Actually, as cos² x + sen² x = 1, and sen x = x, we have: cos² x + x² = 1 So cos x = sqrt{1 - x²} = x. As 1 gives 0, and 0 gives 1, we go in the middle, and 0,5 should give you 0,5.
@@CaoNiMaBi sin(x) = SUM in n with (n in Naturals) of (- x)^(2n + 1) / (2n + 1)! = x - x³ / 3! + x^5 / 5! + ... So, when x is small (|x| < 1), every other thing goes to 0 faster than x itself. This way, with the Fundamental Theorem of Engineering, sin(x) = x.
Papa, would you make a series of videos on complex Analysis. (i have seen a natural teacher in you and would love you to make videos on complex Analysis)
Zach star: You underestimate my power. With my stand, star engineer, I can make EVERY TRIGONOMETRIC RATIO EQUAL TO ITS ARGUMENT Flammy: bakayaro, you failed to consider my final attack, COS X = X Zach star: NANIIIIII
People should check OEIS (The On-Line Encyclopedia of Integer Sequences) sequences A369186 and A369187. The sequences are the denominators and the numerators of an infinite sequence that converges to the Dottie number. I obtained the infinite sequence using an obscure method called "Whittaker's root series formula". I believe that this is the first sequence that converges to the Dottie number that only uses integers. I actually used Whittaker's root series formula to obtain infinite series for other constants (1/e, ln(2), Plastic ratio, Backhouse's constant etc). I am trying to make Whittaker's root series formula more known. It is relatively easy to apply since it just involves the determinants of Toeplitz matrices. The Toeplitz matrices are created using the coefficients of polynomial equations or Taylor/power series.
A fast way to compute it is just to run x = cos(x) over and over, the fixed point will be at x=cos(x) import math x = 1 for i in range(100): x = math.cos(x)
for anyone who is more into JS (or just using the browser's console (Ctrl + Shift + K / F12)): for(var x = 1, i = 0; i < 100; i++) x = Math.cos(x) console.log(x)
I calculated approximate value by Newton's method in Python Python code import math b = True i = 0 x0 = 0 while b: x1 = x0 x0 = x0 - (x0-math.cos(x0))/(1+math.sin(x0)) i += 1 b = not(abs(x1-x0) < 1e-12) For paper and pencil calculations x0=pi/4 would be probably better choice of initial guess
I think the secant method is actually slightly preferred here, since for Newton's method you'd need to calculate both the sine and the cosine, which would essentially lead to each step taking twice as long as a secant method step. So since phi^2 > 2 you end up with better convergence rate with the secant method when time is a factor. That's not to say you're wrong with Newton's method, that would work perfectly fine, just wanted to spread some knowledge
cos(x) = x then -1 ≤ x ≤ 1. x cannot be in [-1, 0[ because for such values cos(x) is strictly positive. We conclude that x must be in [0, 1]. Let g(x) = cos(x) - x. This function is decreasing in [0, 1] and g(1) < 0 < g(0) so there exists a unique c such that g(c) = 0. So, cos(x) = x has a unique real solution in [0, 1] which can be approximated by a dichotomic search (for example) with an error decreasing exponentially.
If you actually do the Taylor series for cosine (not the Maclaurin series) there is a y term. We just always use the Maclaurin and call it the Taylor series for some reason
cos is lipschitz continuous with at most K=sin(1) on the interval [0,1], so you can also use the banach fixed point theorem and just do cos(cos(cos(1))) etc and get the answer too. tho banach is also overkill in this situation, the theorem you used looks like a real monster imo
Indeed complicated. What about to use the Intermediate Value Theorem (Bolzanos) from the start, to check if a solution exists. After that, things will be easy. f(n) = cos(n) - n = 0 and by the use of Bolzanos theorem it will be an easy numerical problem?
Great video! I'm confused at why you refer to it being closed form, however. Being an infinite series, therefore having infinitely many operations, means it is not in closed form, correct?
