Math for fun, sin(sin(z))=1

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The trigonometric equation sin(z)=1 is fairly easy to solve but not sin(sin(z))=1. Here we will be using the complex exponential definition of sine, which is from Euler's formula e^(i*theta)=cos(theta)+i*sin(theta), to solve this equation. We will see sin(sinz)=1 actually has infinitely many complex solutions, just like how we solve sin(z)=2.
sin(z)=2 • Math for fun, sin(z)=2 got over 1M views recently. Thank you all! Enjoy solving sin(sin(z))=1
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29 сен 2024

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Комментарии : 472

@tibees 3 года назад

I was so focussed on the board I didn't see the shirt until I saw a comment 😂Thanks for wearing it!

@ninjacow2200 3 года назад

Tibees gang

@blackpenredpen 3 года назад

😃 😃

@currentlynowshyam1783 3 года назад

😂

@currentlynowshyam1783 3 года назад

😂😂😂

@currentlynowshyam1783 3 года назад

Super 😂💞

@ericbright1742 3 года назад

"This looks... yeah." Sums it up quite nicely.

@breadlegend2480 3 года назад

You know its worse when he has more than 2 pens

@Aaron-gx9gv 3 года назад

Me: using letters as variable Him:😃

@Vampianist3 3 года назад

6:15 and that’s why most people don’t have happy faces when they do maths

@debblez 3 года назад

6:11*

@aadityabhattacharya1811 3 года назад

Lol

@joshmcdouglas1720 3 года назад

4:41 “Alright so it looks.... yeah”😭

@lazarussevy2777 2 месяца назад

I wish I could see more videos like this. Complex numbers are my specialty, and I'd love to learn more about them.

@Henrix1998 2 года назад

The simplest case n=m=0 gives rather nice and tidy solution Z = π/2 - i*ln(π/2 ± √(π²/4-1)) which is quite close to just π/2-i

@egillandersson1780 3 года назад

Ah ! With this t-shirt, I finally understand : you want a beard as long as Tibee's hair !

@blackpenredpen 3 года назад

😆 Fun fact: my beard is growing at a logarithmic rate.

@zacksima8333 3 года назад

@@blackpenredpen legend says if you live to infinity years old your beard size will approach a constant called pen’s constant

@NihilistEmier 3 года назад

The videos on the shorts channel (bprp fast) are so fast that seeing you teach this at normal pace feels very strange .

@muse0622 3 года назад

z is Arg, so we can z+ 2πl is the answer (l is integer)

@Mynthio 3 года назад

I think it would have been a lot better if you cancelled out the 1/2 before substituting :)

@user_08410 5 месяцев назад

z = arcsin(π/2)

@utkarshsingh96 3 года назад

Gahhh!!! I was wishing he would say go pokemon go... Nd the end XD

@sueyibaslanli3519 3 года назад

I have been subscribing u for 3 years , providing that there are infinite unsolved questions and I am slightly less than being as good as u so I should create a math channel on RU-vid, should not I?

@covid-21delta99 3 года назад

You should surely I will support it

@shreyan1362 3 года назад

You should

@stuartyeo5354 3 года назад

If you have nothing better to do and are confident in your teaching skills, then go for it! :D

@LeTtRrZ 3 года назад

God I am such a nerd for enjoying this, but non-nerds will never see the beauty in this accomplishment.

@sharkknight618 3 года назад

Can you verify the answer ???

@liamshelley496 Год назад

I tried to solve this before the video and just ended up discovering that pi/2 = -pi/2

@shalinchakraborty672 3 года назад

Do you have any provision for Maths question doubt solving? I seriously need it for my JEE preperation.

