This might be one of the prettiest proofs in mathematics. It's simple enough for most people to understand, but tricky enough for you to think about for a moment. That was an immaculate presentation, Sal! Godspeed.
Thank you so much Sal! all my book does is state both the sandwich theorem and use that as proof to formally state that therefore lim sinx/x is 1. All my professor did was repeat that in class -___-. for the past week i've just been sitting here like, "i only know this equals 1 because you told me, but WHY does it equal 1!?" It's been frustrating me and this video made so much sense.
Khan academy. Greetings to you from Morocco. You have given me a helping hand, because in our school they are only applying the rule and you gave me proof and thank you very much
Sal, this is why I love you. I'm watching this 5 years after it was uploaded, and every time I was confused between steps, you clarified yourself. You made it seem so SIMPLE. Why couldn't my textbook be like this?
Omg.!! I am attracted.. your videos are like magnet.!! I am watching them since 3 hours continuously like some movie series, and at any point of time I didn't lost my curiosity and interest.!! The are useful even after 12 years.!! True Genius.!! Thank you from the deep of my heart.!!
John Secor at least you now understand where the derivative of sinx how it comes, and from there you can derive other derivatives of trig functions, so its very useful
Thanks! You've squeezed a great deal out of this topic. I did feel pretty x-hausted by the end. Very clear, and informative. Presentation had a nice arc to it. I'll sin out for now. Again, VERY HELPFUL PRESENTATION!
Awesome! My calc teacher decided not to explain this because he said we wouldn't understand and said just to know that lim x->0 of (sinx)/x=1. This makes perfect sense! Thanks!
6:07 If like me, you had a little problem converting the angle x to x radian, they seem to be different. Just remember, when we say angle x we really mean a fraction of a complete circle, which means x of 360 (x/360) by adding the 360 fraction. Now it is easy to convert a fraction of 2piR to a fraction of 360.
@@BBBrasil Hey, can you help me? Still, I can’t understand. In sinx and tanx we are using x as degrees but when we are calculating the are of the sector we are using x as radians. I can’t understand. When we change x degrees to radians I get a completely different thing.
Since x >= 0 for all x in the 1st and 4th quadrants, you don't need the absolute values. You seem to have confused the 2nd and 4th quadrants.Also, your < & > need to be =. The squeeze theorem states that if f(x)≤g(x)≤h(x) for all numbers, and at some point x=k we have f(k)=h(k), then g(k) must also be equal to them.
The area of the 'pie' was derived using the definition of the angle in radians, so the proof of sinx/x = 1 as x->0 is only true when the angle is measured in radians. This limit is used importantly in definining the derivative of Sinx to be Cosx in calculus. This means that derivative also only holds if X is in radians. Power series is based on differentials of sin and cos, so again you must use radians. Everything comes back to how this limit was proved and affects all proof stemming from it.
@@pemifo260 Whether x is in degrees or radians, doesn't effect the way you write Sinx or Tanx, as long as you're clear on the units you're using. However it does make a difference as to what you say the derivative or integral of Sinx and Tanx are. The Derivative of Sinx is only Cosx when you are using x in radians. If using x in Degrees then the derivative of Sinx is instead Cos(x*pi/180).
This is a great explanation of the proof, but in my opinion, the proof is ultimately circular. (heh. circular.) The proof relies on the fact that the area of a disc is pi*r^2. This in turn relies on either integration (which requires a trig substitution and the fact that the cosine is the derivative of the sine, which relies on the fact that sinx/x approaches 1), or Archimedes's method of approximation by polygons, which also essentially relies on the fact that sinx/x approaches 1, or at least relies on an axiom that turns out to be equivalent to this fact. If you pick such an axiom, you can use it directly to prove the limit of sinx/x without having to pass from length to area. The area proof feels more satisfying, but that's ultimately because it hides all the axiomatic heavy lifting inside the formula pi*r^2.
I tried something similar. It was called the "Definition of an Equilateral Polygon" test. As x approaches infinity, Sin(pi/x) * x / cos(pi/x) --> Pi X = number of sides Cos = apothem of equilateral polygon Sin = equal length side divide by 2 of the equilateral polygon Sin(pi/x) * x = half of the perimeter As x approaches infinity, the perimeter starts to resemble that of a circle
Thanks! This is suppose to help my proficiency in mathematics and toward things I would use calculus and trig in! Its something my family never have used in any case of measuring.
this limit is very powerful in spite of its simplicity as its the key to prove that d/dx sin(x)=cos(x) and the other derivatives of trig functions follow. some people prove this limit using L'Hopital rule (LR) but that's wrong because LR requires the derivative of sin(x) which is proved based on the limit itself.