@@stephendonovan9084 I thought as such. I just found it a bit abnormal from Papa who is usually very sound with his mathematical language, but we can all slip-up sometimes. (I was also about 5% unsure that I might have had it wrong too!)
I tried doing it myself and was frustrated about how in the world you calculated the a_n sequence in a closed form. Watched the video and saw that you didn't 👍
I don't think there are any non-trivial solutions. I can see this just by looking at the plot in Desmos for y = x and y = sin(x). So, I think sin(0) = 0 is the fixed point. The fixed point of x = sin(2x) would be more interesting to look at 😁. Looks like there is one trivial solution and two non-trivial points. I guess if you look in a suitable domain around x = 0, a suitable domain around = 1 and a suitable domain around x= -1, then you can find the unique fixed points in each of these domain. The fixed points seems to be (to fourdecimal places) x = 0, x = 0.9477 and x = -09477 respectively.
Yhis has bothered me for some time, I finally figured out why. The domain and range are disjoint sets, specifically, The argument of Cos is an angular measure, radians (if one is sane), and Cos(argument) is dimensionless.
Radians are also dimensionless. Proof: the distance traveled by a rotating circle equals the radius times the radians rotated. Radians times distance yields distance.
The whole underlying idea of radians is that they are the actual dimensionless measure of an angle, being the measure of the arc lengthU* divided by the radius. Two lengths divided by each other will be dimensionless. *i.e. a portion of the circumference of a circle.
@@ZipplyZane Your statement makes good sense. I am out of my area of "expertise", to use the term loosely. Something doesn't taste quite right, but there are lots of things I don't understand. I do try to understand when to not argue, and hence, to not explicitly display my ignorance. Leave 'em in doubt, where ever possible, eh?
Papa Flammy, I think you are confusing the trigonometric functions. Oh sorry, I forgot that sin(x)=tan(x)=sin(x)/cos(x)=x, hence xcos(x)=sin(x)=tan(x)=sin(x)/cos(x)=x/cos(x)=tan(x)/cos(x)=1/sin(x), hence cos(x)=1/xsin(x)=1/x²=(cos²(x)+sin²(x))/x²=(cos(x)/x)²+1=1/tan²(x) +1=(1+x²)/x²=(x+1)²/x² for little values of 1 (every number is closer to 1 that to infinity, so this holds for every value of 1), hence cos(x)=(sin(x)+1)²/sin²(x)=(sin(x)+sin²(x)+cos²(x))²/sin²(x)=(1+x+1/x)²=((x²+x+1)/x)², but this means x²+x+1=x+1-> x²=0, so for little values of 1 we have x²=x-x+1-1=(x-1)+1-x=x+1, so cos(x)=((0+x²)/x)²=x²=x+1=x for little values of 1, Q.E.D.
Who hasn't repeatedly pushed 'cos' on a calculator ? It works in degree mode too, resulting in a different number but with *way* faster convergence. Why ?
Hey flammmy i wanna send you a meme, but i dont have instagram facebook or twitter (basically i live in a cave) do you have any other source from which i can send you that
Could anyone please help me integrate 1/(x^2 * 2^x) ? I'm very stuck. Tried by parts which seems like it could be on the right track but its defeated me thus far.
This is the first video of yours that I watch, do you pronounce "y" as 'wah' on purpose? Your accent is really good so it would seem strange that you would have trouble pronouncing "y". It's pronounced like "why" FYI
Cos(x) = x? That’s easy just let cos(x) = sin(x + pi/2). Then obviously sin(x) = x and cos(x) = 1 by fundamental theorem of engineering. So we have 1 = x + pi/2 and so we solve for x to get x = 1 - pi/2
*You either die a mathematician, or prove long enough to see yourself become an engineer*
lmao this is good
They should write that in all uni xD
😂😂😂
"I just solved sin(x) = x, how much harder could the cosine be?"
