Hyperbolic Functions - Formula Sheet: bit.ly/4eZ5gyo Final Exams and Video Playlists: www.video-tutor.net/ Full-Length Math & Science Videos: www.patreon.com/mathsciencetutor/collections
Complex numbers are usually considered too... _complex_ for simpler topics like this, but they are absolutely necessary to fully appreciate the symmetry between hyperbolic trig and spherical trig. If you already know Euler's Formula exp(ix) = cos(x) + isin(x), you can try taking the average of it with a mirrored copy of itself: ½(exp(ix) + exp(-ix)) = ½(cos(x) + isin(x) + cos(-x) + isin(-x)) = ½(cos(x) + isin(x) + cos(x) - isin(x)) = ½(2cos(x) + 0isin(x) = cos(x) = ½(exp(ix) + exp(-ix)) = cosh(ix) You can do a similar derivation for sin(x) = sinh(ix)/i Hyperbolic and spherical trig are two sides of the same coin, and i is the bridge between them.
@@flipflop1276 That's just substituting x for -x into the original formula. To simplify further would require taking advantage of sine and cosine being an odd and even function respectively. Sine being odd means that its graph is rotationally symmetric around the origin, that sin(-x) = -sin(x), and that its Taylor Series consists only of terms with an odd degree. Cosine being even means that its graph is mirror symmetric across the y-axis, that cos(-x) = cos(x), and that its Taylor Series consists only of terms with an even degree.
Professor Organic Chemistry Tutor, thank you for a Basic Introduction to Hyperbolic Functions in Calculus. Hyperbolic Functions has many applications in Science and Applied Engineering. This is an error free video/lecture on TH-cam TV with the Organic Chemistry Tutor.
Excellent presentation! Thank you. In the unit circle equation (x2 + y2 = 1), the number 1 represents the radius . In the unit hyperbola equation ((x2 - y2 = 1), what does the "1" represent?
The "unit" part of the unit hyoerbola is not the 1, though at least in the case of the unit hyperbola, the distance from the center to the vertex is 1 also. The definition/standard form of a hyoerbola is (sparing the fact it could be y²-x²): x² y² _ - _ = 1 a² b² So if you had x² - y² = 4, really it (x²/4)-(y²/4) =1 => (x²/2²)-(y²/2²) =1 standard form The unit part of the unit hyperbola is that a and b = 1 The foci have distance sqrt(2) from the origin in the unit hyperbola, and generally = sqrt(a² + b²) Honestly, having the focal length being 1 could be consistent still with the regular trig. Heck I would prefer it! This isn't the best description, but I hope this helps somewhat.
That is not true by definition. sinh(x) =/= cosh(x). There are identities relating them, but that is never true. It is equivalent to saying -e^(-x) = 0 or e^(-x) = 0. e^x and e^(-x) are never 0, so that's false. Assume sinh(x) = cosh(x) => sinh(x)-cosh(x) = 0 => ½(e^x + e^-x - e^x + e^-x) = 0 =>½(2e^-x) = 0 =>e^-x = 0 Same for if you had cosh(x)-sinh(x).
Just a second! If definition of the function doesn't tell us that to any element from domain can be assigned only one element from codomain? If yes then: X2 + y2 = 1 Is not a function (x2 means x square)
Domain are the possible x values for a function while the range are the possible y values for the function. for examples as shown, the range for cosh(x) is 1 to infinity because the lowest possible y value (as shown in the graph) is 1 and it can increase indefinitely into infinity, I hope this helps.
Hyperbolic Functions - Formula Sheet: bit.ly/4eZ5gyo
Final Exams and Video Playlists: www.video-tutor.net/
Full-Length Math & Science Videos: www.patreon.com/mathsciencetutor/collections
Finally graduated high school. Moved cities. Started uni. But one thing stays the same. This man teaches me math
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All the way from South Africa 🇿🇦, your videos are the best brother
Been Waiting for this since the beginning of the year.. Thank you so much
Brother you posted this in the very correct time, in need of this.
Complex numbers are usually considered too... _complex_ for simpler topics like this, but they are absolutely necessary to fully appreciate the symmetry between hyperbolic trig and spherical trig.
If you already know Euler's Formula exp(ix) = cos(x) + isin(x), you can try taking the average of it with a mirrored copy of itself:
½(exp(ix) + exp(-ix))
= ½(cos(x) + isin(x) + cos(-x) + isin(-x)) = ½(cos(x) + isin(x) + cos(x) - isin(x))
= ½(2cos(x) + 0isin(x)
= cos(x) = ½(exp(ix) + exp(-ix)) = cosh(ix)
You can do a similar derivation for sin(x) = sinh(ix)/i
Hyperbolic and spherical trig are two sides of the same coin, and i is the bridge between them.
thanksss this is what I've been searching for 😮
Hey! Could you explain why exp(-ix)=cos(-x)+isin(-x) please? I'm trying to do what you just showed :)
This is so cool btw
@@flipflop1276 That's just substituting x for -x into the original formula. To simplify further would require taking advantage of sine and cosine being an odd and even function respectively.