This gave the answer to my question...I wasn't able to prove that lim x -> 0 sin x/x = 1...it is one of these things about limits that I did not know why...LONG LIVE KHAN ACADEMY!!!!
@eileenBrain This isn't supposed to teach anything; it's supposed to be a proof. Which was done admirably. kudos, khanacademy, this was just what I was looking for.
Well, this approach is graphically illuminating but on the other hand laborious. Actually since lim(x→0) sinx/x becomes 0/0, the l'Hôpital's rule can be used, that is, to take the derivative of sinx and x separately. This results in cosx/1. Since cosx=1 at x=0, you get 1/1=1, which is the answer. Probably the squeeze method can be reserved for some extremely difficult ones that are worth the labor.
This may be a dumb question but can you just write sinx as is maclaurin expansion then the limit becomes 1-x^2/(3!) + x^4/(5!) - .... which obviously tends to 1. ( I will cover limits in a few weeks)
@ShneeBnee Sin is defined as opp/hyp where the hyp would be larger than 1 since it goes out of the circle so tan works here since it has 1 as the denominator
Thank you soooooo much. I have been looking sooo long for a good proof of this, and you have finally proved it to me. Thank you thank you thank you!!!!
If you have my attention span and don't mind a less rigorous proof: as x approaches 0, sin(x) gets closer and closer to 0. As x approaches 0, x has nearly the same value as sin(x). So as x = 0.00001, sin(x) also = 0.00001. And 0.00001/0.00001 = 1. The actual value of sin(x)/x at 0 is equal to 0, but recall that the actual value at 0 is not important in calculating the limit at 0.
I see your logic but unfortunately I don't think you can legally do that, and sometimes that logic will be wrong. You are assuming that sin(x) is reaching zero at the same rate as x. If you actually evaluate sin(x) where x=10^-n, you'll find that the sine of x is actually lower than x. You can't divide them out if they're different values. It just so happens to work out this time though :P. Later on you can prove this much easier with L'Hopital's rule (whatever it's called). it states that if the individual functions in the denominator and numerator evaluate to zero, then the limit of the derivative of the equation is equal to the limit of the original (if it evaluates to something rational)! in other words, if lim x->c of f(x) = 0 & lim x->c of g(x) = 0 and lim x->c of f'(x)/g'(x) = L where L just represents a valid result, then lim x->c of f(x)/g(x) = L! Sounds complicated but in this case it's easy. at lim x->0, both sin(x) and x become zero. The first two conditions are already satisfied. the derivative of sin(x) is cos(x), and the derivative of x is 1. as x approaches zero, cos(x) becomes 1 as well. so you get 1/1. That's valid, therefor the limit of the original equation is also 1! I only mention it because I wish I knew about L'Hopital's rule earlier.. it makes every 0/0 case much easier to deal with. Anyway I understand that your way is just a leisurely way of looking at it, I'm just warning that sometimes it may not work :P not to mention that teachers aren't always so nice when it comes to grading :(
+typhoon394 because all sin and cos values are positive in the first quadrant. But only cos values are positive in the forth, making the sin values negative.
It would have been a lot easier to understand if you had made this video on a bigger screen with higher resolution because staring at such bad resolution hurts the eye. But the explanation and video is perfect! Its very helpful! Thanks a lot!
this video was like the harlem shake, my brain was building up tension, and then it went bazaar when Khan asked "What is the limit as x approaches 0 of cosine of x" - 15:04
Great video as I've said before! Just a reminder that a "pie piece" is formally called a "sector." Don't dump inSECTORside on me for being picky. It's just good to get the formal terms (although they can seem inTERMinable) since there are so many of them! I wish SUCCESS to all in your studies!
Intuitive understanding: as the angle approaches zero , the height of the smallest triangle in Sal's drawing (which is exactly sin(x)) and the ARC that subtends angle x (this arc is exactly x radians) are more and more similar, their length tends more and more to be the same. In other words, that ARC tends to be a VERTICAL LINE when the angle is really small, so if the angle approaches 0 , the ARC approaches a VERTICAL LINE and it will be equal to the vertical line represented by sin X
My prof just threw everything at us and suddenly he said he proved sinx/x. I was like O.O UMMM WHAAAT But thanks to you I completely understand it perfectly!
