Calculus 2 Lecture 6.5: Calculus of Inverse Trigonometric Functions

Calculus 2 Lecture 6.5: Calculus of Inverse Trigonometric Functions

[@huiyichen5863]

I used to excel in math, but because of some tough times I had to pull myself out of school for 2 years to work. I forgot everything I knew about math and having to take calc 2 without any refresher, I thought I was completely hopeless. The professor I have just dived right into calc 2 as if all the students took calc 1 with him. Well I didn't. Not being able to understand anything that was written on the board, I wanted to cry. But here you are, an angel disguised as a math professor. All hopes are not lost. :) Thank you.

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+Hui Yi Chen You're asian, youll be ok

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Lol, the stereotype is real

[@cocoa1996]

+Dan Luckman I wish youtube dislikes did something

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Gosh guys... I'm pretty sure he was joking... Its not the end of the world... Its not even a harmful joke...

[@FlamingFlareon]

how did it go? did you succeed?

[@darthboganis1356]

Requirements for a function to have an inverse 0:00 > Trig functions generally aren't 1:1. But restricting the domain allows us to treat them as inverse functions y = sin (x) 4:00 > What does the inverse do? 5:30 y = cos (x) 10:20 y= tan (x) 14:15 y = csc (x) 17:00 y = sec (x) 22:50 y = cot (x) 27:30 w/ Examples 30:00 Compound trig functions 38:00 Properties 46:00 Inappropriate Domain issue 50:00 Derivatives of Inverse trig functions 52:30 Proofs 56:00 cosine inverse [implicit just means chain rule] > using an identity 58:30 [cosine y] > using triangles 1:01:00 [secant y] w/ Examples 1:07:30 Harder problem 1:10:30 Even harder problem 1:16:00 Harder still 1:22:15 Sidebar: using triangles solve cos-1 (x^2) 1:26:00 > 'some fancy maths' 1:30:05 Integrals of inverse trig functions 1:32:20 w/ Examples 1:35:30 Third example rewriting the function into the right form 1:43:00 > In this example the function was rewrite in the 1/1+u^2 form so that it could be integrated using the tables. Important to consider algebraic manipulation, substitution etc... look for patterns. Last example 1:47:00 [using substitution twice]

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hope you have an amazing day and god bless

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danke

[@MrCompenso]

If you could do videos on physics, I think my life would be complete.

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Grayfox772 ilectureonline This guy is the physics version of Prof. Leonard

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NOT IN THE CASE OF BODY STRUCTURE OF COURSE! LOL

[@InvinciblePepe]

Grayfox772 Hell yeah

[@petroslazanis8466]

@@madalincalamanciuc6656 "welcome to ilectureonline" Van Biezen is a God. (almost as Godly as Leonard)

[@rishitsingh2300]

Trust me you are going to love Walter Lewin for physics then

[@SilverArro]

I can't believe I just watched 2 hours worth of Calculus lecture in my spare time. It's truly a testament to how inspiring you are as a professor. Thanks for all the hard work you've put in to making these videos! :D

[@JK-ou8iz]

+Arlo haha literally his lecture is addictive.

[@Peter_1986]

@Jessica Pennington A lot of math teachers have this annoying tendency to be super-formal and super-serious, and obsessed with the most general forms possible of everything so that the actual intuition is lost.

[@adityaekbote8498]

hello fellow mathematician

[@Maladringe666]

His videos should be more popular, this guy really knows how to teach his stuff. I wish my college professor was as good as him :(

[@tenzinkunsang1432]

Professor Leonard teaches math "as Raphael painted pictures, Michelangelo carved marble, Beethoven composed music, or as Shakespeare wrote poetry." In short, thank you for being one of the best educator!

[@chick3n5kin]

Your a gift from god. My Calc 2 teacher is a fool without these videos I would 100% be failing.

[@Pompadour123]

skip to 51:39 for the calculus part and you already know the basic trig

[@honeyraj9302]

You know he is a great teacher when I'm here studying watching his videos, 8 years after he posted this.

[@mr.jceniza]

I understand you a lot better than my professor. I come to his class and take some notes, but ultimately, right after class, It's with you that I learned! THANKS A LOT! Not just an inspiration and a role model mentally, but physically as well! Who in earth could have a nice physique while teaching math? hahhaha

[@sskylark_]

This video is 5 years old today, and I would just like to say that I am thankful for your work. Your videos are crucial to my math education in college. Thank you Professor!

