I've been a Brilliant member for a year and a half now, and it all began with your special offer. I'm delighted with the app! I've learnt a lot and I've enjoyed it so much! So thank you for introducing me to Brilliant and thank Brilliant for sponsoring you!
I actually discovered this when I noticed that cosh and sinh had a structure similar to the inner and outer products of geometric algebra, which are defined as the symetric and antisymetric components of the full product. But the inner and outer products are usually defined with sin and cos... along with i. This is what led me to realize the relation between them and the real and imaginary / even and odd parts of the exponential. The only reason one relation is x^2 + y^2 = 1 and the other is x^2 - y^2 = 1 is because the imaginary factor of y flips the sign when squared. It felt so awesome to find that on my own. Now I kind of want to make a visualization of the complex exponential's even and odd parts to try and get the hyperbolic and spherical trig functions to appear on different axes of the same graph.
An easy way to remember this is that, e^(ix)=cosx+isinx on one hand, on the other, e^ix=cosh(ix)+sinh(ix). Match the even part of one side with the even part of the other side, and do the same with the odd part. You get that cosh(ix)=cosx and sinh(ix)=isinx. Now evaluate these functions at x=it and you get the rest ;)
I always asked myself why the hyperbolic trigs functions and the complex trigs functions looked so similar. Even my teacher didn't showed this relation. Now i have my answer ! Thanks BPRP !
Very interesting. I was wondering about the relation between the hyperbolic trig functions and the complex definitions of the trig functions after seeing one of your videos, and you explained these concepts so clearly.
You solve mathematics like you are hanging out with Ur friends😜😜 And Ur excitement after solving is just awesome. Just because of teacher like u I'm happy of being a mathematic student. Thank you🙏
Before 1 min: There's a 3rd way to interpret the angle - the arc length subtended on a unit circle, whose equation you've written: x² + y² = 1. This may or may not work for the unit rectangular hyperbola; I'm checking into that. It does have the right behavior near 0, and it does go to ∞, but those are no guarantee... Fred
I was searching for these formulae (didn't need the proofs, but they were cool too) for about an hour until I found it here and was able to answer my question.
Great one, thanks! If you could do more parameterization videos it would be great since finding them is always so confusing.. also integrations along a curve with parameterization
What I think is cool is if you were to somehow create a 4D graph and declare your x, y, z, and t axis, and call the z axis the imaginary input and call the t axis the imaginary output, the function x^2+y^2=1 on the t axis looks like x^2-y^2=1 and vice versa. So I kind of think of the hyperbolic function as a complex version of the circle function and vice versa
Rational paramerization of hyperbola is based on observation (1-t^2)^2+(2t)^2=(1+t^2)^2 (2t)^2=(1+t^2)^2-(1-t^2)^2 1=\left(\frac{1+t^2}{2t} ight)^{2} -\left(\frac{1-t^2}{2t} ight)^{2}
Easy way: exp(it)=cos(t)+i*sin(t) But also exp(it)=cosh(it)+sinh(it) Pairing up the odd part with odd part, and even with even we get: cosh(it) = cos(t) sinh(it) = i*sin(t)
Plot them! The hyperbolic sin and cos “jump” off the tops and bottoms of the sin and cos at right angles in the y-i plane. Likewise, when sinh and cosh are real, sin and cos are at right angles in the y-i plane. Etc…. It is beautiful. Next extend to tan and tanh, sec and sech …. Then extend to Bessel and J functions!
Sir at 6:50 u said imaginary looking Theta = it So t is also imaginary so that they can be real Sin theta = real function Sin h x= imaginary function ?????? Sir please please clear this doubt Thank you
0:57 Shouldn't the area = t? The area 0f a unit circle is A = 2π and the area of a sector of a circle is A*(corresponding angle of sector/2π), assuming the angle is in radians. Hence, the area in the diagram should be 2π•t/2π = t.
I noticed that to. But I went the arctan route. cos(i*arctan(3/4))-i*sin(i*arctan(3/4))=1.9031323020709 cos(arctan(i*(3/4)))+sin(arctan(i*(3/4))) =sqrt(7)(4/7+3/7i) for tan= i*3/4 Works just fine this way. You can do geometry with it. It's just pythagorous' theorem with an 'i' in it. [x/sqrt(x^2+y^2), y/sqrt(x^2+y^2)]
ellipse eqn in complex extended x-plane-y-axis will be a hyperbola in the Im-x-side. This so parametric forms cos and cosh are complex and real counterparts
makes me want to say.....this is wonderful short video.... beautiful work...so ...so ..excellent...but i think you need to add a quick physic conclusion to your video.........this certainly is one of the slickest short video in circulation....why?....because you connects non-Euclidean equilateral triangle's surface area(excess/deficit) change to a Euclidean triangle's total energy change and the triangle's inertial mass change dependent on (a function off) the average of the total number of summed ''-' , '+' and '0' Gaussian curved triangle edges counted ......