@jeffly1968 whenever your dividing absolute values, its the same as square rooting or squaring fractions. you can split it up as sq rt x/ sq rt y or x^2/y^2. Its the same for abs values.
because he wanted to simplify the expression. Khan noticed that tan=sin/cos, and he could simplify two of the sides to a simpler expression. (so |sinx| / |sinx| = 1, tanx/|sinx|=1/cosx)
He has used the Squeeze theorem in the last bit to equate sin x/x to 1. See his video in the Calculus playlist on the Squeeze theorem, just before this one.
![Proofs: Lim sinx/x =1 and lim [cosx -1] /x =0 as x goes to zero from geometry](http://i.ytimg.com/vi/W0jiodfVGx0/mqdefault.jpg)
This might be one of the prettiest proofs in mathematics. It's simple enough for most people to understand, but tricky enough for you to think about for a moment. That was an immaculate presentation, Sal! Godspeed.
Thank you so much Sal! all my book does is state both the sandwich theorem and use that as proof to formally state that therefore lim sinx/x is 1. All my professor did was repeat that in class -___-. for the past week i've just been sitting here like, "i only know this equals 1 because you told me, but WHY does it equal 1!?" It's been frustrating me and this video made so much sense.
I know that feel my prof just did that too :O
I've been in similar situations myself.
the squeeeeeeze theorem
SQUEEEEEEEEZE
SKWAIZE
As they would say on screen rant pitch meetings:
"The squeeze theorem is TIGHT"
@@darthsion3844 Bruh even here lmao
Good point. Good thing we made sure that our terms are positive!
Khan academy. Greetings to you from Morocco. You have given me a helping hand, because in our school they are only applying the rule and you gave me proof and thank you very much
@@mohamedrida132 2bac?
Hello to 2008
@@hellensea yankee with no brim
Sal, this is why I love you. I'm watching this 5 years after it was uploaded, and every time I was confused between steps, you clarified yourself. You made it seem so SIMPLE. Why couldn't my textbook be like this?
Here to tell you that 15 years later textbooks are sitll as dissapointing
At the end I was like Woooooooaoaaaaaaaaaaaa magic. Math is awesome sometimes.
😂😂😂😂😂
Irene Crepaldi maths is awasome
Math is always awesome
@@moaydsparklug8311 true
I call it mathematematic
How the hell did anyone even come up with this proof
lol
Maaaaaath . Also they elected to be a virgin their whole life.
Cocaines a hell of a drug.
man you are an osu player, why i found you in so many times on yt comment sections? xD
Lots of curiosity, time, and fundamental math knowledge.
Omg.!! I am attracted.. your videos are like magnet.!! I am watching them since 3 hours continuously like some movie series, and at any point of time I didn't lost my curiosity and interest.!! The are useful even after 12 years.!! True Genius.!! Thank you from the deep of my heart.!!
Ok, i understood it, and this all for a tiny expression, I am going to go take advil now......
+GooseGamingHD Thats pretty funny of you to say
John Secor at least you now understand where the derivative of sinx how it comes, and from there you can derive other derivatives of trig functions, so its very useful
This 'proof' almost makes me wish I hadn't quit drinking.
It was such a big announcement "And now we're ready to use the sqeeze theorem!"
This guy right here is simply awesome...
it's been almost 15 years and this is still super useful. Amazing work
Thanks! You've squeezed a great deal out of this topic. I did feel pretty x-hausted by the end. Very clear, and informative. Presentation had a nice arc to it. I'll sin out for now. Again, VERY HELPFUL PRESENTATION!
This is the only time I have seem this explained in a forthright and logical way without making a bunch of crap assumptions. Thank you.
Awesome! My calc teacher decided not to explain this because he said we wouldn't understand and said just to know that lim x->0 of (sinx)/x=1. This makes perfect sense! Thanks!
Really brilliant... ❤
6:07 If like me, you had a little problem converting the angle x to x radian, they seem to be different. Just remember, when we say angle x we really mean a fraction of a complete circle, which means x of 360 (x/360) by adding the 360 fraction. Now it is easy to convert a fraction of 2piR to a fraction of 360.
