Hyperbolic Functions - Definitions and graph of cosh x

[Vibhrat]

Hyperbola

[demonarex]

horrible diction..

[1214Khan1214]

i saw an asian and i m like uhOh this is gonna suck but you are AMAZINGGGGGG!!

[michalchik]

I know this is terrible immature but I crack up every time him he pronounces sinh as "chink"

[michaelthetaxidriver]

@michalchik same here! i was just going to comment about that. :D tbf it'd be the same if I (a white guy) was lecturing and I kept saying 'cracker'

[maxinator775]

chink.

[pigknuckles211]

Vincenzo Riccati (1707-1775) introduced the hyperbolic functions. Johann Hein-

rich Lambert (1728-1777) further developed the theory of hyperbolic functions in

Histoire de l’acadmie Royale des sciences et des belles-lettres de Berlin, vol. XXIV,

p. 327 (1768).

[BYMYSYD]

this guy really gets into math

[DaGuyFlyinHigh]

I didn't get your video at all, because of your heavy GUK accent.

[xbl2010]

please don't call "sinh" "chink"!!!

[KalleSD]

SUBTITLES PLEASE!

[Chumpster9]

They are arbitrary there is no derivation

[Najam7861]

omd, ive been looking for a proof for these equations and how they were derived for a week and still no clue! anyone who can help will be thanked

[tiggxx]

we take a lot for granted when we're given this shit.. i think you have to be good with infinite series and differential equations to really appreciate where all this comes from not that i'm so delusional to think that would be me.

[MiyukiYuki]

Thanks, for the review.. I have a calculus exam today lol.

[CogitoErgoCogitoSum]

Okay. Hold up. How do you prove that sinh x = [e^x - e^-x ]/2 ??? You are wrong by telling us its definition. It isnt. Do you know how this relationship was derived? Or are you just taking it on faith because your teacher told you? Take the regular circular trig function sine, for example. Definition is opposite/hypotenuse of a unit right triangle. Coordinate pair (cos t, sin t) forms a unit circle. THAT is definition. Saying sin x = [e^ix - e^-ix]/2i IS NOT definition - its consequence.

[CogitoErgoCogitoSum]

My point is simple. Hyp functions are DEFINED based on the unit hyperbola. The coordinate pair (cosh t, sinh t) forms a unit hyperbola. THAT is definition. The Euler's number expression is a consequence of definition, not the definition itself. A single function cannot have more than one separate, seemingly independent, unrelated simultaneous definition without some PROOF of equivalence.

So what is the proof?

[CogitoErgoCogitoSum]

How do you go from a unit hyperbola, defining the hyp trig functions... all the way to their Euler equivalent expressions? I see no way and have yet to see a sound proof. Most mathematicians use circular reasoning, most websites do too. "Its definition" is the age old strategy for weak mathematicians to escape the challenge of proof.

[CogitoErgoCogitoSum]

I have tried to use Taylor expansions to demonstrate a relationship between e^x and the hyp functions. But I cannot expand the Taylor functions without first knowing the derivatives of the hyp functions. So how do you prove the derivatives of the hyp functions? Let me guess... refer back to what Im trying to prove as the basis for a proof? Circular reasoning.

[CogitoErgoCogitoSum]

I am trying to use more fundamental, proven or intuitively true relationships to prove the relationships... Im not going to just sit here and take it on faith. I dont dispute its true... I know that it is. But I dont see a valid proof for it.

[qorilla]

Well. First it's NOT enough that sinh and cosh are functions for which (cosh t, sinh t) points draw a unit hyperbola! Not just the hyp. functions have this property.

You have to state what 't' is. For the hyp. functions 't' is a sector area to the graph of the unit hyperbola. But to define the area, one needs integration. If you integrate the sector, you will find that if you have 't' as area, you have the Euler formula for endpoint coords. I can show if you want. No circularity!

[qorilla]

this was for 'CogitoErgoCogitoSum'

[qorilla]

Hyperbola to Euler expression is not at all a hard proof! Quite easy with integration. But most times people do it the other way round defining them in the exp. form, then proving the hyperbola thing.

[sponsoredwalk1]

(On Hyperbolic Functions)

"Most mathematicians use circular reasoning, most websites do too. "Its definition" is the age old strategy for weak mathematicians to escape the challenge of proof."

Very true, I have been looking for a week now to try to find a way to understand how these functions and Euler's Formula etc... were derived. It'd be a big help if you could pass on any links you have come across (assuming u did lol) that help I'm sick of learning by rote without proof, Gratias Tibi Ago!

[CogitoErgoCogitoSum]

I was a student of only algebra and trig before I saw the equation e^ipi + 1 = 0. It spurred a curiosity and I spent the next two or three weeks trying to understand it. I looked at proof, history, etc... and in the process I taught myself calculus because it was a necessary stepping stone. There is a definition for some things, and proof of everything else, but it requires no circular reasoning. There are can be no more than one unique definition for any concept, not multiple equivalent ones

[CogitoErgoCogitoSum]

I dont have links for you. But if you contact me we may be able to discuss the conceptual and proof-based evolution of these mathematical notions

[michalchik]

@CogitoErgoCogitoSum Why wouldn't it be derived through the series expansion like the Euler expressions for other trig functions?

[tiggxx]

isn't it arc-length paired with the x and y components (cosh(arclength),and sinh(arclength)) respectively based on x^2-y^2=1. just like trig's based on radians..

[senox88]

yo.

just a pin up,

for sinh is called shine,

for cosh is called co-shine.

having a hard time to spell it out yeh?

haha

thanks for the video

[simplystimpy]

i've never heard them called shine and co-shine. I've always heard them as sinch and cosh, tanch, seech, coseech, cotanch.

[neogam209]

in school we used to call them cosh and shine

[largui23]

Thanks a lot! your video really refreshed my mind, it have been about 15 yrs since I studied this. I am preparing for my PE license and your video is very helpful. Thanks!

[donylee]

Not a problem!

PE? Professional Engineering is it?

[cec831]

Very useful video even though the 1 minor mistake,I'm glad u posted it.

[dz4485]

its a verry useful video if you allready learned about h functions, and just need to go over it before a test

[donylee]

Hey Zubb90k, I'm just curious, what does EMO mean?

Anyways, I'll have to applaud lllllrrrrr for spotting the mistake. From my calculus book, it should be:

sinh(x+y)=sinh(x)cosh(y)+cosh(­x)sinh(y) SIMILAR to the trig identity for sin(x+y).

Sorry everybody.

[artcostanza]

he references your emo style hair - simply google it :) anyway great lecture, thanks..

[Zubb90k]

lllllrrrrr Im giving you a thumb down for being an EMO.

[ShaIIowfu]

thansk

[lllllrrrrr]

on 3:08, when you were writing sinh(x+y) I think there is a mistake, you write sinh(x+y)=sinh(x)cosh(y)+cosh(­y)sinh(x), if it is like in trig, then sinh(x+y)=sinh(x)cosh(y)+cosh(­x)sinh(y)