@alejandrofonseca8524 3 года назад

In min 3:17, Didn't you forget a minus for the multiplicative inverse of I? 🤔 P.D. I like your videos are very interesting and educative. 😁

@Sesquipedalia Год назад

multiplicative inverses dont have a minus, those are for additive inverses, np, alot of ppl in my class are sometimes confused about it too

@elias69420 Год назад

@@SesquipedaliaThe multiplicative inverse of i in particular is equal to its additive inverse. Proof: 1/z = -z 1 = -z² z² + 1 = 0 z = ±i Thus, 1/i = -i (and -1/i = i, which is equivalent)

@EduardoHerrera-fr6bd 3 года назад

i do actually like to be on the bottom :)

@blackpenredpen 3 года назад

Oh?!

@obrilanders7694 3 года назад

2 Peyam

@cookieman2028 3 года назад

nice one

@mrmimeisfunny 3 года назад

India: We use colors as variables. Arabia: Well, we don't really want to mess with different pigments while doing math. We're just going to use letters. Europe: The letters are not Christian enough, we will use our own latin and greek letters. Blackpenredpen: 🙂

@chronicsnail6675 3 года назад

Europeans are the pioneers and father of maths . LETS GO EUROPE!!!

@quirtt 3 года назад

@@chronicsnail6675 stfu

@adrianfrauca8118 3 года назад

@@chronicsnail6675 says the guy who uses indo-arabic numerals

@vaxjoaberg9452 3 года назад

@@chronicsnail6675 A stunning display of ignorance.

@chronicsnail6675 3 года назад

@@adrianfrauca8118 and?

@dorian4387 3 года назад

For happy face^2 it should've been drawn as an actual square face, for the true immersion.

@blackpenredpen 3 года назад

Wow! Didn’t think of it. Nice one.

@adamwalker8777 2 года назад

@@blackpenredpen HOW sin(sin(sin(sin(sin(sin(...))))))=1 ???

@shanathered5910 Год назад

what about the cartesian square of a happy face?

@user_cy1er 10 месяцев назад

happy face^3: a cube with happy face on every side

@thechosenone7400 8 месяцев назад

Square root of square face is happy face

@TheMartian11 3 года назад

My guy here rockin' that Tibees merch while solving these unholy equations

@blackpenredpen 3 года назад

Surprised?

@cyrenux 3 года назад

Nope got used to it

@feliciakhansajaidahseno4153 3 года назад

yes :D

@pilopolo5957 3 года назад

Well well

@mathevengers1131 3 года назад

@2C (02) Chan Kwan Yu This formula will give you principal solution. If you want other solutions you can add 2πn. It will give you infinitely many solutions.

@mathevengers1131 3 года назад

@2C (02) Chan Kwan Yu actually I am sending this same comment from last 20 videos so bprp will read but I think km he's not able to notice this comment so many other comments. I hope he reads this comment.

@agabe_8989 3 года назад

You know its terrifying when he giggles time to time.

@SHASHANKRUSTAGII 3 года назад

your beard looks like a perfect binary tree

@jon9103 3 года назад

2:52 "I'm just going to put down a happy face....because a fish is too difficult." 🤔 but a fish can be represented with just one letter: 𝛼

@BlissOn47 3 года назад

Algebra: letters as variables Trigonometric algebra: latin letters as variables Blackpenredpen algebra: emojis

@christopherdyson1158 3 года назад

This is completely unrelated, but I was trying to figure out transistors earlier today since one of the bonus problems in my principles of electrical engineering textbook had them in an example of a monostable vibrator (I havent exactly seen a transistor before in problems... or real life... not even sure why it brought them up because the problems were about basics of DC RC circuits) But apparently the way to calculate the voltage across transistors uses the Lambert-W function and I thought back to your "fish" videos you did on the Lambert-W function. Honestly I didn't know it had much use out of "math for fun".

@blackpenredpen 3 года назад

Wow! I didn’t know!

@caseyleung2985 3 года назад

7:05 Actually, ln(x+sqrt(x^2-1)) = arccosh(x), which can further simply the answer.

@angelmendez-rivera351 3 года назад

This only simplifies one branch of the answer, though.

