Introduction to Hyperbolic Trig Functions

[@tanelkagan]

Just as an observation, when checking to see if cosh² - sinh² =1, as an alternative to expanding out the brackets in full you can use the difference of two squares identity: a² - b² = (a + b)(a - b). Here, a = (e^t + e^-t)/2; and b = (e^t - e^-t)/2. Distributing out the 1/2 you can think of these as:- a = (e^t)/2 + (e^-t)/2 b = (e^t)/2 - (e^-t)/2 So, (a + b)(a - b) reduces quickly to (2(e^t)/2) (2(e^-t)/2) or simply (e^t)(e^-t) which is of course e^0, or 1. You can decide for yourself which method you prefer!

[@ChristAliveForevermore]

You assumed that e^t identity is true. What if you want to derive based solely on the analytical trig intuition and not the logarithmic intuition?

[@surpiers]

@@ChristAliveForevermore it works either way though, and it’s beautiful seeing it in action

[@Abdalrhman_Kilesee]

You know how to write a understandable mathematical comment pretty much 🤩

[@DeeEm2K]

I *always wanted* to know what hyperbolic functions were but was too lazy to actually research it. Thanks man, for researching it and teaching to me

[@armacham]

I didn't want to know, but now I know.

[@DeeEm2K]

*Cosh, the friend of Josh* *Sinh, the brother of Grinch*

[@AAAAAA-gj2di]

Dark Mage, the son of Johnny Cage

[@abdurrahmanlabib916]

Actually its shine

[@Test-ri2kr]

Quick Mafffs Several ways it can be pronounced. I say shine myself. But yah. *Shine, brother of mine* How was that one?

[@Vinny_3041]

Tanh, the friend of Sam

[@drenzine]

@@abdurrahmanlabib916 SHINE OF X=so shiny i cant see anything

[@phosphor6472]

3:39 I'm still waiting for the Drake& Cosh series

[@extendedreal]

Lilanarus hahaha

[@digitig]

I've been using hyperbolic trig functions for forty years plus, and never knew why they were called "hyperbolic".

[@pranavsingla5902]

That is your shortcoming not something to be proud of really

[@setupchess6288]

@@pranavsingla5902 what is wrong with you? How is he proud of it in any way shape or form

[@ripudamansingh2]

@@pranavsingla5902 such arrogance, damn

[@yetii09]

@@pranavsingla5902 he never said he is proud of it...keep your vulgar comment to yourself

[@btsandtxtloverstraykidzfan3486]

@@pranavsingla5902 Wow ever heard of something called " being humble " ?

[@dxk2007]

This is the 1 topic I didn't bother learning in high school... and it turns out Relativity is all based on it. Thank you.

[@lambda2693]

only rapidity is based on hyperbolic trig. otherwise your lorentz transforms and fourvectors require only rudimentary algebra s a mathematical prerequisite

[@david-yt4oo]

the whole "the input is twice as big as the area" really blew my mind away. the whole thing was great!

[@Rocky-me5cw]

"that's pretty much it."

[@Prxwler]

Isn't it?

[@AlvinBalvin321]

I wonder how many times he said that

[@Chaudharys1]

its a done deal.

[@hydropage2855]

@@Prxwler isnet?

[@hydropage2855]

@@Chaudharys1 don dio

[@DavideCanton]

A small suggestion: the check is way faster if you decompose (x²-y²) as (x+y)(x-y). That way you get e^t * e^(-t) = 1.

[@ruhanikhazain7564]

Thanks!

[@blackpenredpen]

Thank you!

[@pierreabbat6157]

If the deck of the bridge is horizontal, the cables are parabolas. If the deck follows the curve of the cables, the cables are weighted catenaries. If you suspend a string at both ends with nothing hanging from the string, it is a catenary, which is the graph of cosh.

[@Apollorion]

If you say all of the cables on the suspension bridge have no mass but the bridge-deck does have, with a homogeneous density and is also -horizontal- straight, then you can easily derive that the curve of the main carrying cables is indeed approximated by a parabola.

[@twwc960]

You are exactly right. It is a very common mistake to assume the curved cables in a suspension bridge are catenaries (hyperbolic cosine curves). In fact, they are not and to a very good approximation they are indeed parabolas. This is true since the road is fairly nearly horizontal and the weight of the road being suspended is generally much greater than the weight of the cables.

