Derivation of cosh and sinh

In this video, I derive the formulas for cosh and sinh from scratch, and show that they are indeed the hyperbolic versions of sin and cos. I also explain what the input x of cosh(x) means. Included is a calculation of the integral of sqrt(x^2-1)

Note: A big thanks to Alex Zorba, who came up with the idea and the proof, thank you 🙏

[triton]

Seriously needed this, one of the reasons I love doing maths is being able to derive the content from the foundations of my own knowledge and you have done just that sir, thank you!

[shahrukhshikalgar]

Wonderful!
I never thought about hyperbolic functions in terms of area parameters.
Now not only I know how hyperbolic functions come into existence from scratch but also got the firm understanding of circular trig functions.
Really appreciated sir.

[henry]

This fills in the gaps of all textbooks which start hyperbolic functions by saying we let cosh x =(e^x +e^-x)/2 and sinh x = (e^x - e^-x)/2 like it is the most natural thing in the world to do.

[nickreeves]

I can tell how much you love what you're talking about. Nothing is better than a teacher that is passionate about what they teach. Thank you so much and please don't stop making videos.

[sriadityasaisurampudi]

Even a boring lecture can be turned into an interesting one when an lecturer explains you with a smiling face and soothing voice like this ❣️❣️

[lowersaxon]

An all American genius, no doubt. Always smiling, short trousers, left handed. Well, no baseball cap but he simply lost it on his way into the studio. These guys prove everything within two minutes.

[srinidhikabra]

Amazing! The explanation i have been searching for a long time, and couldn’t find in any book have been staring me in the face all along! Thank you very much Dr Peyam!

[vasundarakrishnan]

The way you explain things is really amazing. I didn't even realise it was 30 minutes long before the video ended. Really simple and easy to understand. Thanks for this.

[kelvinsenteza]

Wow; I just loved this: was doing complex numbers in my first year course and suddenly had sing and cosh with no idea where they even came from; this has really helped! Keep up the great work

[doug]

I've always had trouble wrapping my head around the hyperbolic trig functions, but this really helped me understand them. Thank you for a great presentation!

[niloyroy]

Love you, really thought about the idea for 5-6 hours but i was unsuccessful.. I couldn't find any perfect answers on internet explaining the reason of the values of coshx, sinhx...
Then i found this.
Amazing.... I'm very much pleased now...keep up good works, sir...
Again, lot's of love and respect who found the idea to derive the formulas.. ❤️❤️❤️❤️❤️❤️🙏🙏🙏🙏🙏

[neilmccafferty]

superb. Finally someone has derived these odd looking expo functions from scratch. many thanks.

[lexvegers]

Just stunning how elegant geometry can be. I love it. Thank you very much.

[freepointsgals]

I wish I could half an hour of calculus and algebra with a smile on my face. I love math, but this man is something to aspire to.

[idolgin]

Great lecture! I might've seen this derivation, but forgot it, and you explained it perfectly. Thanks!

[charlesromano]

Incrediblely well explained. Now I feel I have a stronger understanding of the HB fns. The algebra/calculus used is truly "magical".

[shiina_mahiru_]

At least my thought on sinh and cosh is: mathematicians came across the function (e^x+e^-x)/2 and (e^x-e^-x)/2 for whatever reason --> then they realize they have lots of identities similar to sin and cos --> then they realize they are exactly the parametric form of points on hyperbola --> let's called them sinh and cosh

[sopheadou]

Thank you Dr. Peyam!! My doubt has been solved

[soniaaali]

Wow i was searching for this proof thumbs up man!!

[RieMUisthegoaT]

YES THANKS FOR THIS VIDEO!!

I really needed this