@caseyleung2985 3 года назад

Probably another branch ln(x-sqrt(x^2-1)) can be written as ln(-1)+arccosh(-x), but the domain might be tricky

@DrQuatsch 3 года назад

actually, you can simply extract the +/- sign outside the ln. ln(x - sqrt(x^2 - 1)) = ln[(x - sqrt(x^2 - 1))(x + sqrt(x^2 - 1)) / (x + sqrt(x^2 - 1))] = ln[(x^2 - (x^2 - 1))/(x + sqrt(x^2 - 1))] = ln[1/(x + sqrt(x^2 - 1))] = - ln(x + sqrt(x^2 - 1)). So the +/- can be extracted out of the ln. So in the end it would be +/- arccosh(x).

@spencergrogin1074 Год назад

We don't allow trig functions in the simplification... Otherwise the answer to the whole problem can be trivially reduced at step 2 to "z = arcsin(pi/2+2npi)"

@Memzys 9 месяцев назад

@@spencergrogin1074except the input is outside the domain of arcsin

@matthewstevens340 3 года назад

The video on sin(z)=2 was the first video I saw from your channel! I've been following you since that video was uploaded ;)

@blackpenredpen 3 года назад

Thank you!

@Eichro 3 года назад

"We'll have to go to the complex world" *[screams in agony]*

@stevengu1253 3 года назад

The complex realm is called complex for a reason. 😃

@xevira 3 года назад

"Technically, should have written pi m." ... pi m... as in... Dr Peyam? XD

@jplikesmaths 3 года назад

That’s deep

@erikkonstas 3 года назад

LMAO that's not anything new...!

@nombreusering7979 3 года назад

I remember u used this technique/Other way to write arcsin in ur sin(?)=2 vid. Amazing

@blackpenredpen 3 года назад

Yea. This is a continuation video and also a little celebration (since sinz=2 got over 1M views recently).

@nombreusering7979 3 года назад

@@blackpenredpenI was a follower of your channel since then I think, I really like the content of yours man. Keep it up!

@mathevengers1131 3 года назад

@@blackpenredpen *BPRP please please please read this comment.* Your videos are very amazing. I have a request, can you please please please make a video on what I have derived. I have derived a formula for sin inverse of x. The proof is as follow: y=sin^-1(x) sin(y)=x e^(iy)-e^-(iy)=2ix (e^(iy))^2-2ixe^(iy)-1=0 Using quadratic formula: e^(iy)= ix+-√(1-x^2) y= -iln(ix+-√(-(x^2-1)) y= -iln(i(x+-√(x^2-1))) Using ln(ab)=ln(a)+ln(b) y= -i(ln(i))-i(ln(x+-√(x^2-1))) sin^-1(x)= π/2 - iln(x+-√(x^2-1)) To check this formula put x=2 and you will get: sin^-1(2)=π/2-iln(2+-√3) You have proved that sin(π/2-iln(2+-√3)=2 in one of your previous videos. I also request you to put sin^-1(x)=π/180 and put formula of sin^-1(x) which I derived and solve for x so we will get value of sin(1°) or sin(π/180), I had tried to find value of sin(1) this way but I failed. I hope you will make a video on this formula. My name is Kathan Parikh and I am 16 years old. And if you want one more golden equation which includes Phi,π,i,e and even Fibonacci series(All five in one equation) then just reply me so I will give my phone number and you can call me as it is difficult to type the equation, so I will be easily able explain the equation and it's proof to you by sending you a pic or on call.

@mathevengers1131 3 года назад

@2C (02) Chan Kwan Yu This formula will give you principal solution. If you want other solutions you can add 2πn. It will give you infinitely many solutions.

@mathevengers1131 3 года назад

@neilgerace355 10 месяцев назад

Happy face, meet fish. Fish, meet happy face.

@MeepMu 3 года назад

Finally, emojis in math

@WackyAmoebatrons 3 года назад

x/0 should be the poop emoji :-)

@7he_5tranded_4stronaut 3 года назад

Two..? Two... And then it got intense when another pen approached the board

@hassanalihusseini1717 3 года назад

Yhank you for that video. It was quite interesting! Even the solution was a little bit too "complex" :-)

@wpbn5613 3 года назад

sin sin sin sin sin x = 1

@blackpenredpen 3 года назад

Lol. I pass.

