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

Рет қаралды 165,583

3 жыл бұрын

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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Пікірлер: 472

@tibees 3 жыл бұрын

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

@ninjacow2200 3 жыл бұрын

Tibees gang

@blackpenredpen 3 жыл бұрын

😃 😃

@currentlynowshyam1783 3 жыл бұрын

😂

@currentlynowshyam1783 3 жыл бұрын

😂😂😂

@currentlynowshyam1783 3 жыл бұрын

Super 😂💞

@mrmimeisfunny 3 жыл бұрын

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 жыл бұрын

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

@quirtt 3 жыл бұрын

@@chronicsnail6675 stfu

@adrianfrauca8118 3 жыл бұрын

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

@vaxjoaberg9452 3 жыл бұрын

@@chronicsnail6675 A stunning display of ignorance.

@chronicsnail6675 3 жыл бұрын

@@adrianfrauca8118 and?

@agabe_8989 3 жыл бұрын

You know its terrifying when he giggles time to time.

@ericbright1742 3 жыл бұрын

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

@dorian4387 3 жыл бұрын

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

@blackpenredpen 3 жыл бұрын

Wow! Didn’t think of it. Nice one.

@adamwalker8777 Жыл бұрын

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

@shanathered5910 Жыл бұрын

what about the cartesian square of a happy face?

@user_cy1er 7 ай бұрын

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

@thechosenone7400 5 ай бұрын

Square root of square face is happy face

@TheMartian11 3 жыл бұрын

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

@breadlegend2480 3 жыл бұрын

You know its worse when he has more than 2 pens

@Aaron-gx9gv 3 жыл бұрын

Me: using letters as variable Him:😃

@BlissOn47 3 жыл бұрын

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

@jon9103 3 жыл бұрын

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

@Vampianist3 3 жыл бұрын

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

@debblez 3 жыл бұрын

6:11*

@aadityabhattacharya1811 3 жыл бұрын

Lol

@SHASHANKRUSTAGII 3 жыл бұрын

your beard looks like a perfect binary tree

@blackpenredpen 3 жыл бұрын

Surprised?

@cyrenux 3 жыл бұрын

Nope got used to it

@feliciakhansajaidahseno4153 3 жыл бұрын

yes :D

@pilopolo5957 3 жыл бұрын

Well well

@mathevengers1131 3 жыл бұрын

@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 жыл бұрын

@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.

@Eichro 3 жыл бұрын

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

@stevengu1253 3 жыл бұрын

The complex realm is called complex for a reason. 😃

@neilgerace355 7 ай бұрын

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

@caseyleung2985 3 жыл бұрын

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

@angelmendez-rivera351 3 жыл бұрын

This only simplifies one branch of the answer, though.

@caseyleung2985 3 жыл бұрын

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 жыл бұрын

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 Жыл бұрын

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 7 ай бұрын

@@spencergrogin1074except the input is outside the domain of arcsin

@imaginaryangle 5 ай бұрын

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 😄

@AnCoSt1 3 жыл бұрын

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?

@MeepMu 3 жыл бұрын

Finally, emojis in math

@WackyAmoebatrons 3 жыл бұрын

x/0 should be the poop emoji :-)

@keerthanarao3184 3 жыл бұрын

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

@joshmcdouglas1720 3 жыл бұрын

4:41 “Alright so it looks.... yeah”😭

@matthewstevens340 3 жыл бұрын

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 жыл бұрын

Thank you!

@lazarussevy2777 3 күн бұрын

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

@peterromero284 3 жыл бұрын

Wow. Gears grinding on this one! Chapeaux.

@christopherdyson1158 3 жыл бұрын

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 жыл бұрын

Wow! I didn’t know!

@shivansh668 3 жыл бұрын

Loving these Math Content 😊

@hassanalihusseini1717 3 жыл бұрын

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

@egillandersson1780 3 жыл бұрын

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

@blackpenredpen 3 жыл бұрын

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

@zacksima8333 3 жыл бұрын

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

@jandro8787 3 жыл бұрын

i love your t-shirt!! tibees

@joshuapaulorigenes1936 3 жыл бұрын

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

@rudrarocks3056 3 жыл бұрын

Actually it was basic

@7he_5tranded_4stronaut 3 жыл бұрын

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

@Gamiboi612 3 жыл бұрын

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 жыл бұрын

Though he is doing something very wrong, it seems reasonable

@GabrielLira267 Жыл бұрын

@@karryy01 what he did that was wrog?

