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

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  • Опубліковано 1 лют 2021
  • 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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КОМЕНТАРІ • 470

  • @tibees
    @tibees 3 роки тому +479

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

  • @mrmimeisfunny
    @mrmimeisfunny 3 роки тому +797

    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
      @chronicsnail6675 3 роки тому +12

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

    • @quirtt
      @quirtt 3 роки тому +46

      @@chronicsnail6675 stfu

    • @adrianfrauca8118
      @adrianfrauca8118 3 роки тому +104

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

    • @vaxjoaberg9452
      @vaxjoaberg9452 3 роки тому +34

      @@chronicsnail6675 A stunning display of ignorance.

    • @chronicsnail6675
      @chronicsnail6675 3 роки тому

      @@adrianfrauca8118 and?

  • @dorian4387
    @dorian4387 3 роки тому +300

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

    • @blackpenredpen
      @blackpenredpen  3 роки тому +96

      Wow! Didn’t think of it. Nice one.

    • @adamwalker8777
      @adamwalker8777 Рік тому +1

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

    • @shanathered5910
      @shanathered5910 Рік тому

      what about the cartesian square of a happy face?

    • @user_cy1er
      @user_cy1er 6 місяців тому

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

    • @thechosenone7400
      @thechosenone7400 4 місяці тому

      Square root of square face is happy face

  • @agabe_8989
    @agabe_8989 3 роки тому +156

    You know its terrifying when he giggles time to time.

  • @breadlegend2480
    @breadlegend2480 3 роки тому +96

    You know its worse when he has more than 2 pens

  • @ericbright1742
    @ericbright1742 3 роки тому +29

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

  • @TheMartian11
    @TheMartian11 3 роки тому +178

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

  • @BlissOn47
    @BlissOn47 3 роки тому +42

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

  • @Aaron-gx9gv
    @Aaron-gx9gv 3 роки тому +63

    Me: using letters as variable
    Him:😃

  • @jon9103
    @jon9103 3 роки тому +15

    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

  • @blackpenredpen
    @blackpenredpen  3 роки тому +158

    Surprised?

    • @cyrenux
      @cyrenux 3 роки тому +13

      Nope got used to it

    • @feliciakhansajaidahseno4153
      @feliciakhansajaidahseno4153 3 роки тому +1

      yes :D

    • @pilopolo5957
      @pilopolo5957 3 роки тому

      Well well

    • @mathevengers1131
      @mathevengers1131 3 роки тому

      @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
      @mathevengers1131 3 роки тому +1

      @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.

  • @Vampianist3
    @Vampianist3 3 роки тому +64

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

  • @caseyleung2985
    @caseyleung2985 3 роки тому +92

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

    • @angelmendez-rivera351
      @angelmendez-rivera351 3 роки тому +10

      This only simplifies one branch of the answer, though.

    • @caseyleung2985
      @caseyleung2985 3 роки тому +7

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

    • @DrQuatsch
      @DrQuatsch 3 роки тому +9

      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
      @spencergrogin1074 Рік тому

      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
      @Memzys 5 місяців тому +1

      @@spencergrogin1074except the input is outside the domain of arcsin

  • @SHASHANKRUSTAGII
    @SHASHANKRUSTAGII 3 роки тому +84

    your beard looks like a perfect binary tree

  • @christopherdyson1158
    @christopherdyson1158 3 роки тому +49

    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".

  • @Gamiboi612
    @Gamiboi612 3 роки тому +46

    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
      @karryy01 3 роки тому +4

      Though he is doing something very wrong, it seems reasonable

    • @GabrielLira267
      @GabrielLira267 11 місяців тому

      @@karryy01 what he did that was wrog?

  • @Eichro
    @Eichro 3 роки тому +20

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

    • @stevengu1253
      @stevengu1253 3 роки тому +2

      The complex realm is called complex for a reason. 😃

  • @matthewstevens340
    @matthewstevens340 3 роки тому +10

    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 ;)

  • @nombreusering7979
    @nombreusering7979 3 роки тому +42

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

    • @blackpenredpen
      @blackpenredpen  3 роки тому +20

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

    • @nombreusering7979
      @nombreusering7979 3 роки тому +3

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

    • @mathevengers1131
      @mathevengers1131 3 роки тому +2

      @@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
      @mathevengers1131 3 роки тому +1

      @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
      @mathevengers1131 3 роки тому

      @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.

  • @AnCoSt1
    @AnCoSt1 3 роки тому +7

    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?

  • @keerthanarao3184
    @keerthanarao3184 3 роки тому +6

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

  • @shivansh668
    @shivansh668 3 роки тому

    Loving these Math Content 😊

  • @MeepMu
    @MeepMu 3 роки тому +16

    Finally, emojis in math

  • @Amoeby
    @Amoeby 3 роки тому +6

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

    • @plislegalineu3005
      @plislegalineu3005 2 роки тому

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

  • @imaginaryangle
    @imaginaryangle 4 місяці тому

    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 😄

  • @7he_5tranded_4stronaut
    @7he_5tranded_4stronaut 3 роки тому +15

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

  • @joshmcdouglas1720
    @joshmcdouglas1720 3 роки тому +2

    4:41 “Alright so it looks.... yeah”😭

  • @peterromero284
    @peterromero284 3 роки тому

    Wow. Gears grinding on this one! Chapeaux.

