The complex relationship between regular and hyperbolic trig functions

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Hyperbolic Trig, intro, • Introduction to Hyperb...
Euler's Formula, FAST, • Euler's Formula (but i...
complex definition of sine and cosine: • Complex definitions of...
This is why the area is t/2 • Hyperbolic trig functi...
Even and Odd part of e^x, • Write e^x as a sum of ...
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blackpenredpen | 曹老師
tags:
connections between sin and sinh, connections between cos and cosh, hyperbolic trig functions, parameterization of the hyperbola, parameterization of x^2-y^2=1,

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17 сен 2018

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Комментарии : 196

@msolec2000 5 лет назад

9:27 isn't it? or isin(it)? ;)

@naremarcusmogakala8121 5 лет назад

damn

@theforgemaster1688 5 лет назад

Oh my god that was pure gold.

@Prxwler 5 лет назад

Best comment

@Sean-of9rs 3 года назад

you should be the winner

@kennethx7801 2 года назад

Loool

@fernandogaray1681 5 лет назад

I love this kind of videos. I love all the proof videos! Thanks!

@blackpenredpen 5 лет назад

Fernando Garay thank you

@retired5548 5 лет назад

complex relationship well played, good sir, well played

@barthennin6088 3 года назад

WoW! I haven't been this blown away since when I was shown Euler's identity!

@plaustrarius 5 лет назад

looking at the series expansions for exp(x) cosh(x) and sinh(x) is what really drove this point home for me.

@andresxj1 5 лет назад

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!

@blackpenredpen 5 лет назад

Thank you Andy! Glad to hear that you like it!!!

@nomadr1349 2 года назад

this is by far the best take on hyperbolic functions I found on youtube so far. And I looked far and wide too!

@Riiisuu 5 лет назад

Give this problem a try and when you’re ready, continue the video. Did *You* figure it out?

@markuswilliams4475 5 лет назад

Reece 5..4..3..2..1

@markuswilliams4475 5 лет назад

Math meanies 😡

@davidadegboye773 5 лет назад

Hey guys it's presh talwaker making sure you mind your decisions

@idolgin776 Год назад

I've been fascinated by these patterns for a while, and yours is an excellent explanation. Thanks!

@zralok 5 лет назад

Did you saw the joke isn't it? So the similarity of "i sin(it)" to isn't it.

@blackpenredpen 5 лет назад

der Ultrahero Nice catch!!!

@mike4ty4 5 лет назад

Or "I sign it"?

@angeldude101 2 года назад

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.

@rafaellisboa8493 5 лет назад

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!

@hemanthkotagiri8865 5 лет назад

Your videos are pretty amazing man. Keep going. 👌

@rockapedra1130 2 года назад

Excellent video! This is super interesting! Thanks for making these videos!

@borg972 5 лет назад

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

@MaChanceSevapore 5 лет назад

I just discovered your channel. Your videos are brilliant! Good thing they're your sponsor :D

@theomegaspec7923 6 месяцев назад

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.

@muriatik_ 5 лет назад

please continue this series!

@kennethx7801 2 года назад

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 ;)

@axelreispereiravaz1699 5 лет назад

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 !

@carlosraventosprieto2065 Год назад

Wow!! Thank you for the video!

@tm89681 2 года назад

Nice lecture👍

@kostantinos2297 5 лет назад

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?

@cuzeverynameistaken1283 5 лет назад

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

@filyb 4 года назад

@@cuzeverynameistaken1283 did you find one?

@joea-497kviews2 3 года назад

@@filyb he’s still working on it

@filyb 3 года назад

@@joea-497kviews2 lmao

@paulniziolek9200 2 года назад

@@joea-497kviews2 eta perhaps?

@ramkrishnapandey7737 3 года назад

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🙏

@aidan8858 5 лет назад

cos(it) + cosh(t) = cosh(it)

@cringy7-year-old5 7 месяцев назад

that implies cosh(t) = cos(t)/2

@penguincute3564 6 месяцев назад

So 2cosh(t) = cosh(it)?