[@realcygnus]

This is quite interesting. Somehow I never covered this topic adequately. Is there a function that interpolates between the two(catenaries & parabolas) ? l suppose based on the weight ratios &/or the suspended platforms straightness(to horizontal). I'd guess it must assume an infinite # of vertical hangers?

[@twwc960]

realcygnus Google "suspension bridge catenary" and there are links to a few papers which do that. The Wikipedia page on "catenary" has a brief discussion under "Suspension bridge curve" with links to a couple of papers.

[@realcygnus]

thanks

[@eric_welch]

"It's like your friend Josh, but with a C, so cosh" ....pure gold right there :)

[@RetroGamingClashOfClans]

7:32 - the legendary marker switching skills omg

[@ayoubsbai6339]

One of the best maths channels on KZbin :)

[@lordofkeebs8424]

10/10 like the Doramon theme in background

[@ashutoshojha4244]

Thanks so much man you just saved me for my viva tomorrow

[@d1v1212]

老哥讲挺好啊，终于搞懂了

[@overlordprincekhan]

There is a quote "The teachers who complicates the study is the biggest state criminal" This 4 minute is enough to understand me the lesson taught by by teacher of a whole month. Thanks for that nice explanation!

[@alberteinstein3612]

Thanks for sharing this video with me!! These make a lot more sense to me now 😁

[@urluberlu2757]

Wow, i'm just climbing to the next level in mathematics, and re-discover it's beauty and real, and complex pleasure with it, thanks of you ;-)

[@quahntasy]

Love you for listening to us!

[@blackpenredpen]

Quahntasy - Animating Universe : )

[@Apollorion]

Each good teacher needs to do that.

[@guliyevshahriyar]

how you switch the pens is unnoticable👏👏👏 genius person!

[@pablojulianjimenezcano4362]

I always wondered a lot of things about hyperbolic trigonometry and I think your videos will help me a lot!!!^-^

[@ImSomebady]

Currently just finished calc 3 and starting "advanced calculus and applications" and didn't know where the trig and hyperbolic functions relation came from. Thank you so much!

[@DatBoi_TheGudBIAS]

Everybody gangsta till matmaticians invent sech, csch and coth

[@kaistrandskov]

I absolutely love any connection between pi and e (not to mention i and phi).

[@DatBoi_TheGudBIAS]

What's the relation between i and φ? Idk that one lol

[@mukkupretski]

i*i+sqrt(2)^2=phi-phi+1

[@DatBoi_TheGudBIAS]

@@mukkupretski ¦:| Bruh, Dat doesn't count, the i turns into -1 and the φ is canceled

[@canyon_online]

This is awesome. Never seen cosh and sinh in my life until I was asked to integrate it last week for Calc 2. Could not tell you for the life of me what they meant until now. #YAY

[@zohar99100]

"Never seen cosh and sinh in my life until I was asked to integrate it last week for Calc 2."... What?!? Be like: "Never seen a girl until I was married"

[@heavennoes]

@@zohar99100 those are very different, maybe he was never taught hyperbolic trig and then suddenly he saw a question maybe by a different teacher who assumed the class knows hyperbolic trig and take the derivative of it

[@pigman6954]

this explains everything i was looking for. thanks so much! i'll have to show this one to my math teacher :)

[@summerishere5146]

6:25 BLEW MY MIND!!!!

[@trueriver1950]

8:50 that's not true. A free hanging chain or rope does form a cosh curve. However that depends on the rope or chain having constant mass power unit length. In other words it depends on the mass of the straight line of the deck of the bridge being zero (if you are a mathematician) or being negligible (if you are a physicist or engineer). Likewise, if we make the opposite approximation and treat the rope or chain as having negligible mass per unit length, compared to the mass of the deck, then the rope does indeed form a parabola (to within the approximation we made when we ignored the mass of the rope or chain). If we do the fully accurate version, allowing for an appreciable mass per unit length for both the rope and the deck, then the shape of the rope is somewhere between a cosh and a parabola.

[@giacomocasartelli5503]

Great video, just leaves me a question: why are Hyperbolic functions so important and not the Elliptical ones, for example?

