❖ Taylor / Maclaurin Series for Sin (x) ❖

📚 Maclaurin Series for sin(x) – Step-by-Step Example 📚

In this video, I show how to find the Maclaurin series expansion for the function f(x) = sin(x), centered at a = 0. This example walks through the process of calculating the necessary derivatives and using them to construct the series. The same process can also be applied to find the Maclaurin series for cos(x)!

🚀 What’s covered:

Understanding the Maclaurin series as a special case of the Taylor series when a = 0.
Step-by-step calculation of the derivatives of sin(x) at x = 0.
Constructing the series using the Maclaurin series formula and identifying the pattern for sine functions.
A quick note that a similar process works for cos(x) as well!
By the end of this video, you’ll understand how to calculate the Maclaurin series for both sin(x) and cos(x), essential concepts in calculus for approximating trigonometric functions near the origin.

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Hashtags:
#MaclaurinSeries #TaylorSeries #sinx #CosineSeries #Calculus #MathTutorial #Trigonometry #SeriesExpansion #Derivatives #MathHelp #STEM #Mathematics #LearningCalculus #SeriesMath

[patrickjmt]

Hi all! Wanna help a Youtube education OG? Please post comments, questions and anything else on your mind in the comment section! so, don’t forget to LIKE, THUMBS UP, and SUBSCRIBE! I’d appreciate it greatly as it helps me :)

[sardarhedayati]

You can also use the exponent 2n+1 as long as the series begins at 0.

[ClarkPianoManKent]

I really enjoy the nice pace you go at, not too fast and not too slow. Thank you so much for great help.

[AustinDoggie]

Ohhhh that feeling when math finally makes sense.... I LOVE YOU!! I have learned more from you than from the whole quarter in my math class! i wish I had known about you earlier!

[fawwazmunir]

Bro u saved my life, got a perfect score on my second midterm with this example

[Juniorchi]

I just watched all three of your video's about finding taylor/maclaurin series, and i just have to say thank you so much! I spent weeks trying to understand it but couldn't, and just by watching your video's it's now all so clear to me :). I have a test in 2 hours, but thanks to you i'm now at ease. Keep up the good work!

[WhoArrrrghYou]

You made me love math even more than I already did before. I'm literally never bored when studying new math topics because of your videos! Thnx

[schoolissoboring]

Dear Patrick,
You save me so many times. My professor loves to derive and I never understand how to do any of the problems he assign. You help me through it countless times. Thank you.
Sincerely,
1 in hundreds of thousands of students you helped. :)

[linkny]

It depends entirely on where your summation notation begins. In patrickJMT's example, we start at n=1 going to infinity. If you started at n=0 going to infinity, then the summation notation would change to sum n=0 to infinity of (-1)^(n) * (x^(2n+1))/(2n+1)!. Your notations will shift depending on what n is equal to.

[nairameam]

Your channel and the videos in it are the reasons why I didn't drop my Math55 subject. Thank you so much. you're way better than my professors here in terms of making me understand the lessons.

[ricola]

Wow weeks of trying to understand this in class.. it's all so simple when explained in this way! I wish all teachers would explain things simply in this way! Thank you so much, definitely going to help on my final tomorrow.

[phdan]

Hey Patrick, I just want to say thanks for making these awesome videos, and making them all for free. That being said, I have a suggestion. Maybe you could make a video on Newton's cooling law. I see it a lot in first year calc, and I bet a lot of people could benefit from it.

Thanks again!

[MakTony]

I cannot thank you enough, my final is tomorrow and I'm ready now.

[patrickjmt]

@PANIONIOSG3 well, when using finitely many terms, there is an error; you also can to justify that the function does have a taylor series expansion (that is, for which values of x it is valid).
yes, my point in this was simply to show the series expansion, assuming one exists.

[kyavvall]

You rock man! After trying to view solutions and other tutorials to understand the material, I watched the first four minutes of this video and have a much better understanding. 

[Virusruneblazer]

Profs have their heads in the clouds with their research that they cannot explain things without going through way too much detail. Great vids, they help alot

[Kajehart]

Patrick, you so rock! My Calc 2 instructor lectured on writing the sigma series the other night and I was so lost! My notes had f(x)=sin3x and he used n=0 and 2n+1 but I couldn't understand how he was coming up with the terms! Thanks so much!

[turtleenaitor]

What type of person would dislike this?? Guaranteed this video has helped countless people like myself.

[BLUFFIRL]

Had to review this for Differential Equations, great video!

[thearm]

I believe it can be either 2n+1 or 2n-1 because the point of the these expressions is to generate an odd number so that the Taylor series only outputs x terms to an odd integer and terms with odd factorials.