+Marc Abelha If you use agnles you get Lim x--0 sin(x)/x=3.14/180
Yep, radians makes this theorem really elegant.
@@BBBrasil Hey, can you help me? Still, I can’t understand. In sinx and tanx we are using x as degrees but when we are calculating the are of the sector we are using x as radians.
I can’t understand. When we change x degrees to radians I get a completely different thing.
Since x >= 0 for all x in the 1st and 4th quadrants, you don't need the absolute values.
You seem to have confused the 2nd and 4th quadrants.Also, your < & > need to be =.
The squeeze theorem states that if f(x)≤g(x)≤h(x) for all numbers, and at some point x=k we have f(k)=h(k), then g(k) must also be equal to them.
Ok, i've just finished the 48th vid in this series, and i finally understand this
thank you soo much, i would not hav e gotten that in a million years
i iove this guy. he just made me understand everything. keep up the good work please
this example is badass. The squeeze theorum states that f(x)
15:30 That was an "Ooohhhhhhh, I get itttttt" moment at its finest.
Thank you very much teacher , this is even much better than Arabic , I haven't found anything about this proof in Arabic
The area of the 'pie' was derived using the definition of the angle in radians, so the proof of sinx/x = 1 as x->0 is only true when the angle is measured in radians. This limit is used importantly in definining the derivative of Sinx to be Cosx in calculus. This means that derivative also only holds if X is in radians. Power series is based on differentials of sin and cos, so again you must use radians. Everything comes back to how this limit was proved and affects all proof stemming from it.
So, if we are talking about 𝑥 in radians, how we are writing sin𝑥 or tan𝑥? Is it possible?
@@pemifo260 Whether x is in degrees or radians, doesn't effect the way you write Sinx or Tanx, as long as you're clear on the units you're using. However it does make a difference as to what you say the derivative or integral of Sinx and Tanx are. The Derivative of Sinx is only Cosx when you are using x in radians. If using x in Degrees then the derivative of Sinx is instead Cos(x*pi/180).
I feel the excitement in your voice when you started talking about the squeeze theorem LOL .. Thank you soo much this was very helpful!
SIR SIR SIR , I FREAKING LOVE YOU , THE AMOUNT OF HELP YOU PROVIDE IS INCREDIBLE
Beautiful work Sal! This is needed to show that the derivative of sin(x)=cos(x).
This is a great explanation of the proof, but in my opinion, the proof is ultimately circular. (heh. circular.)
The proof relies on the fact that the area of a disc is pi*r^2. This in turn relies on either integration (which requires a trig substitution and the fact that the cosine is the derivative of the sine, which relies on the fact that sinx/x approaches 1), or Archimedes's method of approximation by polygons, which also essentially relies on the fact that sinx/x approaches 1, or at least relies on an axiom that turns out to be equivalent to this fact. If you pick such an axiom, you can use it directly to prove the limit of sinx/x without having to pass from length to area. The area proof feels more satisfying, but that's ultimately because it hides all the axiomatic heavy lifting inside the formula pi*r^2.
I tried something similar. It was called the "Definition of an Equilateral Polygon" test.
As x approaches infinity, Sin(pi/x) * x / cos(pi/x) --> Pi
X = number of sides
Cos = apothem of equilateral polygon
Sin = equal length side divide by 2 of the equilateral polygon
Sin(pi/x) * x = half of the perimeter
As x approaches infinity, the perimeter starts to resemble that of a circle
I refused to watch these videos when the teacher told us to, but now I regret it!! They're VERY helpful! Thank you so much!
I always love your videos but this one is my favorite. Superb job!
Thanks! This is suppose to help my proficiency in mathematics and toward things I would use calculus and trig in! Its something my family never have used in any case of measuring.
this limit is very powerful in spite of its simplicity as its the key to prove that d/dx sin(x)=cos(x) and the other derivatives of trig functions follow. some people prove this limit using L'Hopital rule (LR) but that's wrong because LR requires the derivative of sin(x) which is proved based on the limit itself.
This gave the answer to my question...I wasn't able to prove that lim x -> 0 sin x/x = 1...it is one of these things about limits that I did not know why...LONG LIVE KHAN ACADEMY!!!!