[@Jono.]

Covered the exact same content that I had in my first class except this time my head didn't explode. I'll be using your videos all semester and I hope you know what a great service you are doing for everyone!

[@MrMegaHamSandwich]

I'm going to show this to my grandchildren one day.

[@TheCrushenator28]

I wanted to state what a fantastic professor you are. My current professor seems to gloss over topics without ever going over edge cases or techniques to use for solving more difficult problems. You have saved me from a very difficult situation. These lessons are excellent! Very clear, and we'll explained

[@Sniclepants]

Thank you so much for posting all of these. Excellent teacher.

[@sandlertossone1813]

I totally love your simplicity as a teacher. The inverse trig intergration method is way better than the a and u method where you have to memorize two separate formulas for each. Thanks

[@user-so1zn3gc7s]

I have to say thank you so much you really have helped me so much with eng maths.. honestly can't say it enough thank you. Your teaching style is amazing.

[@nishalliniarasu9409]

He`s a legend!!!!Finally i understand inverse function!!wow!!you`re really have a lot of positive vibes!!thanks a lot!!

[@fatimazaghab5855]

Thank you so much Professor Leonard! You are awesome.

[@Phineastutti]

you have absolutely no idea of how much the trig lecture helped me! thank you so so much

[@mr.obiachrispus3076]

Am really glad because the presentation was marvelous and I wish you could also do it for other subjects like Chemistry.

[@troyblast3941]

hello prof. i am from Philippines ive always watched your tutorial video..'' and it make sense to me''...i learned a lot from you honestly.

[@Ireemo1]

keep on uploading videos ur a superman disguised as a math teacher

[@rapidsfire1]

Literally a life saving video

[@abdullahmohammad5613]

This lesson isn't hard at all as long as you stick to what has said and memorized the graphs and the domains of the functions that he just drew :) Thanks again Professor Love

[@cssfriendly]

You are the greatest teacher i ever had till now. Your classes are even better than those of Khan Academy!! You are the true saviour of The CalculusVerse.

[@Cam._S]

Derivatives start at 1:07:28 Integrals start at 1:32:17

[@marieb.1372]

I have a calc 2 exam tomorrow. I barely did any hw because the wow xpac release. I realized this morning that I am a fool as I sat down to "study." Now, after finding this video I feel there still may be hope. And yes, I think I have learned my lesson.

[@marianpalko2531]

Summer B. Did you pass?

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Summer B. Rip

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You're awesome and very talented, wish my prof taught like this!!

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Dude thanks for saving my academic year!! I salute you

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Such a life saver...

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I’ve sat through your classes practicing problems on my whiteboard. Thank you for these

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This teacher teaches stuff better than my profrssor. I did not even know that i have spend 4 and half hours doing these. Until i finishespd two videos

[@hufflepuffphoenix2898]

math final in 3 days, will be binge watching! great explanations!

[@Stantonesque]

Thank you! My instructor put up the triangle, and I had zero clue where he got it from! Thank you thank you thank you!!!

[@heatherhewett8973]

THANK YOU!!!! Especially for showing the proofs. I feel like I finally understand!! :)

[@gxglee5616]

i like how a flick of a button can bring captain america in a Calculus class, u are awesum thank u

[@chemicalengineeringbd7614]

You're the mathematical legend of the world 🙏🏿👍🏿

[@juanortegon7838]

Extremely good professor.

[@alejandrodiaztamayo6221]

WOW. You are amazing man. Thank you so much for these videos.

[@henryponnefz1419]

What a Great Teacher!

[@hardguy_gh1929]

This is an appreciation post..if you found yourself here... u wont regret it!! stick to him... after 2 Fs in calculus i got a C+ after watching his videos... thankyou!! ❤❤❤❤❤❤❤❤❤

[@amirahsaifuladli9350]

Thank you so muchhh for teaching me calculus...it means a lot.

[@radhikamadan376]

Sir , a video providing *detailed explanation* on applying *properties of inverse trigonometric functions* to solve equations would be really helpful. Thank you.

