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!
only rapidity is based on hyperbolic trig. otherwise your lorentz transforms and fourvectors require only rudimentary algebra s a mathematical prerequisite
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.
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.
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.
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?
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.
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!
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
"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"
@@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
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!
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!
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.
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.
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.
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.
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.
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.
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?
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.
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.
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.
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.
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!
So you kind of glossed over the result on Brilliant. The shape of the arch on an ideal suspension bridge is, in fact, a parabola, because the cable is not holding its own (negligible) weight, but is holding the weight of the road below it, which can be assumed to provide a uniform force density downward.
Please give us some examples showing how to use the hyperbolic functions. Everyone has had physics problems where we use the trig functions to decompose vectors into their constituent parts along directions that are more convenient. How do we use hyperbolic functions?
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
What is the physical significance of hyperbolic trig functions ? Like trig functions are the trig ratios of the different sides of a triangle having the angle defined. @blackpenredpen pls reply
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?
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)
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
I didn't want to know, but now I know.
*Cosh, the friend of Josh*
*Sinh, the brother of Grinch*
Dark Mage, the son of Johnny Cage
Actually its shine
Quick Mafffs Several ways it can be pronounced. I say shine myself. But yah. *Shine, brother of mine*
How was that one?
Tanh, the friend of Sam
@@abdurrahmanlabib916 SHINE OF X=so shiny i cant see anything
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!
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?
@@ChristAliveForevermore it works either way though, and it’s beautiful seeing it in action
You know how to write a understandable mathematical comment pretty much 🤩
Thanks!
Thank you!
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.
only rapidity is based on hyperbolic trig. otherwise your lorentz transforms and fourvectors require only rudimentary algebra s a mathematical prerequisite
I've been using hyperbolic trig functions for forty years plus, and never knew why they were called "hyperbolic".
That is your shortcoming not something to be proud of really
@@pranavsingla5902 what is wrong with you? How is he proud of it in any way shape or form
@@pranavsingla5902 such arrogance, damn
@@pranavsingla5902 he never said he is proud of it...keep your vulgar comment to yourself
@@pranavsingla5902 Wow ever heard of something called " being humble " ?
the whole "the input is twice as big as the area" really blew my mind away. the whole thing was great!
3:39
I'm still waiting for the Drake& Cosh series
Lilanarus hahaha
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.
"that's pretty much it."
Isn't it?
I wonder how many times he said that
its a done deal.
@@Prxwler isnet?
@@Chaudharys1 don dio
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.
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.
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.
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?
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.
thanks
老哥讲挺好啊，终于搞懂了
Thanks for sharing this video with me!! These make a lot more sense to me now 😁
One of the best maths channels on UA-cam :)
Thanks so much man you just saved me for my viva tomorrow
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 ;-)
how you switch the pens is unnoticable👏👏👏
genius person!
Love you for listening to us!
Quahntasy - Animating Universe
: )
Each good teacher needs to do that.
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!
Everybody gangsta till matmaticians invent sech, csch and coth
I always wondered a lot of things about hyperbolic trigonometry and I think your videos will help me a lot!!!^-^
7:32 - the legendary marker switching skills omg
this explains everything i was looking for. thanks so much! i'll have to show this one to my math teacher :)
I absolutely love any connection between pi and e (not to mention i and phi).
What's the relation between i and φ? Idk that one lol
i*i+sqrt(2)^2=phi-phi+1
@@mukkupretski ¦:|
Bruh, Dat doesn't count, the i turns into -1 and the φ is canceled
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
"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"
@@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
10/10 like the Doramon theme in background
"It's like your friend Josh, but with a C, so cosh" ....pure gold right there :)
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!
6:25 BLEW MY MIND!!!!
Brilliant.org is awesome. I’m glad I saw the site from your video.
Glad you like it!!!
Brilliant is really very concept-oriented website. Keep the good work up. Thankyou
you are the best in what you are doing Sir
Yestarday i was really curious what exactly is coshx now two of my favourite youtubers (you one of them) made a video about it!
Best explanation of cosh x and sinh x ever! I’ll be looking for your other hyperbolic function videos.