I Really Like The Video From Your Using the squeeze theorem to prove that the limit as x approaches 0 of (sin x)/x =1
that was so revealing and helpful on so many levels
@eileenBrain This isn't supposed to teach anything; it's supposed to be a proof. Which was done admirably.
kudos, khanacademy, this was just what I was looking for.
Khan, you´re our hero!
You get my infinite ´thank you´
Khan is the patron saint of people cramming for exams
Thanks 😊
What a beautiful proof, why isn’t this taught in school?
u are jus simply the best teacher ever.. how would i understand maths without you :)
Thanks so much! I love the added visual and graphical portions to make the proof easier to understand and see.
@hubomba It's only to simplify. You may know that tan x = sin x/cos x. Thus when you divide it by sin x it becomes 1/cos x.
Well, this approach is graphically illuminating but on the other hand laborious. Actually since lim(x→0) sinx/x becomes 0/0, the l'Hôpital's rule can be used, that is, to take the derivative of sinx and x separately. This results in cosx/1. Since cosx=1 at x=0, you get 1/1=1, which is the answer. Probably the squeeze method can be reserved for some extremely difficult ones that are worth the labor.
This may be a dumb question but can you just write sinx as is maclaurin expansion then the limit becomes 1-x^2/(3!) + x^4/(5!) - .... which obviously tends to 1. ( I will cover limits in a few weeks)
You explained it so clearly, I like your explanation Mr.Khan
Best proof till date covers almost everything that need to know 👍
2:40 not sure I like the fact the variable x is being used both as a coordinate and as the angle of the triangle
Thank you, that really helped clear my foggy understanding of that proof.
That was very helpful! omg i wish you were my teacher
@ShneeBnee Sin is defined as opp/hyp where the hyp would be larger than 1 since it goes out of the circle so tan works here since it has 1 as the denominator
That is so you can get |x|/|sin x| from which we can get |sin x|/|x|, which is the expression we want to find the limit of as x tends to 0.
Thank you soooooo much. I have been looking sooo long for a good proof of this, and you have finally proved it to me. Thank you thank you thank you!!!!
If you have my attention span and don't mind a less rigorous proof: as x approaches 0, sin(x) gets closer and closer to 0. As x approaches 0, x has nearly the same value as sin(x). So as x = 0.00001, sin(x) also = 0.00001. And 0.00001/0.00001 = 1.
The actual value of sin(x)/x at 0 is equal to 0, but recall that the actual value at 0 is not important in calculating the limit at 0.
I see your logic but unfortunately I don't think you can legally do that, and sometimes that logic will be wrong. You are assuming that sin(x) is reaching zero at the same rate as x. If you actually evaluate sin(x) where x=10^-n, you'll find that the sine of x is actually lower than x. You can't divide them out if they're different values. It just so happens to work out this time though :P. Later on you can prove this much easier with L'Hopital's rule (whatever it's called). it states that if the individual functions in the denominator and numerator evaluate to zero, then the limit of the derivative of the equation is equal to the limit of the original (if it evaluates to something rational)! in other words, if lim x->c of f(x) = 0 & lim x->c of g(x) = 0 and lim x->c of f'(x)/g'(x) = L where L just represents a valid result, then lim x->c of f(x)/g(x) = L! Sounds complicated but in this case it's easy. at lim x->0, both sin(x) and x become zero. The first two conditions are already satisfied. the derivative of sin(x) is cos(x), and the derivative of x is 1. as x approaches zero, cos(x) becomes 1 as well. so you get 1/1. That's valid, therefor the limit of the original equation is also 1! I only mention it because I wish I knew about L'Hopital's rule earlier.. it makes every 0/0 case much easier to deal with. Anyway I understand that your way is just a leisurely way of looking at it, I'm just warning that sometimes it may not work :P not to mention that teachers aren't always so nice when it comes to grading :(
Mine was just explaining why it works for people who didn't want to watch a long video. Wasn't meant to be a rigorous proof haha
No I understand! I guess I just wanted an excuse to use a fancy new rule I found out about, sorry :P
***** NO :- Sin(0)=0 so when x=0 Sin(x)/x=0/0 which is indeterminate.
yeah that was a typo
I was going to edit it but for got lmao
Thank you very much !!! I have my oral exam on next wednesday...so one more theorem I understood^^
all these squares make a circle
all these squares make a circle
didn't think I'd see a tfs reference in a maths tutorial video
lol
Anonymous i
Kami, I need you to tell me that I may leave the lookout if I want to!