QUESTION: With -i out in front of sin(it), [-isin(it)], doesn't the proof fail?
6 років тому+4
Cool!!!! So one can get the derivative of sinh and cosh using chain+product rule from the equal sin/cos statement, never thought on that :-O I always did that from the definition of sinh/cosh only ("e-stuff").
Tbh I've always felt like this is true because I can say integral of 1/1-x^2 dx = integral of 1/1+(ix)^2 and then use u-sub. But at the same time, the integral is tanh(x)
hey bprp, there was an integral video involving cos's and sin's I think and you solved it with a creative way of adding 2 solutions of 2 integrals together and I can't find that video, any ideas? thank you :D
Still unsatisfied how can you say that this equation can make the area t/2 and is there a way to come up with this formula using calculus like what you can do with sin and cosine by knowing the derivatives first then using taylor then coming up with a formula like eulers identity
to simplify the work or notation of multiple real life problems, instead of putting an enormous amount of digits you simply use hyperbolic functions, same as trigonometry in general
cosh(it) = cos(t) Like, it's simply inserting i*t into x for cosh(x) and get cos(t) as the result just like as if you have inserted i*t for cos(x) and get cosh(t) as a result.
Is it possible to actually understand complex numbers, or are they simply an abstract tool in the world of mathematics? Like, I can understand integers as "counting things", real numbers as "measuring things", and sines and cosines as "the vertical and horizontal weights of a diagonal line". But imaginary numbers and complex numbers ... ? I don't see them in the real world anywhere. Maybe I'm not looking hard enough?
I think of them as vectors that rotate at a certain frequency, or phasors. They are used all the time in electrical engineerig, in the field of power systems.
I don't doubt the utility of complex numbers in real-world calculations, but they seem to be a way to arrive at a real-world result rather than a representation of the real-world result: for example, if two electrical signals are out of phase with one another, the result is the superposition of sine curves, with no visible evidence of any imaginary components. So I'm wondering if they appear visibly in the real world anywhere. If someone asked where I can see Fibonacci Numbers I could point to a sunflower. Is there anywhere I could see complex numbers?
Complex numbers are as real as the rotation you are doing in your problem. For most engineering and physics applications those rotations are very real. Hell, 3d rotation can't even be mathematically described properly without quaternions, which are basically more complex imaginary numbers.
Show me an imaginary number. Don't just tell me that they factor into the arithmetic of sine waves; show me an imaginary number in the real world. I'm not sure it can be done, but if it can, I'd like to see it.
Showing an imaginary number is no more possible than showing a real number. Numbers are not real physical things that you can point to or reach out and touch, even our real numbers are just abstractions used to represent reality. So if I tell you to stand up and turn and face to your left and tell you that is the number 'i' it is just as valid as saying telling you to count the number of fingers on your hand and telling you that it is the number '5'.
9:27 isn't it? or isin(it)? ;)
damn
Oh my god that was pure gold.
Best comment
you should be the winner
Loool
WoW! I haven't been this blown away since when I was shown Euler's identity!
I've been a Brilliant member for a year and a half now, and it all began with your special offer.
I'm delighted with the app! I've learnt a lot and I've enjoyed it so much! So thank you for introducing me to Brilliant and thank Brilliant for sponsoring you!
Thank you Andy! Glad to hear that you like it!!!
I love this kind of videos. I love all the proof videos! Thanks!
Fernando Garay thank you
this is by far the best take on hyperbolic functions I found on youtube so far. And I looked far and wide too!
looking at the series expansions for exp(x) cosh(x) and sinh(x) is what really drove this point home for me.
I've been fascinated by these patterns for a while, and yours is an excellent explanation. Thanks!
complex relationship
well played, good sir, well played
I actually discovered this when I noticed that cosh and sinh had a structure similar to the inner and outer products of geometric algebra, which are defined as the symetric and antisymetric components of the full product. But the inner and outer products are usually defined with sin and cos... along with i. This is what led me to realize the relation between them and the real and imaginary / even and odd parts of the exponential. The only reason one relation is x^2 + y^2 = 1 and the other is x^2 - y^2 = 1 is because the imaginary factor of y flips the sign when squared.