Thanks for the simple explanation
dude u are intelligent and funny too
and I love ur learning
Woow, I always wanted to learn about hyperbolic trig functions!!! Thank you, sir for making this so much easier
Can I please know what is
Amazing, please continue with the series.
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.
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!
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.
at 2:03 he says automatically,and its the funniest thing I"ve ever heard
Great video, just leaves me a question: why are Hyperbolic functions so important and not the Elliptical ones, for example?
Well we already have the most simple ellipse: the unit circle
Djdjcjcjcj Jfnfjfidnf Actually, hyperbolas are in a way stretched out circles, where a = 1 & b = i.
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.
Angel Mendez-Rivera multiplying by i is NOT a ‘scale’. It is more of a rotation in an argand diagram.
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.
This is what im waiting for
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!
Still amazing ! Thank you for your work ! You make me love math even more with each video !
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.
And because therefore the load on it is proportional to the x-length not the arc length
Thanks for this, man.
You're an amazing teacher
This is REALLY well explained
"Enjoyment of learning mathematics" That is what I'm here for.
I was taught to pronounce it as “shine” and “than” but that was in the 70s in Australia.
I remember learning the same, Stephen.
Nice to have someone else confirm what I recall.
Kind regards from the Shoalhaven.
boi ur awesome ❤️
Twitter ftw! Nicely done can you please make an introduction video with differential equations?
SiR PuFFaRiN was j
Mind blown🤯
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.
Thankyou. Quality lecture.
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.
Nice work well explained ...might add a more detailed explanation of Radian measure ???
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.
Ohhh my God ! What's that I see here ....I thought it's too complicated but it's really funny .thnxxx bro
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?
At Uni in the 1990s we were taught to pronounced sinh as 'shine' in Australia.
You're a genius kid!
Great job!
Math kicks ass
The Tauist says:
In 5:35 to 6:25 - the area formulae in the circle get more concise when you use tau := 2pi
Pretty cool but e^(iτ/2)+1=0 isn't really cool
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.
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.
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.
COOL! Area(θ)=θ/2 is interesting.
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.
Thank u sir for solving my great problem...... Awesome 😍
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.
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
I thought the same thing as well but I’m not sure
Nope.
06:20
t=2.area
So, area=t/2
That was a great video... thank you so much!
Please make a video on how " e"( irrational number) is related with hyperbola
1MILLION SUBS!
Thanks 😍
I gather from watching that e in example is Euler's number and not any variable. Would any variable other than e still work?
Great video
Thank you
Thank you !!!
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!
Would the cables on power lines or telephone masts be a better example, since they hang freely?
Geez that cable problem of the Golden Gate Bridge was on my pset for physics. Hardest thing
So you kind of glossed over the result on Brilliant. The shape of the arch on an ideal suspension bridge is, in fact, a parabola, because the cable is not holding its own (negligible) weight, but is holding the weight of the road below it, which can be assumed to provide a uniform force density downward.
Great brother
it also works for y=1
Please give us some examples showing how to use the hyperbolic functions. Everyone has had physics problems where we use the trig functions to decompose vectors into their constituent parts along directions that are more convenient. How do we use hyperbolic functions?
I would think about the cables on the suspense bridges
Finally! Waited for it for so long #YAY
Double angle identity
Tan(2@)=2tan@/1-tan^2@
Can we verify it for @=45 degree.?
where did those e come from, everyone just jumps to the formula, I want to know were those e come from?
is t an angle? does it have special angle like in trigonometry like 30 degree, 45 degree, 60 degree, 90 degree?
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
When I was an architecture student I designed a catenary structure ;-)
What is the physical significance of hyperbolic trig functions ? Like trig functions are the trig ratios of the different sides of a triangle having the angle defined.
@blackpenredpen pls reply
Great video!
both are equal to 1 is that mean we can simply equate this two identities
"Isn't it?" ……
My brain: Yes
Me: No
"RIGHT???"
"WRONG!!!"
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?
cosh(t) = cos(t). Euler's expression pretty much sums up that. BTW, bprp has made a video on it
@@astudent9206 cosh(t)=cos(it).
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)
Thank you!
Does taking (sec t, tan t) as parameters of unit hyperbola lead anywhere? Why prefer cosh, sinh over simple trig functions?