+typhoon394 because all sin and cos values are positive in the first quadrant. But only cos values are positive in the forth, making the sin values negative.
Khan academy is too much helpful
It would have been a lot easier to understand if you had made this video on a bigger screen with higher resolution because staring at such bad resolution hurts the eye. But the explanation and video is perfect! Its very helpful! Thanks a lot!
dude you are simply amazing !!!
thank you so much....
I loved loved loved how you explained the theorem sir. Thankyou.
you just saved my weekend - the way it's explained in my textbook, i couldn't get my head round it, and you've explained it really well! thanks! :)
how is life nowadays its been 13 years
Finally the proof as promised in your earlier video. Thanks very nice.
this video was like the harlem shake, my brain was building up tension, and then it went bazaar when Khan asked "What is the limit as x approaches 0 of cosine of x" - 15:04
Great video as I've said before! Just a reminder that a "pie piece" is formally called a "sector." Don't dump inSECTORside on me for being picky. It's just good to get the formal terms (although they can seem inTERMinable) since there are so many of them! I wish SUCCESS to all in your studies!
Intuitive understanding: as the angle approaches zero , the height of the smallest triangle in Sal's drawing (which is exactly sin(x)) and the ARC that subtends angle x (this arc is exactly x radians) are more and more similar, their length tends more and more to be the same. In other words, that ARC tends to be a VERTICAL LINE when the angle is really small, so if the angle approaches 0 , the ARC approaches a VERTICAL LINE and it will be equal to the vertical line represented by sin X
this was how euler thought of it
That moment you get this .... MY GOD THIS MAKES SO MUCH SENSE
Why cant my math teachers ever be this good
It's been 13 years. I'm seeing this video 13 years after it was uploaded and....just wow.
waw i can believe this equation i took roday in my cal class. thank you very much my math teacher Khan.
The melody of logic always plays the notes of truth
Sal you 're a GENIUS!!!!!!!!!!!!
I just fell in love with you. We weren't taught this, hence why I had to find this video!!! Thank you!
It's one of the most amazing theorems and proofs I've ever seen
Nothing like some classic Khan
😍😍 Beauty of maths
Man, you needed to really think out of the box for this one. Really cool though.
great video!
But shouldn't the area's be smaller or equal to? And later on greater or equal to?
indeed, it should. logically, 1 < 1 is not true. it works for
Thanks Sal, very clear!
My prof just threw everything at us and suddenly he said he proved sinx/x. I was like O.O UMMM WHAAAT But thanks to you I completely understand it perfectly!
Awesome - the book was very confusing on this.. you made it awesome-er
very nicely explained
thank u
Three words:
You are amazing!!!!
Thank you so much ♥️♥️♥️♥️♥️
@jeffly1968 whenever your dividing absolute values, its the same as square rooting or squaring fractions. you can split it up as sq rt x/ sq rt y or x^2/y^2. Its the same for abs values.
Thankyou so much teacher,
If the limit (x/sin(x)) happened to evaluate to 0, then inverting the signs would be incorrect right? I'm referring to the step at 12:39
because he wanted to simplify the expression.
Khan noticed that tan=sin/cos, and he could simplify two of the sides to a simpler expression.
(so |sinx| / |sinx| = 1, tanx/|sinx|=1/cosx)
incredible explanation. Soothing and calm voice, coherent explanation...what more could the student want? Thanks a lot:)
Just a question: isn't strange dividing a segment by an angle? Like, sin(x) and x aren't in the same unit measure, so there is no division of sinx/x.
wow, and my mathematical analysis class finally becomes understandable
Thanks so much! My professor sucked at explaining this
He has used the Squeeze theorem in the last bit to equate sin x/x to 1. See his video in the Calculus playlist on the Squeeze theorem, just before this one.
Excellent video. I need to learn this apparently for my syllabus, and I'm glad to say I can reproduce this myself after your video. Thanks!
you are really wonderful, i wish u and your family the best.
pretty sure ur never gonna see this but thanx a lot for these videos!
AWESOME explanation! Thank you soooooo much.
Great explanations. I had studied it in college without appreciating it like this!
Great explanation!! Hopefully this helps me on my calculus test!!
how are 10 years later. Did you pass the test
KHAN ACADEMY at 11:20 what is the basis why everything is divided by absolute value sin x?