[@georgesadler7830]

Professor Leonard, thank you for an awesome explanation and analysis of the Inverse Trigonometric Functions in Calculus Two. The graphs and derivation are important in understanding Inverse Trigonometric Functions. Instead of memorizing these important Transcendental Functions, students should derive them from scratch with the aid of Trig Identities and the Right Triangle. There are no errors in this KZbin video.

[@xndndndjdxnxj6175]

my saviour !!! thank u sm

[@flechiertutoringservices2949]

Job well done professor!

[@fatimamahmoud519]

Thank you very much ❤️❤️❤️you’re the best

[@ROZAY305BOSS]

Amazing stuff yet again!

[@statelessturk]

ringing phones make him crazy but u r the best Mr. Leonard

[@ppxccw]

Prof: "We'll start off easy and gradually build up to where its harder" Student: "You mean more fun?" 🤣🤣🤣

[@abdullahmohammad5613]

An easy way of memorizing the inverses of CSC and SEC is comparing them to the derivatives of cos and sin and appyling the changes which are : we change the signs of the outside and the inside which is the sign under the root and don't forget to multiply x in the denominator

[@celinevazquez3299]

dude YOU ARE THE BEST

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BIG THUMB UP !!!!!!

[@martinchishoko9252]

Really impresive

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I just stumbled upon ur channel and holy HELL who do I pay to have you teach me math lol

[@felido2040]

great video

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You are a very fine teacher!

[@s.o.k.1393]

If I pass calculus it is ENTIRELY because of you

[@suhadandis9383]

Thank you generous man

[@khamiskholafa8817]

you are legend !!!!!

[@Amandaleenyc]

i'm not sure why I'm learning this in calculus 1 but regardless,i'm happy this video is here!!!!!!

[@sandlertossone1813]

I have a question on your last problem. I put it in my tnspire cas cx and get arctan sqrt(9x^2-1). Is there another way of doing these and if so, do you have a video on how to do it? Btw I want to thank you for unknowing helping me get through this first year of Calculus. I was going to give up half way through until I found your videos and turned my grade completely around. Next week is the final. Thanks

[@teddy_bear0018]

Before the y=cot(csc) prof. Leonard goes what do you see someone in the room goes “2chainz” i literally laughed out loud😂

[@berendmichielsen3842]

Show hands if you actually show hands when he asks you to show hands.

[@hawadrammeh8870]

For those who know your pre calc and trig well, the video starts at 51.35

[@Aa11qq]

When your math teacher doesn't speak English and you have to get KZbin to teach you smh.

[@Lina-kl7hj]

my prof is left handed and stands in front of his work while hes writing and explaining it so i can't see anything and thus dont understand anything so i feeeeeeeel this

[@Razzybeets]

LMAO SAME

[@sarahlolo2100]

SAME

[@tawneeens436]

literally same omg

[@lubna7639]

felt that when i first been taught math in english in college lol

[@The1Goby]

Hey Professor Leonard, thank you so much for having these videos up! I'm starting to understand Calc 2 a lot more than I used to thanks to you! I have one quick question on derivatives of inverse trig functions. Does the chain rule still apply if there are no x's [Ex: sin^-1 (1/2)]?

[@ebob0531]

I know this is pretty late, but yes you still apply the chain rule in this instance. the derivative of your example ends up being 0 though, as it is a constant

[@joblong9605]

Professor, for your proof in the inverse cosine, could you please explain how you get positive square root of (1-x^2), but not the negative? Is it because the inverse cosine has negative gradient?

[@sorinpanciuc5712]

for the exercice at 1:22:30 => is it possible to do it otherwise ? I did it by setting COT(....) = (Y) into COT^-1 (Y) = (....) and taking the derivative with respect to x of both sides. We will then have an implicit derivative on the left side and a normal derivative on the right side. We will use the formulas on the board for both sides and we solve for dy/dx and then replace Y by COT(....). This way I got (dy/dx)=[2x(COT^2(COS^-1(x^2))+1)]/[sqrt(1-x^4)]. It may seem different but it's the same derivative function as the one found in the video but just wrote in another way....

[@angelamarie1771]

I was trying to understand around the time 34:46 the numbers you were saying on the unit circle. Tell me where u have a video on an explanation on that

[@greense65]

Unit circles are used to derived the trig functions, and so are discussed in a trigonometry class or in the trigonometry section of a "Precalculus" class. Unfortunately, Prof. Leonard has not posted videos of him teaching the Precalculus class.