Sine being odd means that its graph is rotationally symmetric around the origin, that sin(-x) = -sin(x), and that its Taylor Series consists only of terms with an odd degree.
Cosine being even means that its graph is mirror symmetric across the y-axis, that cos(-x) = cos(x), and that its Taylor Series consists only of terms with an even degree.
Really thank you for sharing your insights.
Only 50 comments?! I literally learn everything above my boards from this man!!
really out here saving lives
Please show us how to prove hyperbolic identities as well , and identity derivations 🎉
Chemistry silently crying in the corner
Bro you're amazing Thank you for every thing❤
Thanks great tr. ❤❤🎉 Getting the whole concept here 🎉🎉
Professor Organic Chemistry Tutor, thank you for a Basic Introduction to Hyperbolic Functions in Calculus. Hyperbolic Functions has many applications in Science and Applied Engineering. This is an error free video/lecture on TH-cam TV with the Organic Chemistry Tutor.
next hyperbolic identities !!
Thanks man u have really helped me alot
This is what i needed !!!
Bruh you've got that thumbnail wrong 😅
He didn’t
Never mind I think you meant it was eye catching or something
@@BingW369nah it's just wrong
@@BingW369look at cosh it’s minus when it should be plus
@@BingW369I think he meant coshx..it's (e^x+e^(-x))/2 ..
In the thumbnail, cosh x should be plus, not minus, between e^x and e^(-x)
Thank you!
Thank you very much.
Thanks,,
Which software do you use for displaying this concepts..
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Nice video
From the thumbnail, cosh(x) should be (exp(x) + exp(-x)) /2
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Excellent presentation! Thank you. In the unit circle equation (x2 + y2 = 1), the number 1 represents the radius
. In the unit hyperbola equation ((x2 - y2 = 1), what does the "1" represent?
The "unit" part of the unit hyoerbola is not the 1, though at least in the case of the unit hyperbola, the distance from the center to the vertex is 1 also.
The definition/standard form of a hyoerbola is (sparing the fact it could be y²-x²):
x² y²
_ - _ = 1
a² b²
So if you had x² - y² = 4, really it
(x²/4)-(y²/4) =1
=>
(x²/2²)-(y²/2²) =1 standard form
The unit part of the unit hyperbola is that a and b = 1
The foci have distance sqrt(2) from the origin in the unit hyperbola, and generally =
sqrt(a² + b²)
Honestly, having the focal length being 1 could be consistent still with the regular trig. Heck I would prefer it! This isn't the best description, but I hope this helps somewhat.
I've definitely walked into a party I wasn't invited, pardon me.
On the thumbnail associated with this video, both sinh and cosh have the same value (ex - e-x)/2 (with the minus sign in both).
That is not true by definition.
sinh(x) =/= cosh(x).
There are identities relating them, but that is never true.
It is equivalent to saying -e^(-x) = 0 or e^(-x) = 0. e^x and e^(-x) are never 0, so that's false.
Assume
sinh(x) = cosh(x)
=>
sinh(x)-cosh(x) = 0
=>
½(e^x + e^-x - e^x + e^-x) = 0
=>½(2e^-x) = 0
=>e^-x = 0
Same for if you had cosh(x)-sinh(x).
thanks sir
Just a second!
If definition of the function doesn't tell us that to any element from domain can be assigned only one element from codomain?
If yes then:
X2 + y2 = 1
Is not a function
(x2 means x square)
Sir , why and how hyperbolic function are written in terms of exponential function.
Please explain .
Ah yes the most perfect circle xD
Thanks
For hyp
Sir
Thanks
can you please do videos on Cayley Hamilton , eigen values and vectors and triple integration :)
Yeah
Excuse me sir , would you please explain what domain and range implies ?
Domain are the possible x values for a function while the range are the possible y values for the function. for examples as shown, the range for cosh(x) is 1 to infinity because the lowest possible y value (as shown in the graph) is 1 and it can increase indefinitely into infinity, I hope this helps.
Hyperbolic function basic mcq
The comments are so fresh lol
Hi could you please do a video on how to find first order Lagrange conditions for a function like rk+wl and constraint K^1/4L^2/3
Why is it that the cosec and sec don’t follow the formula of the unicircle? If you were to shift, it’ll be x^2 - y^2 rite?
Why didnt my teacher lectured like this
Did you make the thumbnail incorrect on purpose lol?
Wow, I'm really early
Sup brother 😂
Same
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Ew, a lot of people on here are minors.
@@sebas31415 bruh obvi? 😭
Why’s a 10 year old watching these videos
Have you even hit puberty
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Double check your thumbnail.
😑it is correct, just the sign is wrong