It felt so awesome to find that on my own. Now I kind of want to make a visualization of the complex exponential's even and odd parts to try and get the hyperbolic and spherical trig functions to appear on different axes of the same graph.
An easy way to remember this is that,
e^(ix)=cosx+isinx on one hand, on the other, e^ix=cosh(ix)+sinh(ix). Match the even part of one side with the even part of the other side, and do the same with the odd part. You get that cosh(ix)=cosx and sinh(ix)=isinx. Now evaluate these functions at x=it and you get the rest ;)
I always asked myself why the hyperbolic trigs functions and the complex trigs functions looked so similar. Even my teacher didn't showed this relation. Now i have my answer ! Thanks BPRP !
Very interesting. I was wondering about the relation between the hyperbolic trig functions and the complex definitions of the trig functions after seeing one of your videos, and you explained these concepts so clearly.
You solve mathematics like you are hanging out with Ur friends😜😜
And Ur excitement after solving is just awesome. Just because of teacher like u I'm happy of being a mathematic student. Thank you🙏
Is there a geometrical representation of tanh(t), coth(t) etc, just like cosh(t) and sinh(t) are the x and y values of the points of the hyperbola?
Putting this comment just so if someone else finds it. Right now its late where Im from so I'll try and see if there is one in the morning
@@cuzeverynameistaken1283 did you find one?
@@filyb he’s still working on it
@@joea-497kviews2 lmao
@@joea-497kviews2 eta perhaps?
It's like you can read my mind comrade! second time I was studying some maths and you made a vid exactly about what I was studying, great vid!
Give this problem a try and when you’re ready, continue the video.
Did *You* figure it out?
Reece 5..4..3..2..1
Math meanies 😡
Hey guys it's presh talwaker making sure you mind your decisions
Before 1 min: There's a 3rd way to interpret the angle - the arc length subtended on a unit circle, whose equation you've written: x² + y² = 1.
This may or may not work for the unit rectangular hyperbola; I'm checking into that. It does have the right behavior near 0, and it does go to ∞, but those are no guarantee...
Fred
I was searching for these formulae (didn't need the proofs, but they were cool too) for about an hour until I found it here and was able to answer my question.
Great one, thanks!
If you could do more parameterization videos it would be great since finding them is always so confusing.. also integrations along a curve with parameterization
Your videos should be required viewing for most math classes. Do you do anything for dicrette algebra?
Omg.. The biggest problem of my life.. Finally solved 😱😱😱😱😱impressed
DD
Yup!!!! : )
@@blackpenredpen I have plenty of calculus books and have never seen that one around, weird :)
Throw in a bit of an explanation of Eulers formula in terms of the Taylor series of e^x polynomial... love your passion...😊
I just discovered your channel. Your videos are brilliant! Good thing they're your sponsor :D
cos(it) + cosh(t) = cosh(it)
that implies cosh(t) = cos(t)/2
So 2cosh(t) = cosh(it)?
@@cringy7-year-old5that is seriously wrong…
Cos(it)=0?
I hope you're gonna be a math teacher because yours videos are so clear and precises
What I think is cool is if you were to somehow create a 4D graph and declare your x, y, z, and t axis, and call the z axis the imaginary input and call the t axis the imaginary output, the function x^2+y^2=1 on the t axis looks like x^2-y^2=1 and vice versa. So I kind of think of the hyperbolic function as a complex version of the circle function and vice versa
Your videos are pretty amazing man. Keep going. 👌
Rational paramerization of hyperbola is based on observation
(1-t^2)^2+(2t)^2=(1+t^2)^2
(2t)^2=(1+t^2)^2-(1-t^2)^2
1=\left(\frac{1+t^2}{2t}
ight)^{2} -\left(\frac{1-t^2}{2t}
ight)^{2}
I'm headed back to your channel to find the link to "even" and "odd" parts of e^t, described at 15:18. Not sure where to look....
Ah, found it here. ua-cam.com/video/oLZoGEcJ2YE/v-deo.html
It's here: ua-cam.com/video/oLZoGEcJ2YE/v-deo.html
Excellent video! This is super interesting! Thanks for making these videos!
Easy way:
exp(it)=cos(t)+i*sin(t)
But also
exp(it)=cosh(it)+sinh(it)
Pairing up the odd part with odd part, and even with even we get:
cosh(it) = cos(t)
sinh(it) = i*sin(t)
9: 26 "isin(it)"?
nice one :)
please continue this series!
