero negative 1 zero then you're going to go back to positive 1 so on and so forth so let's find it's polynomial representation using the Maclaurin series and just reminder this one up here this was approximately cosine of X and you'll get closer and closer to cosine of X I'm not rigorously showing you how close and then it's definitely the exact same thing as cosine of X but you get closer and closer and closer to cosine of X as you keep adding terms here and if you go to infinity you're going to be pretty much at cosine of X now let's do the same thing for sine of X so I'll pick a new color the screen should be nice so this is our new P of X so this is approximately going to be sine of X as we add more and more terms and so the first term here f of 0 that's just going to be 0 so we're not even going to need to include that the next term is going to be F prime of 0 which is 1 times X so it's going to be X then the next term is f prime the second derivative at 0 which we see here is 0 let me scroll down a little bit it is 0 so we won't have the second term this third term right here the third derivative of sine of X evaluated at 0 is negative 1 so we're now going to have a negative 1 negative 1 let me scroll down so you can see this negative 1 this is negative 1 in this case times X to the third over 3 factorial so negative X to the third over 3 factorial and then the next term is going to be 0 because that's the fourth derivative the fourth that's the fourth derivative evaluated at 0 is the next coefficient we see that that is going to be 0 so it's going to drop off and what you're going to see here and actually maybe I haven't done enough terms for you for you to feel good about this let me do one more term right over here just so it becomes clear F of the fifth derivative of X is going to be cosine of X again the fifth derivative let me do it in that same color just so it's consistent the fifth derivative the fifth derivative evaluated at zero is going to be one so the fourth derivative evaluated zero zero then you go to the fifth derivative evaluated zero is going to be positive one and if I kept doing this it would be positive one time I have to write the 1 as a coefficient times X to the fifth over 5 factorial so there's something interesting going on here and for cosine of X I had 1 essentially 1 times X to the 0 then I don't have X to the first power I don't have X to the odd powers actually and then I just said essentially have X to all of the even powers and whatever power it is I'm dividing it by that factorial and then the signs keep switching and this is this is I shouldn't say this is an even power because you were really isn't well I guess you can view it as an even number because it's no I won't go into all of that but it's essentially 0 2 4 6 so on and so forth so this that this is interesting especially when you compare it to this this is all of the odd powers this is X to the first over 1 factorial I didn't write it here this is X to the third over 3 factorial plus X to the fifth over 5 factorial yeah 0 would be an even number anyway I don't that's almost my brain is in a different place right now and you could keep going if we kept this process up you would then keep switching signs X to the seventh over 7 factorial plus X to the ninth over 9 factorial so there's something interesting here you once again see this kind of this kind of complementary nature between sine and cosine here you see almost this you know they kind of they're filling each other's gaps over here cosine of X is all of the even powers of X divided by that powers factorial sine of X when you take it's polynomial representation is all of the odd powers of X divided by its factorial and you switch signs in the next video I'll try I'll do either the X and what's really fascinating is that e to the X starts to look like a little bit of a combination here but not quite and you really do get the combination when you involve imaginary numbers and that's when it starts to get really really mind-blowing