  • @hassanalihusseini1717
    @hassanalihusseini1717 3 роки тому +8

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

  • @nevenazivic5237
    @nevenazivic5237 3 роки тому +8

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

  • @neilgerace355
    @neilgerace355 6 місяців тому +3

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

  • @NihilistEmier
    @NihilistEmier 3 роки тому +8

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

  • @jandro8787
    @jandro8787 3 роки тому

    i love your t-shirt!! tibees

  • @carterwoodson8818
    @carterwoodson8818 3 роки тому +4

    Love the Tibees shirt!!

  • @xevira
    @xevira 3 роки тому +34

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

  • @egillandersson1780
    @egillandersson1780 3 роки тому +18

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

    • @blackpenredpen
      @blackpenredpen  3 роки тому +18

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

    • @zacksima8333
      @zacksima8333 3 роки тому +1

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

  • @NonTwinBrothers
    @NonTwinBrothers 3 роки тому +2

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

  • @poo2uhaha
    @poo2uhaha 3 роки тому +1

    Love the Tibees shirt!

  • @joshuapaulorigenes1936
    @joshuapaulorigenes1936 3 роки тому +9

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

  • @HAbarneyWK
    @HAbarneyWK 3 роки тому +4

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

  • @muse0622
    @muse0622 3 роки тому +1

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

  • @aravinds3846
    @aravinds3846 3 роки тому +8

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

  • @Henrix1998
    @Henrix1998 Рік тому

    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

  • @theimmux3034
    @theimmux3034 3 роки тому +4

    The infamous C and R axes

  • @shalinchakraborty672
    @shalinchakraborty672 3 роки тому

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

  • @MuPrimeMath
    @MuPrimeMath 3 роки тому +6

    8:15 lol

  • @jackhandma1011
    @jackhandma1011 3 роки тому +1

    Holy moly! I thought this was impossible.

  • @afeathereddinosaur
    @afeathereddinosaur 3 роки тому

    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
      @angelmendez-rivera351 3 роки тому

      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
      @afeathereddinosaur 3 роки тому

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

  • @aashsyed1277
    @aashsyed1277 3 роки тому +1

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

  • @BriceLavorel
    @BriceLavorel 2 роки тому

    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
      @jsjsjjsud9556 2 місяці тому

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

    • @BriceLavorel
      @BriceLavorel 2 місяці тому

      @@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

  • @TheBlackMinds
    @TheBlackMinds 3 роки тому +1

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

  • @Tomaplen
    @Tomaplen 3 роки тому +6

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

    • @blackpenredpen
      @blackpenredpen  3 роки тому +7

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

  • @ghsdftfm
    @ghsdftfm 3 роки тому +2

    inb4 this question comes out for my finals in 3 weeks

  • @NonTwinBrothers
    @NonTwinBrothers 3 роки тому +1

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

  • @dft163
    @dft163 3 роки тому +1

    Could use arcosh for the second logarithm?

  • @spontidakisnikolas3313
    @spontidakisnikolas3313 3 роки тому

    U got me at the first minute not gonna lie

  • @spudhead169
    @spudhead169 8 місяців тому

    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.

  • @alimimir
    @alimimir 3 роки тому

    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
      @MichaelRothwell1 3 роки тому +1

      There is no real solution, only complex solutions.

  • @Alrescha
    @Alrescha 3 роки тому

    Whaaaaah... long time no see..

  • @robfrohwein2986
    @robfrohwein2986 4 місяці тому

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

  • @abmxnq
    @abmxnq 3 роки тому

    thats awesome

  • @TheMrcoolguy1998
    @TheMrcoolguy1998 3 роки тому

    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.

  • @amaarquadri
    @amaarquadri 3 роки тому

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

  • @ranjitsarkar3126
    @ranjitsarkar3126 3 роки тому +1

    6:12
    Regret at its peak

  • @eckhardtdom
    @eckhardtdom Рік тому +2

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

  • @rohankhubchandani3694
    @rohankhubchandani3694 3 роки тому

    easy. inverse sine (pi/2)

  • @tsarhyrantine
    @tsarhyrantine 3 роки тому

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

  • @danialdunson
    @danialdunson 3 роки тому

    Yeah. Bravo

  • @Mynthio
    @Mynthio 3 роки тому +1

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

  • @jasimmathsandphysics
    @jasimmathsandphysics 3 роки тому

    That's impressive

  • @pigeonbass11
    @pigeonbass11 3 роки тому +4

    One can only imagine

  • @onlymeok
    @onlymeok 3 роки тому

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

  • @greece8785
    @greece8785 3 роки тому +8

    10:39: He: Very Nice
    Me: 😫

    • @tzonic8655
      @tzonic8655 3 роки тому +2

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

    • @greece8785
      @greece8785 3 роки тому +2

      @@tzonic8655
      Unfortunately 😪😪😪

    • @geosalatast5715
      @geosalatast5715 3 роки тому +2

      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
      @greece8785 3 роки тому +2