@poo2uhaha 3 жыл бұрын

Love the Tibees shirt!

@nevenazivic5237 3 жыл бұрын

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

@nombreusering7979 3 жыл бұрын

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

@blackpenredpen 3 жыл бұрын

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

@nombreusering7979 3 жыл бұрын

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

@mathevengers1131 3 жыл бұрын

@@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 жыл бұрын

@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 жыл бұрын

@Amoeby 3 жыл бұрын

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

@plislegalineu3005 3 жыл бұрын

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

@NonTwinBrothers 3 жыл бұрын

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

@xevira 3 жыл бұрын

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

@jplikesmaths 3 жыл бұрын

That’s deep

@erikkonstas 3 жыл бұрын

LMAO that's not anything new...!

@Henrix1998 Жыл бұрын

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

@carterwoodson8818 3 жыл бұрын

Love the Tibees shirt!!

@blackpenredpen 3 жыл бұрын

Thanks, Carter.

@tsarhyrantine 3 жыл бұрын

That stress at the end as ya were running out of space 😂

@NihilistEmier 3 жыл бұрын

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

@abmxnq 3 жыл бұрын

thats awesome

@shalinchakraborty672 3 жыл бұрын

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

@spontidakisnikolas3313 3 жыл бұрын

U got me at the first minute not gonna lie

@aashsyed1277 3 жыл бұрын

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

@jackhandma1011 3 жыл бұрын

Holy moly! I thought this was impossible.

@NonTwinBrothers 3 жыл бұрын

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

@HAbarneyWK 3 жыл бұрын

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

@MuPrimeMath 3 жыл бұрын

8:15 lol

@Stephen_2330 3 жыл бұрын

«I don't know if you use happy face in math, is very hard»😂

@Alrescha 3 жыл бұрын

Whaaaaah... long time no see..

@eckhardtdom Жыл бұрын

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

@theimmux3034 3 жыл бұрын

The infamous C and R axes

@blackpenredpen 3 жыл бұрын

😂

@spudhead169 9 ай бұрын

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.

@aravinds3846 3 жыл бұрын

Can you do IITJEE math questions? Those are terrifying when you read them but are fun to solve and give you tons of views

@TheBlackMinds 3 жыл бұрын

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

@muse0622 3 жыл бұрын

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

@dft163 3 жыл бұрын

Could use arcosh for the second logarithm?

@robfrohwein2986 5 ай бұрын

It was a bit messy ... but i understood... thanks again !

@jasimmathsandphysics 3 жыл бұрын

That's impressive

@ranjitsarkar3126 3 жыл бұрын

You actually became much more funnier

@ghsdftfm 3 жыл бұрын

inb4 this question comes out for my finals in 3 weeks

@danialdunson 3 жыл бұрын

Yeah. Bravo

@bludermate38 3 жыл бұрын

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

@jonaslaww 3 жыл бұрын

I don't even know the first step but it's seems like very cool solutions so here's a like

@gouharmaquboolnitp 3 жыл бұрын

I'm 100th like of your video 😜

@alicewyan 3 жыл бұрын

Nice

@TheMrcoolguy1998 3 жыл бұрын

How about: sqrt(happyface^2-1)=sqrt(sin(z)^2-1)=sqrt(-cos(z)^2)=sqrt(i^2cos(z)^2)=i*cos(z) then rewrite in exp form and then try to workout z. Note this was just a quick thought. Probably doesnt take some stuff in to account like multiple branches of solutions. However I though perhaps there was a way to get a bit nicer form. Not sure though, didnt work anything out. The form and considering trig identities just made me think of this.