@penguincute3564 6 месяцев назад

@@cringy7-year-old5that is seriously wrong…

@Questiala124 6 месяцев назад

Cos(it)=0?

@thalesbastos3915 5 лет назад

Thank you sooooo much!!!

@koltonjones866 5 лет назад

Your videos should be required viewing for most math classes. Do you do anything for dicrette algebra?

@6root91 2 года назад

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.

@leoarzeno 3 года назад

great video

@edgardojaviercanu4740 3 года назад

Beautiful!

@user-kw5qv6zl5e Месяц назад

Throw in a bit of an explanation of Eulers formula in terms of the Taylor series of e^x polynomial... love your passion...😊

@leeluu998 4 года назад

I hope you're gonna be a math teacher because yours videos are so clear and precises

@debaprasadparui4757 4 года назад

Sir you are awesome....!!!!

@ugursoydan8187 4 года назад

thanks

@irrelevantgaymer6195 4 года назад

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

@ffggddss 5 лет назад

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

@_DD_15 5 лет назад

Omg.. The biggest problem of my life.. Finally solved 😱😱😱😱😱impressed

@blackpenredpen 5 лет назад

DD Yup!!!! : )

@_DD_15 5 лет назад

@@blackpenredpen I have plenty of calculus books and have never seen that one around, weird :)

@alejrandom6592 26 дней назад

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)

@a.a.sunasara9202 5 лет назад

Bruh😍awesome.... Love ot

@Thoalfeqargamer 4 года назад

i love you man 💕💕💕💕

5 лет назад

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").

@alaba5085 5 лет назад

¡¡Lo máximo!!

@ian-ht1nf 5 лет назад

9: 26 "isin(it)"?

@crappypoopycrap9800 5 лет назад

nice one :)

@afafsalem739 5 лет назад

Yes it's very cool

@nishasharma-gk5bo 2 года назад

Look at this cute face he is blushing while playing with Maths 😍 ,maths must be his love.

@thomasolson7447 11 месяцев назад

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)]

@tunneloflight 2 года назад

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!

@bernardfinucane2061 5 лет назад

Very cool

@blackpenredpen 5 лет назад

Yay!

@nicholaslau3194 5 лет назад

Damn clickbait title! I wish professors can use clickbait to make lectures more interesting

@khaled014z 5 лет назад

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

@holyshit922 5 лет назад

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}

@h4c_18 5 лет назад

What about x=sec(t) and y=tan(t) for 0

@gwalla 8 месяцев назад

Are there other conic section analogues of the trigonometric functions? Parabolic sine? Elliptical cosine?

@davidmorochnick498 4 года назад

QUESTION: With -i out in front of sin(it), [-isin(it)], doesn't the proof fail?

@siddharthsengar8859 5 лет назад

after all i've been through in last year , "Imaginary" is a inappropriate title.

@comingshoon2717 4 года назад

Tengo sueño ... pero igual veo estos videos aunque hayan sido contenidos que vi hace muchos años!....

@Anonymous-rr5cx 4 года назад

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

@abdonecbishop 8 месяцев назад

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 ......

@spelunkerd 5 лет назад

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....

@spelunkerd 5 лет назад

Ah, found it here. ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-oLZoGEcJ2YE.html

@blackpenredpen 5 лет назад

It's here: ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-oLZoGEcJ2YE.html

@mehwishbhatti6207 Год назад

Can you please make a video relating tan and tanh

@billazz9176 4 года назад

RIGHT HERE, RIGHT HERE, RIGHT HERE

@Koisheep 5 лет назад

Well to some extent you can also use x(t)=sqrt(1-t) and y(t)=sqrt(t) I mean (?)

@helloitsme7553 5 лет назад

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)

@armchairtin-kicker503 6 месяцев назад

Then there is Osborn's Rule, a very useful relationship between trigonometric and hyperbolic functions and identities.