@shreyan1362 3 года назад

@@blackpenredpen this is an imo compendium question ... I am not joking

@mathevengers1131 3 года назад

@2C (02) Chan Kwan Yu This formula will give you principal solution. If you want other solutions you can add 2πn. It will give you infinitely many solutions.

@mathevengers1131 3 года назад

@2C (02) Chan Kwan Yu and this is not the same formula. You can get sin inverse of any number you want with this formula. For example he found sin inverse of π/2 in this video and he took so much time. But with my formula sin inverse of π/2 can be found in some seconds.

@Gamiboi612 3 года назад

I’m surprised how I’m slowly starting to understand these types of videos as I learn. I still remember how I would not understand any statements in these videos a few years ago.

@karryy01 3 года назад

Though he is doing something very wrong, it seems reasonable

@GabrielLira267 Год назад

@@karryy01 what he did that was wrog?

@Tomaplen 3 года назад

Does this have more solutions than sin(z)=2 or its the same infinite?

@blackpenredpen 3 года назад

Good question! I believe they are both “countable” infinity.

@keerthanarao3184 3 года назад

Just recently saw that sin(z)=2 video of urs.Amazing. Love ur videos🤗

@Amoeby 3 года назад

This is what I call fun! From now on I'll use pi*(4n+1)/2 instead of =)

@plislegalineu3005 3 года назад

Great! [ π(4n+1)/2, m є Z, n є Z ] is funnier (idk why I used [] )

@eckhardtdom Год назад

6:10, there are like 26 letters in english alphabet, around a infinite many symbols and other stuff, and he choose a happy face 🤣🤣🤣🤣🤣

@AnCoSt1 3 года назад

I think it'd be cool if you quickly calculated the solution for n,m=0, to show what value for z that would present. I absolutely love these complicated problems that you keep showing!! Any plans to consider more AIME or even USAMO/IMO videos?

@novidsonmychanneljustcomme5753 3 года назад

Next one: sin(sin(sin(z)))=1. ;) And while we're on it, why not also the general case with sin n times in itself: sin(sin(sin(...(sin(z))))=1. 😛 Let's see if finding a solution to this still can give a happy face to the one trying to solve it... 😁

@MuPrimeMath 3 года назад

8:15 lol

@emilianososa1252 3 года назад

What about sin(sin(sin z))=1?

@blackpenredpen 3 года назад

No, thanks lol

@nevenazivic5237 3 года назад

I really love complex formula for sin(z), really fun to do

@HAbarneyWK 3 года назад

Well, that was my guess to begin with. It's quite intuitive.

@joshuapaulorigenes1936 3 года назад

ah yes me at late in the evening pretending to understand these kind of maths

@rudrarocks3056 3 года назад

Actually it was basic

@imaginaryangle 8 месяцев назад

I love how you use basic examples to train all the little concerns and caveats that must be observed when solving a specific class of problems. Very effective and fun to watch, especially with emoji substitution 😄

@maulin_shah44 3 года назад

I got the vid to 69 likes. Rule #1 enjoy the little things in life...

@maxreenoch1661 3 года назад

2*pi*n 2 pi en 2 pen black pen red pen :0

@willie333b 3 года назад

Happy face lol

@blackpenredpen 3 года назад

😃

@mathevengers1131 3 года назад

@2C (02) Chan Kwan Yu This formula will give you principal solution. If you want other solutions you can add 2πn. It will give you infinitely many solutions.

@mathevengers1131 3 года назад

@josephgeorge2949 3 года назад

Here I was thinking that I could solve it by looking at the thumbnail...how wrong I was...

@stevengu1253 3 года назад

Well, it is solveable by staring if the question was asking only for real solutions.