Wow!! Thank you for the video!
Look at this cute face he is blushing while playing with Maths 😍 ,maths must be his love.
Did you saw the joke isn't it?
So the similarity of "i sin(it)" to isn't it.
der Ultrahero
Nice catch!!!
Or "I sign it"?
Plot them! The hyperbolic sin and cos “jump” off the tops and bottoms of the sin and cos at right angles in the y-i plane. Likewise, when sinh and cosh are real, sin and cos are at right angles in the y-i plane. Etc…. It is beautiful. Next extend to tan and tanh, sec and sech …. Then extend to Bessel and J functions!
Sir at 6:50 u said imaginary looking
Theta = it
So t is also imaginary so that they can be real
Sin theta = real function
Sin h x= imaginary function
??????
Sir please please clear this doubt
Thank you
Thank you sooooo much!!!
0:57 Shouldn't the area = t? The area 0f a unit circle is A = 2π and the area of a sector of a circle is A*(corresponding angle of sector/2π), assuming the angle is in radians. Hence, the area in the diagram should be 2π•t/2π = t.
Area of circle with r = 1: pi * r ^ 2 = pi. Not 2pi.
Nice lecture👍
I noticed that to. But I went the arctan route.
cos(i*arctan(3/4))-i*sin(i*arctan(3/4))=1.9031323020709
cos(arctan(i*(3/4)))+sin(arctan(i*(3/4))) =sqrt(7)(4/7+3/7i) for tan= i*3/4
Works just fine this way. You can do geometry with it.
It's just pythagorous' theorem with an 'i' in it.
[x/sqrt(x^2+y^2), y/sqrt(x^2+y^2)]
Are there other conic section analogues of the trigonometric functions? Parabolic sine? Elliptical cosine?
Cool again! Love your content!
ellipse eqn in complex extended x-plane-y-axis will be a hyperbola in the Im-x-side. This so parametric forms cos and cosh are complex and real counterparts
Oh You mentioned it? Just now say it.
What about x=sec(t) and y=tan(t) for 0
Beautiful!
Sir you are awesome....!!!!
makes me want to say.....this is wonderful short video.... beautiful work...so ...so ..excellent...but i think you need to add a quick physic conclusion to your video.........this certainly is one of the slickest short video in circulation....why?....because you connects non-Euclidean equilateral triangle's surface area(excess/deficit) change to a Euclidean triangle's total energy change and the triangle's inertial mass change dependent on (a function off) the average of the total number of summed ''-' , '+' and '0' Gaussian curved triangle edges counted ......
QUESTION: With -i out in front of sin(it), [-isin(it)], doesn't the proof fail?
Cool!!!! So one can get the derivative of sinh and cosh using chain+product rule from the equal sin/cos statement, never thought on that :-O I always did that from the definition of sinh/cosh only ("e-stuff").
thanks
Wait... Are the trig functions C -> N? Or can some input a+bi give imaginary output?
they're C->C. Also, you put C->N, pretty sure you meant C->R or C-> [-1,1].
@@justacutepotato2945 yeah, you're right. And I meant C->R.
great video
RIGHT HERE, RIGHT HERE, RIGHT HERE
You don't even need Euler's formula to show that cos(it) = cosh(t). You can also show that their power series are the same.
Yeah, Tibees just did that Bob Ross style!
You mean her newest video? I don't see any cosh in there.
@@artey6671 yeah guess you are right. May have misremembered.
What we do with the part of the hyperbola on the left side
i love you man 💕💕💕💕
We just went over sinh and cosh in my calc class today. What are the chances? This is a much more complete explination than we got.
How's your stock of whiteboard pens ? 😊
Tbh I've always felt like this is true because I can say integral of 1/1-x^2 dx = integral of 1/1+(ix)^2 and then use u-sub. But at the same time, the integral is tanh(x)
What's the usage of sinh and cosh？
Can you please make a video relating tan and tanh
Ever heard of gd(x), the Gudermannian function?
Bruh😍awesome.... Love ot
¡¡Lo máximo!!
Well to some extent you can also use x(t)=sqrt(1-t) and y(t)=sqrt(t) I mean (?)
after all i've been through in last year , "Imaginary" is a inappropriate title.
Very cool
Yay!
Tengo sueño ... pero igual veo estos videos aunque hayan sido contenidos que vi hace muchos años!....