[@friedkeenan]

Well we already have the most simple ellipse: the unit circle

[@angelmendez-rivera351]

Djdjcjcjcj Jfnfjfidnf Actually, hyperbolas are in a way stretched out circles, where a = 1 & b = i.

[@angelmendez-rivera351]

Djdjcjcjcj Jfnfjfidnf In fact, by allowing complex numbers, any equation for any of the conic sections can be written in the form of (x/a)^2 + (x/b)^2 = 1.

[@tomgraham7168]

Angel Mendez-Rivera multiplying by i is NOT a ‘scale’. It is more of a rotation in an argand diagram.

[@angelmendez-rivera351]

Tom Graham Yes, technically, but if your scalar field of a vector space with a complex coordinates is the set of complex numbers, then that still counts as scaling.

[@scathiebaby]

The Tauist says: In 5:35 to 6:25 - the area formulae in the circle get more concise when you use tau := 2pi

[@dystotera77]

Pretty cool but e^(iτ/2)+1=0 isn't really cool

[@geoffhuang2438]

Brilliant.org is awesome. I’m glad I saw the site from your video.

[@blackpenredpen]

Glad you like it!!!

[@DRUCVSKAMAU]

at 2:03 he says automatically,and its the funniest thing I"ve ever heard

[@davidawakim5473]

4:28 shouldn't the area be 2t? Because the input is the area divided by 2 and 2t/2 = t. Whereas with the t/2 that he put t/2 * 1/2 = t/4

[@simonwalthers9951]

I thought the same thing as well but I’m not sure

[@kseriousr]

Nope. 06:20 t=2.area So, area=t/2

[@kingsbarber0008]

you are the best in what you are doing Sir

[@peterhui7023]

8:52 The shape of the cables at both sides of the bridge is incorrect. It should be nearly a straight line since it should provide a force against the tower from pulling inwards and the cables are anchored into the massive RC foundation on both sides.

[@billharm6006]

I wish my college math teacher had taught hyperbolics this way. I went from, "memorize the formula" to OH! in about one-quarter of a class period's duration. And I do love that Ah Ha! moment.

[@yashikakaushal645]

dude u are intelligent and funny too and I love ur learning

[@lorostotos5647]

the bridge cable is a parabola because the cable is practically weightless comparing to the road it holds underneath.the road is horizontal so the load is linear.

[@AlecBrady]

And because therefore the load on it is proportional to the x-length not the arc length

[@sunandachaudhary9936]

Brilliant is really very concept-oriented website. Keep the good work up. Thankyou

[@That_One_Guy...]

why dont we call sinh as shine ? then cosh as coshine lol

[@technux5382]

comme au Portugal ou en Auvergne :p

[@Vinny_3041]

Cotanshent Arcshcoshine

[@LetsSink]

I love this

[@anirudh7137]

Thanks for the simple explanation

[@sirface7951]

Yestarday i was really curious what exactly is coshx now two of my favourite youtubers (you one of them) made a video about it!

[@surrindernayar]

Hyperbolic function applies to a freely suspended cable called catenary. However, the curve of the suspension bridge cable which is uniformly loaded (road) and negligible cable weight is indeed a parabola. Check it out. Lots of people make this mistake.

[@eta3323]

Woow, I always wanted to learn about hyperbolic trig functions!!! Thank you, sir for making this so much easier

[@sgiri2012]

Can I please know what is

[@matchedimpedance]

The shape of a suspension bridge cable would only be a catenary if the weight of the bridge to be supported was negligible compared to the weight of the cable. But in general that is not the case. Usually the weight of the bridge is more significant than the weight of the cable so in that case the shape of the cable would in fact be more like a parabola.

[@laurensiusfabianussteven6518]

This is what im waiting for

[@g.v.3493]

Best explanation of cosh x and sinh x ever! I’ll be looking for your other hyperbolic function videos.

[@krishnasarmavenkatrao6020]

"Enjoyment of learning mathematics" That is what I'm here for.

[@blacknoir2404]

This inspired me to invent the parabolic trigonometry functions. I have cosp(t) = (3t)^⅔ and sinp(t) = (3t)^⅓. These aren’t very exciting so far.