@utkarshsingh96 3 жыл бұрын

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

@jeelthesiya5995 3 жыл бұрын

Gotta catch em all pokemon 👍👍👍😎

@anujpandey2083 3 жыл бұрын

thumbnail excited me to here

@BriceLavorel 2 жыл бұрын

a little late but in your final formula you do not treat the case when inside the "ln" the value is negative (ln of a negative value is undefined). This case happens for n

@jsjsjjsud9556 4 ай бұрын

ln of a negative value is just ln(abs(x))+i*pi it isn't undefined

@BriceLavorel 4 ай бұрын

@@jsjsjjsud9556 the equation exp(z)=-x has an infinite solutions : ln(abs(-x))+(2k+1)*i*pi with k integer. So there is no clear extension of the ln function on the real negative. Just writing ln(abs(-x))+i*pi is a shortcut and there are interesting exercises using this abuse to lure the reader

@samp-w7439 3 жыл бұрын

Beard game going stronggggg!!!

@ranjitsarkar3126 3 жыл бұрын

6:12 Regret at its peak

@alimimir 3 жыл бұрын

i just checked it with grapher and it seems like the function doesn’t intersect y = 1 at all. what’s the deal? i plugged y = sin(sin(x)) in desmos

@MichaelRothwell1 3 жыл бұрын

There is no real solution, only complex solutions.

@moofymoo 2 жыл бұрын

does this prove than if I commit two sins in a row, then that still counts as one?

@afeathereddinosaur 3 жыл бұрын

I'm so happy the only part I didn't get was the intrusion of (e^iz - e^-iz)/2i and of course the ln log, my weakness.

@angelmendez-rivera351 3 жыл бұрын

e^(i·z) = cos(z) + i·sin(z) for all complex z. If you substitute z |-> -z, you derive e^(-i·z) = cos(z) - i·sin(z). Subtract the second equation from the first, and this results in 2·i·sin(z) = e^(i·z) - e^(-i·z). Since 2·i is not equal to 0, you can divide, obtaining sin(z) = (e^(i·z) - e^(-i·z))/(2·i), and this is where that substitution in the video came from. In fact, this formula is often taken to be the definition of sin(z) for all complex z.

@afeathereddinosaur 3 жыл бұрын

@@angelmendez-rivera351 Oh, so that's where it comes from, thanks for breaking it down for me

@amaarquadri 3 жыл бұрын

So it's like a warped 2d lattice of points in the complex plane!

@kaisersouzei 3 жыл бұрын

I love when the answer looks more like a question then the question did... ;p

@onlymeok 3 жыл бұрын

Does the solution plot to a point or to some interesting pattern? Too lazy to do this myself.

@Mynthio 3 жыл бұрын

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

@nicogehren6566 3 жыл бұрын

Mr. bprp can u help me : integral of cos (cot x - tan x) dx from 0 to pi

@aymanehar4386 3 жыл бұрын

blackpenredpen rule broken (blue pen)

@LeTtRrZ 3 жыл бұрын

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

@greece8785 3 жыл бұрын

10:39: He: Very Nice Me: 😫

@tzonic8655 3 жыл бұрын

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

@greece8785 3 жыл бұрын

@@tzonic8655 Unfortunately 😪😪😪

@geosalatast5715 3 жыл бұрын

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 жыл бұрын

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

@tzonic8655 3 жыл бұрын

@@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

@marienbad2 3 жыл бұрын

A tibees T-shirt! cool!

@DanBurgaud 6 ай бұрын

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

@prashantmishra7507 3 жыл бұрын

That beard gives us a Confucius vibe.

@Tomaplen 3 жыл бұрын

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

@blackpenredpen 3 жыл бұрын

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

@oxygen2623 7 ай бұрын

doing all that and the happy face turns into a sad face

@flamingworldz7663 3 жыл бұрын

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

@runneypo 3 жыл бұрын

Hello i am programmer and wish to learn this. how can I learn this starting from scratch

@sabriath 3 жыл бұрын

You should have gone a little further and shown that 'n' cannot be between -.152 to -.508 (about)...and since no integers lie within that range, THEN you can deduce that it works for all integers.

@pigeonbass11 3 жыл бұрын

One can only imagine

@devincustodio2258 3 жыл бұрын

beard is so effin cool

@sueyibaslanli3519 3 жыл бұрын

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 KZfaq, should not I?

@covid-21delta99 3 жыл бұрын

You should surely I will support it

@shreyan1362 3 жыл бұрын

You should

@stuartyeo5354 3 жыл бұрын

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