@pigeonbass11 3 года назад

One can only imagine

@theimmux3034 3 года назад

The infamous C and R axes

@blackpenredpen 3 года назад

😂

@aurelienhermant2242 3 года назад

Simple sin x ~= x so you have sin(x) which is also x so x=1 Cqfd

@MatteoDolcin-ye8xm 2 месяца назад

Easy! arcsin(arcsin(1))

@NonTwinBrothers 3 года назад

The sin(?)=2 vid is such a good video, lol

@greece8785 3 года назад

10:39: He: Very Nice Me: 😫

@tzonic8655 3 года назад

Fun fact: complex nunbers are not taught anymore in greek high schools

@greece8785 3 года назад

@@tzonic8655 Unfortunately 😪😪😪

@geosalatast5715 3 года назад

I am greek and I indeed was not taught complex numbers because when I was at the last grade of high school, complex numbers had stopped being taught already for 2 years... However thanks to uni and youtube videos I think I have a decent understanding of complex numbers!

@greece8785 3 года назад

@@geosalatast5715 Είναι κρίμα Ένας τόσο ωραίος τομέας των μαθηματικών να διδάσκεται μόνο στο πανεπιστήμιο...

@tzonic8655 3 года назад

@@geosalatast5715 yeap,m2! Next semester i have to choose between discrete math or arithmetic analysis(αριθμητική ανάλυση δεν ξέρω αν είναι έτσι στα αγγλικά) or complex analysis.complex analysis looks so interesting but I'm not sure yet

@TheBlackMinds 3 года назад

Why don't you use the simplified quadratic formula for even b coefficient?

@NonTwinBrothers 3 года назад

8:16 He's so done with it, lmao

@carterwoodson8818 3 года назад

Love the Tibees shirt!!

@blackpenredpen 3 года назад

Thanks, Carter.

@RyanLucroy 3 года назад

Well, I'll just stick to cos(cos(x))=1 :D

@aashsyed1277 3 года назад

Now this video has 100k views. So do another video like this

@yahianattouf3454 3 года назад

Hello I can't speak English very will

@prernasingh5865 3 года назад

Oh he has worn t-shirt of tibees( RU-vidr).

@bludermate38 3 года назад

Something very serious is going on when blue pen is involved.

@thefantasicm_2407 3 года назад

Sorry to interrupt, but your actually saying that sin(sin z) = 1 is equivalent to sin z = pi/2 + 2pi*n for z complex number and n an integer ? This is not true, because sin z for z complex is not necessarily a real number. Because with z = a + ib we have : e^(iz) = e^(-b)e^(ia) e^(-iz) = e^(b)e^(-ia) So e^(-iz) is not necessarily the conjugate complex number associated to e^(iz). Because of that, e^(iz) - e^(-iz) may have a non negative real part so sin(z) = (e^(iz) - e^(-iz)) / 2i may have a non negative imaginary part, hence sin(z) may be a non real number, so we cannot use our trigonometric cycle :-( :-( You may ask : when is sin(z) a real number ? Well you can prove that we must have b = 0 so z mus be real ...

@angelmendez-rivera351 3 года назад

Let sin(z) = y. Hence sin(y) = 1 = (exp(i·y) - exp(-i·y))/(2·i) ==> exp(i·y) - exp(-i·y) = 2·i ==> exp(i·y)^2 - 1 = 2·i·exp(i·y) ==> exp(i·y) = [2·i + sqrt(-4 - 4·(-1))]/2 = 2·i/2 = i, because sqrt[-4 - 4·(-1)] = 0. exp(i·y) = i ==> y = π/2 + 2·n·π, and y = sin(z), so sin(z) = π/2 + 2·n·π. So you are wrong. sin(sin(z)) = 1 is indeed equivalent to sin(z) = π/2 + 2·n·π, and BPRP was correct in saying this. You argument that the conjugate of e^(i·z) is not e^(-i·z) is a moot point, because the conjugate is actually a scalar multiple of e^(-i·z) anyway. In fact, you literally proved this yourself.