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

    • @tzonic8655
      @tzonic8655 3 роки тому +2

      @@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

  • @gouharmaquboolnitp
    @gouharmaquboolnitp 3 роки тому

    I'm 100th like of your video 😜

  • @jonaslaww
    @jonaslaww 3 роки тому

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

  • @anujpandey2083
    @anujpandey2083 3 роки тому +2

    thumbnail excited me to here

  • @tmfan3888
    @tmfan3888 3 роки тому

    BPRP: 2:58
    also BPRP: 3:19

  • @bludermate38
    @bludermate38 3 роки тому +2

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

  • @utkarshsingh96
    @utkarshsingh96 3 роки тому +1

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

  • @sueyibaslanli3519
    @sueyibaslanli3519 3 роки тому +3

    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 UA-cam, should not I?

    • @covid-21delta99
      @covid-21delta99 3 роки тому +1

      You should surely I will support it

    • @shreyan1362
      @shreyan1362 3 роки тому +1

      You should

    • @stuartyeo5354
      @stuartyeo5354 3 роки тому

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

  • @ranjitsarkar3126
    @ranjitsarkar3126 3 роки тому +2

    You actually became much more funnier

  • @marienbad2
    @marienbad2 3 роки тому

    A tibees T-shirt! cool!

  • @tejveersingh6148
    @tejveersingh6148 3 роки тому +1

    Hello bprp..please tell about sin^-1(2) + cos^-1(2) =?

    • @angelmendez-rivera351
      @angelmendez-rivera351 3 роки тому +1

      arccos(z) = π/2 - arcsin(z) for all complex z ==> arcsin(z) + arccos(z) = π/2 for all complex z ==> arcsin(2) + arccos(2) = π/2.

  • @LUKAS-bb4jc
    @LUKAS-bb4jc 2 роки тому

    Apparently it’s also asin(pi/2) and -asin(pi/2)+pi

  • @nicogehren6566
    @nicogehren6566 3 роки тому

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

  • @sabriath
    @sabriath 3 роки тому

    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.

  • @alicewyan
    @alicewyan 3 роки тому

    Nice

  • @samp-w7439
    @samp-w7439 3 роки тому

    Beard game going stronggggg!!!

  • @B4UTechTips1
    @B4UTechTips1 3 роки тому

    try to evaluate Gamma (1/3).

  • @kaisersouzei
    @kaisersouzei 3 роки тому

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

  • @moofymoo
    @moofymoo 2 роки тому

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

  • @Stephen_2330
    @Stephen_2330 3 роки тому

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

  • @runneypo
    @runneypo 3 роки тому

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

  • @LeTtRrZ
    @LeTtRrZ 3 роки тому

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

  • @flamingworldz7663
    @flamingworldz7663 3 роки тому

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

  • @georgekirby7656
    @georgekirby7656 3 роки тому +1

    Anyone know some good videos to understand the use of complex numbers here? I have no clue where he got eiz - e-iz / 2i from either.
    Thankyou

    • @angelmendez-rivera351
      @angelmendez-rivera351 3 роки тому +2

      Do you know of Euler's formula? Euler's formula states exp(i·z) = cos(z) + i·sin(z). You can use this to derive the formula he used there.

    • @lily_littleangel
      @lily_littleangel 3 роки тому +1

      Do you know what sin(x) means? It's the height of a point at an unit circle (=circle with radius 1 starting at the rightmost point going counterclockwise) with angle x in radians.
      One way to look at it is to remember what e^xi means. It means rotating a point on the unit circle with am angle of x in radians. For the e^-xi we have the same angle just in reverse direction.
      So by subtracting this two values, the real parts cancel out as they are the same but the imaginary parts get added together, as they are equal bit with opposite signs. We then divide by 2 because we are twice too high and we divide by I to get rid of the imaginary part.

    • @georgekirby7656
      @georgekirby7656 3 роки тому

      @@lily_littleangel thank you for the help!

    • @georgekirby7656
      @georgekirby7656 3 роки тому

      @@angelmendez-rivera351 hi, it’s the formula I haven’t come across, thanks

  • @Ricardo_S
    @Ricardo_S 8 місяців тому

    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

  • @jeelthesiya5995
    @jeelthesiya5995 3 роки тому +1

    Gotta catch em all pokemon 👍👍👍😎

  • @wpbn5613
    @wpbn5613 3 роки тому +11

    sin sin sin sin sin x = 1

    • @blackpenredpen
      @blackpenredpen  3 роки тому +13

      Lol. I pass.

    • @shreyan1362
      @shreyan1362 3 роки тому +1

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

    • @mathevengers1131
      @mathevengers1131 3 роки тому

      @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
      @mathevengers1131 3 роки тому +1

      @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.

    • @mathevengers1131
      @mathevengers1131 3 роки тому

      @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.

  • @abdula1717
    @abdula1717 3 роки тому +1

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