@rishinandha_vanchi 4 года назад

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

@rishinandha_vanchi 4 года назад

Oh You mentioned it? Just now say it.

@TheNerd484 5 лет назад

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.

@drshamajain4149 4 года назад

What we do with the part of the hyperbola on the left side

@user-kw5qv6zl5e Месяц назад

How's your stock of whiteboard pens ? 😊

@KwongBaby 5 лет назад

What's the usage of sinh and cosh？

@nicolasinostrozamoreno4248 5 лет назад

Why you don't have spanish subtitles ?? Its so interesting

@coisasdemacho1594 5 лет назад

If I stick a string at (0,0) with lenght=1 and make a complete rotation it will give me x^2 + y^2=1 and all trig. magic. Following the same "string" example, how can I define hiperbolic graphic x^2 - y^2=1????

@kaszimidaczi 5 лет назад

Easy, you just need a stretchy string to follow the hyperbola with ;d

@Harkmagic 5 лет назад

I think you need to draw that circle in the imaginary world.

@user-vj7uc9tj7c 5 лет назад

You can do something similiar. In a spherical world, that has radius of one, if you draw a specific line, you'll get the unit circle. In a specific hyperbolic world, do the same thing, and get the unit hyperbola. These 2 worlds' geometry work in a non-flat world. They (and some others) are called Non-Euclidean Geometries. The Euclidean one works only on a flat surface/world, and is the one we're most familiar with.

@mathteacher2651 4 года назад

Another great video - kid!

@bullinmd 3 года назад

Ever heard of gd(x), the Gudermannian function?

@artey6671 5 лет назад

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.

@koenth2359 5 лет назад

Yeah, Tibees just did that Bob Ross style!

@artey6671 5 лет назад

You mean her newest video? I don't see any cosh in there.

@koenth2359 5 лет назад

@@artey6671 yeah guess you are right. May have misremembered.

@zralok 5 лет назад

That's in my textbook xD

@dwarkeshdhamechahiteshdham7336 5 лет назад

Wow

@snejpu2508 5 лет назад

What do you need hyperbolic functions for in math? Of course, except defining them and solving equations with them?

@nicolastroncoso1791 5 лет назад

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

@rafaellisboa8493 5 лет назад

Could you make a vid about Lobachevsky space pleaseee? don't make me beg

@rot6015 5 лет назад

OMG!!!! @&@&&@&@&@; THIS IS SO EXTREME LIKE THE TITLE!!!

@geoffstrickler 2 года назад

3^2 - 2^2 = 1 too. 😎

@ny6u 3 года назад

Gorgeous

@decay2__ 5 лет назад

You probably don't know this but you made a pun at 5:49

@canaDavid1 3 года назад

Wait... Are the trig functions C -> N? Or can some input a+bi give imaginary output?

@justacutepotato2945 3 года назад

they're C->C. Also, you put C->N, pretty sure you meant C->R or C-> [-1,1].

@canaDavid1 3 года назад

@@justacutepotato2945 yeah, you're right. And I meant C->R.

@shoobadoo123 3 года назад

What about cosh(it)

@Lucky10279 2 года назад

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.

@maximkarpov5123 2 года назад

Area of circle with r = 1: pi * r ^ 2 = pi. Not 2pi.

@AayushDuttaStanish 5 лет назад

Can u Integrate xtan(x)? 😛 Help me if u can. I really love ur Videos. I learn a lot from them. Thanks

@azmath2059 5 лет назад

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!!

@jimallysonnevado3973 5 лет назад

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

@blackpenredpen 5 лет назад

Jim Allyson Nevado as I said in the video. I will do a proof for that. So stay tuned!

@jimallysonnevado3973 5 лет назад

blackpenredpen waiting for that

@jimallysonnevado3973 5 лет назад

blackpenredpen oops i apologize for not listening carefully