[@lambda2857]

An explanation of the elliptic functions sn, tn, cn, dn, and so on, from a geometric standpoint, would be a very good video to make.

[@wduandy]

Amazing, please continue with the series.

[@preetjitsingh328]

Easier way to proof @ 13:12 Cosh x ^2 -sinh x ^2 = (cosh x+sinh x) ( cosh x -sinh x) = ( e^t /2 + e^-t /2 + e^t /2 - e^-t/2) = ( e^t /2 + e^-t /2 - e^t /2 + e^-t/2) = e^t * e^ -t = 1 ##

[@jagatkumartudu]

Ohhh my God ! What's that I see here ....I thought it's too complicated but it's really funny .thnxxx bro

[@Paraselene_Tao]

Around 7:00, why do we square the radius for the area of a circle but we don't square theta/(2(pi))? I mean, these are definitions but why square one and not square the other?

[@ajantamondal3725]

8:54 min this graph is Transcendental functions graph,is it???

[@mango417]

"Isn't it?" …… My brain: Yes Me: No

[@snyfalcryo524]

"RIGHT???" "WRONG!!!"

[@RichardCorongiu]

Nice work well explained ...might add a more detailed explanation of Radian measure ???

[@allannunez9464]

How to get the enjoyment of leaning mathematics? By watching ALL the videos! #YAY!

[@antoniocampos9721]

Thanks for this, man.

[@mathteacher2651]

You're a genius kid! Great job!

[@rafaellisboa8493]

I enjoyed this video very much comrade, I never knew what hyperbolic trig functions where and they sound very cool and I have been curious about this for a week, thanks!

[@NXT_LVL_DVL]

4:14 why the input is not angle but half of the area . Is there any intuitive or logical explanation ?

[@louf7178]

Thankyou. Quality lecture.

[@Chaosdude341]

Thank you! Missed out on these functions in Pre-Calc and Calc I, so I'm figuring this out in Calc II. Love the analysis!

[@darkiiboii5855]

boi ur awesome ❤️

[@zack_120]

COOL! Area(θ)=θ/2 is interesting.

[@stephentrouse2069]

I was taught to pronounce it as “shine” and “than” but that was in the 70s in Australia.

[@CrystalClearMaths]

I remember learning the same, Stephen. Nice to have someone else confirm what I recall. Kind regards from the Shoalhaven.

[@rubensenouf1813]

Still amazing ! Thank you for your work ! You make me love math even more with each video !

[@nonamenoname6921]

At Uni in the 1990s we were taught to pronounced sinh as 'shine' in Australia.

[@mattmackay76]

That was a great video... thank you so much!

[@wildmonkiesJR]

Mind blown🤯

[@M4TT4TT4CK]

Math kicks ass

[@machobunny1]

Just wondering, where does the exponential identity for cosh and sinh come from? Does looking at Euler's identity for sin and cos derivation answer that?

[@astudent9206]

cosh(t) = cos(t). Euler's expression pretty much sums up that. BTW, bprp has made a video on it

[@Namchha1]

@@astudent9206 cosh(t)=cos(it).

[@YorangeJuice]

for cosh: suppose you wanted to calculate cos(i). start with the maclaurin series for cosine and plug in i. you will find that cos(i) is equal to the sum from 0 to infinity of 1/(2n)!, which I will call S for brevity. Recall the maclaurin series for e^x, which i will call exp(x). notice S looks similar to exp(1), but there are a bunch of extra 1/[odd factorial] terms in exp(1). we can get rid of these extra terms by adding exp(-1) to exp(1). this will cancel all of the 1/[odd factorial] terms, but we will be left with extra 1/[even factorial] terms. we can divide by 2 to get rid of these extra terms, and after all this, we see that S is equal to (exp(1)+exp(-1))/2, which means cos(i) is equal to (exp(1)+exp(-1))/2. this can be generalized by instead doing cos(ix) to find that it will be equal to (e^x + e^-x)/2 and define this to be cosh(x). we can then find cosh(ix) using this definition of cosh and euler's formula to see cosh(ix)=cos(x)

[@mattyjackson3857]

This is REALLY well explained

[@OhlordyOh]

You're an amazing teacher

[@jackiekwan]

Finally! Waited for it for so long #YAY

[@jaldo7364]

teacher, how can something that has two values of y for a value of x be a function ? isnt that a relationship ?