@thefantasicm_2407 3 года назад

@@angelmendez-rivera351 here you proved that in this case it is indeed equivalent, what i was trying to say is that in this video he took it as granted from the real number case while in fact it is not that obvious.

@ChuiKing Год назад

unfortunately, you forgot to add 2npi.

@jamesondasilva6502 2 года назад

Shouldn't there also be another variable (lets say s since n and m are already used) which represents the period of the outer sin function? Let BPRP's solution equal J: to get all of the values of z, z = J + 2πs Which is to say: z = π ( 4m + 1 ) / 2 - i ln [ ( π ( 4n + 1 ) / 2 ) ± √ [ (π ( 4n + 1 ) / 2 )^2 - 1 ] ] + 2πs Simplify by factoring out pi : z = π ( 2s + ( 4m + 1 ) / 2 ) - i ln [ ( π ( 4n + 1 ) / 2 ) ± √ [ (π ( 4n + 1 ) / 2 )^2 - 1 ] ] where n ∈ Z, m ∈ Z, s ∈ Z

@ryanmarcus3970 Год назад

I think so! Just a correction however, s would be accounting for the *inner* sine functions period, as n is already accounting for the outer sine function’s (the outer sin is collapsing all possible values for sin(z), which bprp denotes with 2n(pi), whereas the inner sin function is collapsing all possible values of z itself, which you denoted with 2s(pi).

@pe3akpe3et99 3 года назад

x=arcsin(arcsin(1)), solved

@PpTheBest 3 года назад

x = arcsin(arcsin(1)) Probably the easiest question ever.

@blackpenredpen 3 года назад

U won!

@mathevengers1131 3 года назад

@2C (02) Chan Kwan Yu This formula will give you principal solution. If you want other solutions you can add 2πn. It will give you infinitely many solutions.

@mathevengers1131 3 года назад

@angelmendez-rivera351 3 года назад

You guys seriously need to stop it with the spam.

@abdula1717 3 года назад

Sin after sin I have endured Yet the wounds I bear Are the wounds of COMPLEX ANALYSIS

@jeelthesiya5995 3 года назад

Gotta catch em all pokemon 👍👍👍😎

@DanBurgaud 9 месяцев назад

8:50 at this point the happy face 😀is now becoming 😭 8:51 2 Pie Em? emmm... Payem! Payem! calling Payem!

@samibaheru4029 3 года назад

Yo you made a mistake again.look at arcsine 1=90 or Pi/2.Again arcsine Pi/2=Unknown or Nothing further can be done after this.

@Ricardo_S 11 месяцев назад

You missed infinite answers After all, remembering that we are inside the sin function, at the end you must add +2πk K being an integer Very good video

@ranjitsarkar3126 3 года назад

You actually became much more funnier

@poo2uhaha 3 года назад

Love the Tibees shirt!

@bobby4976 3 года назад

Wondering, are all the commenters here math geeks? And of course the way he explained it is amazing but kinda scary when he was giggling. 😂

@DatBoi_TheGudBIAS 2 года назад

wait cant we do sin(sinx)=1, so sinx must be pi/2? edit: oh right, pi/2 exceeds the range of sin XD

@spudhead169 Год назад

I understood all of that and was able to follow along, but, if I were given that problem to solve from scratch I'd not have a chance.

@flamingworldz7663 3 года назад

6:11 Because it's hard to be happy when doing maths jkjkjk i luv maths

@ActionJaxonH 3 года назад

Respectfully, that beard is a little scraggly. Might wanna cut that bush off 😉

@Apple-qy3sc 3 года назад

he i m from morocco this false cause :sin (sinx ) =1 .sinx= π/2 .بما ان sin معرف في للمجال 1> sinx >-1 and π/2=1,57 so imposible

@jamesondasilva6502 2 года назад

The expression 1 > sinx > -1 is only true when x is a real number. However, in the video he uses z instead of x because z is a complex number. And the complex definition of sine isn't restricted by 1 > sinx > -1