[@monnashady1678]

Still 2 years later, Professor Leonard saves me again.

[@KM-ww8gz]

I wish you were my professor i love you thank you pray for me on finals I am currently failing calc 2

[@tanyafitchett7480]

great one

[@drewallison7191]

At 38:00 when you talk about inverse COS, would you not have you calculator in degrees for COS(.6)? You said radians and im pretty sure you would use degree.

[@Aiden-yx9ht]

went over this in calc 1 for maybe 40 minutes. wow....

[@ShivamYadav-ys3bp]

Thank you sir thanks🙏🙏🙏🙏🙏🙏🌹🌹🌹🌹🌹🌹❤🌹

[@pain8117]

Thank you buff math Chud. Very cool

[@axelgonzalez5963]

So, is this the only video he covers inverse trigonometric functions with differentiation & integration?

[@yoshida_chad2393]

the feeling when you solve the hard examples before professor do😇

[@theirreghoular8435]

About the derivative of the inverse trigonometric function of sinx (sin^-1 x) there is another proof (I guess): So the derivative of a an inverse function is equal to: (f^-1)' = 1 / (( f' o f^-1)) = 1 / ((sin)' o (sin^-1))(x) = 1/(cos(sin^-1(x)) = 1/(sqrt(1 - sin^2(sin^-1x)) = 1/(sqrt(1-x^2)) Please correct me if i'm wrong

[@michaelpiotto4988]

why is there no absolute on the (1-x^4) in the last derivative example?

[@geeo93]

Do you have a video on work using integrals

[@AmitKumar-ho3mr]

how to find the value of sine inverse of sine sixty where sixty is in radian?

[@ThePerfect1077]

are the inverse trig differentials only for radians?

[@keotshepilemandona9882]

I CANT BELIEVE THAT I FAILED CALCULUS 1 ........ IF IVE WATCHED THIS PROFESSOR FROM THE BEGINING I WAS GOING TO SCORE BETTER GRADES

[@zeldwaugh5703]

Does anyone know where the actual lessons are located? If it's close to me I'mma head my ass over. I've been wanting to finish my math forever but actually want to have the credits. I'm done highschool and wanna do academic upgrading but most places around me have horrible teachers.

[@jaridwilliams739]

if all my profs were like him id have a 4.0 in my major fr

[@anthonynunez5068]

for the graph y=cosx, wouldn't our range be [1, -1] since the graph begins at 0,1 and ends at pi, -1? He wrote the range as [-1,1]. Also, when he draws the inverse function of y=cosx, wouldnt the knew domain and range change? since the graph begins at [-1, pi] and ends at 1,0? So the domain should be [-1,1] and range is [pi,0]. Leonard wrote that the domain and range stayed the same, help

[@vladyslavkorenyak4093]

In 59:51 you didn't put the plus and minus in front of the root. I don't understood why :(

[@ladylonly2013]

?prof wher we can get extra exsercise about calculusbook

[@isobar5857]

At 1:09:53...isn't d/dx [5x]^2= d/dx [25x^2]=50x, by the chain rule ? And at 1:14:42 isn't the square of the square root [2T+1] just equal to 2T+1, and therefore it's derivative is equal to 2 ? I am confused !

[@tuxfighter]

it’s 5 because we calc just the differential of 5x

[@clarkwingfield1768]

hero

[@ieatpaintchips72]

Thanks for saving my butt. My Calc II teacher is so manic and fast talking. I can't understand a thing in that class. Here's hoping I don't flunk the midterm.

[@ProfessorLeonard]

You can do it!!

[@jakubpuchatek3029]

I love u papa leonard

[@alytechnology]

He is very good at teaching. If you want to understand better take notes and try to solve examples before he does. And I think in 1:15 he missed a little square power.

[@srinivasans5748]

fabously taught i love the way u enjoy and teach maths u r the best u r my mentor

[@robloxtv6937]

I love you

[@xrisku]

~ 13:00, that graph of y=cos^-1 x cannot be correct. something seems wrong. opinions anyone?

[@raymondxu8961]

how would you graph manipulated inverse trig functions like f(x) = 2arccos(x-2)?

[@greense65]

This is an old question and not a calculus one. But, this looks like the regular arccos graph shifted 2 units to the right and vertically stretched by a factor of 2.