[@MathIguess]

Great question! And yes, it is considered to be a relationship, but it can be parameterised as a function of time (which might be why he used t as the variable, not theta). When rewriting the unit circle with parameter t, it will be a function in R2. If that doesn't make sense, don't worry! The point is just that relations can be written as higher dimensional functions.

[@Arjun-fy6jy]

Great video! Can someone please explain why the coordinates on a hyperbola are (cosh t, sinh t) where t is twice the area of the region bounded by x-axis and the line joining the point and origin? Is there like a proof or definition for it?

[@SirPuFFaRiN]

Twitter ftw! Nicely done can you please make an introduction video with differential equations?

[@SalamenceKidd2000]

SiR PuFFaRiN was j

[@benhbr]

The cables on a suspension bridge carry not only their own weight, but also the road. This load is much heavier and horizontally uniform, so the cables actually ARE parabolas!

[@erynn9770]

Would the cables on power lines or telephone masts be a better example, since they hang freely?

[@ashishpandey5583]

Thank u sir for solving my great problem...... Awesome 😍

[@jimallysonnevado3973]

how do you derive the exponential formula form that hyperbola

[@jimallysonnevado3973]

but we how can we know the integral based only from the hyperbola

[@karolakkolo123]

@@jimallysonnevado3973 Hyperbola is just a rotated y=1/x function. I'm not going to go into why this is the case, but trust me, it's just 1/x rotated by -45 degrees. The area he talks about is between the hyperbola, the x axis, and the y=x line. So likewise, the same area is going to be equal to the area between the function y=1/x, the line y=x and the y axis this time. Now, the further you go on the hyperbola, the closer it goes to the y=x line, so analogously, the further you go up the 1/x function (positive y direction), the closer you will get to the y axis. This translates into an integral where the further you go on the hyperbola, your lower bound of your definite integral will approach zero. The upper bound will always be 1. To find the *exact* number for the lower bound, you need to set up an equation with the definite integral, with the lower bound being x (unknown). You basically set it equal to the area, and by integrating you will be able to find x, which will be basicaly related to cosh(x). It's actually more complicated than that because you need to account for extra area that you accidentally add in the process, and you have to convert to polar coordinates to prove that it's equal to the cosh(x). However that's too long for a youtube comment, so I won't explain it here. (While working in the polar coordinates, your expressions of numbers will be in the form re^(i*theta), which is a hint to where the exponential comes from

[@arjavgarg5801]

00:10 doraemon

[@MushookieMan]

Assuming the weight of the bridge is negligible compared to the weight of the cable is the most insane thing I've ever seen in a derivation. A bridge cable assumes the shape of a parabola, it's easy to show.

[@walter9029]

I wonder, if I will be able to figure out the area t/2 in the hyperbolic case. I think of the area of the triangle minus the integral of the squ.root function.

[@АлександрИгнатьев-и1д]

Спасибо большое за это видео.) Узнал о том, о чем не рассказывали в моем вузе на математике)

[@AwareHK]

Hi professor Cho, I am a foolish on mathematics, can you explain why cosh and sinh is defined as exponential functions? I tried to search the proof, but most web only give out the definition only. Thanks.

[@angelmendez-rivera351]

Aware Wong Because sin and cos are also defined in such a way it

[@VilemJankovsky]

www.quora.com/Whats-the-derivation-of-the-hyperbolic-functions/answer/S-H%C3%A1jek?ch=10&share=59a50601&srid=3RS2i

[@drenzine]

6:14-6:29 Someone who only know degrees: So if the area is theta/2, and the angle is 90, so 90/2 is 45, this means the area of the circle is 45×4 which is 180, which means.... Pi×1 is 180 so Pi is not irrational at all!

[@MrBobbybrown7]

I gather from watching that e in example is Euler's number and not any variable. Would any variable other than e still work?