Damn clickbait title! I wish professors can use clickbait to make lectures more interesting
Yes it's very cool
What about cosh(it)
Then there is Osborn's Rule, a very useful relationship between trigonometric and hyperbolic functions and identities.
We have theta be real and t be real also. So, how can we put theta = it ??
We're extending the theta and t to complex world
hey bprp, there was an integral video involving cos's and sin's I think and you solved it with a creative way of adding 2 solutions of 2 integrals together and I can't find that video, any ideas? thank you :D
You probably don't know this but you made a pun at 5:49
Why you don't have spanish subtitles ?? Its so interesting
Still unsatisfied how can you say that this equation can make the area t/2 and is there a way to come up with this formula using calculus like what you can do with sin and cosine by knowing the derivatives first then using taylor then coming up with a formula like eulers identity
Jim Allyson Nevado as I said in the video. I will do a proof for that. So stay tuned!
blackpenredpen waiting for that
blackpenredpen oops i apologize for not listening carefully
What do you need hyperbolic functions for in math? Of course, except defining them and solving equations with them?
to simplify the work or notation of multiple real life problems, instead of putting an enormous amount of digits you simply use hyperbolic functions, same as trigonometry in general
great video. but try starting from first principles and proving that for a hyperbola x=cosht and y=sinht and see how long that takes you!!
What does E equal?
2.71828182... it is transcendental.
Approximately 2.7182818
Its a irrational and transcendental number like pi, its related to the exponential function
(1+1/n)^n if n goes to infinity
e=3=pi *isn't* *it* *?*
Sum from 0 to infinity of 1/n! 😃
Why you don't advatise for patrion
3^2 - 2^2 = 1 too. 😎
Could you make a vid about Lobachevsky space pleaseee? don't make me beg
Can u Integrate xtan(x)? 😛 Help me if u can. I really love ur Videos. I learn a lot from them. Thanks
sinh(x) = -i sin(ix)
cosh(x) = cos(ix)
tanh(x) = -i tan(ix)
sinh(ix) = i sin(x)
cosh(ix) = cos(x)
tanh(ix) = i tan(x)
lim x → 0 sin2x^(tan2x) ²
Btw, is this a newly discovered relation?
Yes, considering that the history of mathematics goes back millennia. Wikipedia dates them to the 1760s.
Another great video - kid!
cos(it)=cosh(t) is interesting. cosh(it)=? is a more interesting question though.
cosh(it) = cos(t)
Like, it's simply inserting i*t into x for cosh(x) and get cos(t) as the result just like as if you have inserted i*t for cos(x) and get cosh(t) as a result.
Is it possible to actually understand complex numbers, or are they simply an abstract tool in the world of mathematics? Like, I can understand integers as "counting things", real numbers as "measuring things", and sines and cosines as "the vertical and horizontal weights of a diagonal line". But imaginary numbers and complex numbers ... ? I don't see them in the real world anywhere. Maybe I'm not looking hard enough?
I think of them as vectors that rotate at a certain frequency, or phasors. They are used all the time in electrical engineerig, in the field of power systems.
I don't doubt the utility of complex numbers in real-world calculations, but they seem to be a way to arrive at a real-world result rather than a representation of the real-world result: for example, if two electrical signals are out of phase with one another, the result is the superposition of sine curves, with no visible evidence of any imaginary components. So I'm wondering if they appear visibly in the real world anywhere. If someone asked where I can see Fibonacci Numbers I could point to a sunflower. Is there anywhere I could see complex numbers?
Complex numbers are as real as the rotation you are doing in your problem. For most engineering and physics applications those rotations are very real. Hell, 3d rotation can't even be mathematically described properly without quaternions, which are basically more complex imaginary numbers.
Show me an imaginary number. Don't just tell me that they factor into the arithmetic of sine waves; show me an imaginary number in the real world. I'm not sure it can be done, but if it can, I'd like to see it.
Showing an imaginary number is no more possible than showing a real number. Numbers are not real physical things that you can point to or reach out and touch, even our real numbers are just abstractions used to represent reality.
So if I tell you to stand up and turn and face to your left and tell you that is the number 'i' it is just as valid as saying telling you to count the number of fingers on your hand and telling you that it is the number '5'.
OMG!!!! @&@&&@&@&@; THIS IS SO EXTREME LIKE THE TITLE!!!
ALGEBRAIC EXPRESSIONS HATE HIM.
Wow
Notice how close they are to co shit.
Gorgeous
*slaps theta* TAG YOU'RE (it)
The return of black shirt red shirt.