[@littlebobbytables6841]

Are these functions related to the complex sine and cosine? They look suspiciously similar with the (e^t ± e^-t)/2

[@yaeldillies]

Yes they are! cosh(x) = cos(ix) and sinh(x) = isin(ix)

[@GoldrushGaming0107]

Genuine question: Ive been having trouble finding out how they found the e^x definitions of hyperbolic trig functions, but i found on wikipedia that they're the even and odd parts of e^x. I was very relieved to find that out, but then I started wondering how they knew that it was sinh and cosh because it immediately goes on to say that cosh(x) + sinh(x) = e^x. Now I am just wondering how they knew to name them the hyperbolic trig functions because they seem like pretty random exponential functions that have nothing to do with hyperbolas. I just dont see the connection of (e^x + e^-x) / 2 being a hyperbolic trig function and the same for sinh. Can anyone help me out on this? Thanks.

[@neilmccafferty7830]

i have the same concern. cannot find the derivation of these formulae anywhere.

[@tiemen9095]

It makes some sense to look at the hyperbola, which from a geometric sense is related to the circle. Note that e^ix = cos(x) +i*sin(x) is related to a circle and cosh(x) + sinh(x) = e^x is related to the hyperbola. The link between circles and hyperbolas is found geometrically in conic sections (cross-sections of the cone), for which there are 4 possible shapes: - a circle (if you slice the cone straight through) - an ellipse (if you slice the cone under a slight angle) - a parabola (if you slice the cone parallel to its slanted side, so to say) - a hyperbola (if you slice even steeper than parallel) Their links appear more often in mathematics and geometry, for example in astronomy and orbital mechanics. Satellite / moon / planet orbits are typically ellipses. A useful parameter to describe the shape of an ellipse is its eccentricity "e". if e=0 it is a circle; the bigger e, the longer the major axis is compared to the minor and if e=1 the major axis would be infinite. However, in orbital mechanics we can easily "set" initial conditions that would satisfy an orbit with e>1. It turns out that these orbits will now follow a hyperbolic trajectory (with some imagination, you could think of a hyperbola as an ellipse that is so big that it wraps around infinity and minus infinity to put its "far end" in the opposite quadrant). This is just another example of a mathematical link between the circle and the hyperbola that was well known to people like Newton and Euler. So considering the fact that circles and parabolas more often show up as possible solutions to the same type of problem, it makes some sense to look for the involvement of the hyperbola and play around with unit-hyperbola equivalent concepts and see what happens.

[@YorangeJuice]

to show why hyperbolic trig functions have anything to do with e, we must make 2 key assumptions: 1. cosh^2(t) - sinh^2(t) = 1 2. the input "t" is twice the area "A" as shown in this video, which is the same property as sine and cosine on the unit circle we can find A using integration. notice A is equal to: the integral from zero to cosh(t) of tanh(t)x with respect to x, minus the integral from 1 to cosh(t) of sqrt(x^2 - 1) with respect to x. if you do this integration, you'll find that the area is equal to 1/2 times ln(cosh(t)+sinh(t)), which is equal to 1/2 times ln(1/(cosh(t)-sinh(t))) by property 1. from property 2, we also know this area is t/2. we can equate t/2 with two expressions for the area we determined by integration and then solve the system for cosh(t) and sinh(t) to get the expressions for each in terms of e

[@arjyadeep1818]

Please make a video on how " e"( irrational number) is related with hyperbola

[@WahyuNurudin]

is t an angle? does it have special angle like in trigonometry like 30 degree, 45 degree, 60 degree, 90 degree?

[@becalmandquiet881]

Great video

[@holyshit922]

Try to parametrize both circle and hyperbola with rational functions It can be useful in integration I try to reduce integrand to rational function if possible

[@KUYAJRIP]

1MILLION SUBS!

[@dimosthenisvallis3555]

-what about sinht. whats does it represent?- whould love to see other hyp trigs like tanht. and maybe hyper sec if it exists. i mean what they represent on a cartesian

[@dimosthenisvallis3555]

yes i rewatched it and i got that thnx.

[@scathiebaby]

I would like to know, too, if you can depict tanh(x) - analogous to how you can draw tan(x) on the unit circle, as the tangent (yay!) at y=1

[@BennettAustin7]

Geez that cable problem of the Golden Gate Bridge was on my pset for physics. Hardest thing

[@ElectronicsPeddler]

Thank you so very much for posting this; it may not have millions of views but to those who have watched this video, it is immeasurably valuable.