90 minutes and much blood later ...
try:
sin(x) = sqrt(1-x^2)
0
Newton Raphson was More practical, I think...
@@DrDeuteron it's the same number as in above video
Did you get more solutions than obvious one x = 0
Cos(x) = x always holds true by the fundamental theorem of engineering.
Actually, it's cos(x)=1
@@Lamiranta and x=1 for very x values of 1
And by the fundemental theorem of engineering this becomes simply x=1
Wtf bro, that's sin x =x and cosx =1
Dude, you are slowly becoming a villain.
b r u h
Why?
LMAO
@@shirosurfer8864 Why??????????
Implying he isn't already one
Easy, since sin(x) = x it holds that cos(x) = (sin(x))' = (x)' = 1, so we just let x = 1
Differentiation goes brr.
What??
Actually, as cos² x + sen² x = 1, and sen x = x, we have: cos² x + x² = 1
So cos x = sqrt{1 - x²} = x.
As 1 gives 0, and 0 gives 1, we go in the middle, and 0,5 should give you 0,5.
why would sin(x) = x?
@@CaoNiMaBi
sin(x) = SUM in n with (n in Naturals) of (- x)^(2n + 1) / (2n + 1)!
= x - x³ / 3! + x^5 / 5! + ...
So, when x is small (|x| < 1), every other thing goes to 0 faster than x itself.
This way, with the Fundamental Theorem of Engineering, sin(x) = x.
2:43 let function f be a n a l.
cos(x)=1 for smol enough x and now if we check, x=1 is sufficiently smol, compared to for example 69^420, and thus the solution is x=1.
i would call that lemma 1 for fundamental theorem of engineering
Huh?
@@nikhilnagaria2672 Im only in pre algebra. What the heck is all this.
@@sebasthecrab1345lol nvm
What's lemma?
Papa, would you make a series of videos on complex Analysis. (i have seen a natural teacher in you and would love you to make videos on complex Analysis)
Yes please! Im taking a Complex Analysis course this semester, and it would be very helpful if you made more videos on Complex Analysis!!
Yeah. Like spicy meme free though, then professors can recommend it too. As much as we love the spice, most can't handle it.
Zach star: You underestimate my power. With my stand, star engineer, I can make EVERY TRIGONOMETRIC RATIO EQUAL TO ITS ARGUMENT
Flammy: bakayaro, you failed to consider my final attack, COS X = X
Zach star: NANIIIIII
This deserves to be the top comment! They should actually play such a skit.
"Let f be anal."
I've never laughed so much in a Math video.
Yeah. I was scrolling through the comment section just to see this comment.
3:31 Was waiting for that word, since you had already written it
:^)
3:30 that is a sentence I never expected to hear.
I love you Flammable Maths! Even as a twelve year old, your content is extremely entertaining.
Thanks Emil
cos(whaa) = whaa
When I was a kid, I've not only discovered how to find the answer with a calculator, but also I've came up with an approximate form (pi/160)^(1/13).
Yes, please make a video with derivation of that huge formula. Your explenations look so clear. You're the best on YT. Best regards
How the hell did UA-cam even let you publish this? Normally it throws out an ERROR 666: TITLE IS LUCIFERIAN in situations like this.
:'D
I‘m an engineer and can confirm the title, before even watching this video.
People should check OEIS (The On-Line Encyclopedia of Integer Sequences) sequences A369186 and A369187. The sequences are the denominators and the numerators of an infinite sequence that converges to the Dottie number. I obtained the infinite sequence using an obscure method called "Whittaker's root series formula". I believe that this is the first sequence that converges to the Dottie number that only uses integers. I actually used Whittaker's root series formula to obtain infinite series for other constants (1/e, ln(2), Plastic ratio, Backhouse's constant etc).
I am trying to make Whittaker's root series formula more known. It is relatively easy to apply since it just involves the determinants of Toeplitz matrices. The Toeplitz matrices are created using the coefficients of polynomial equations or Taylor/power series.
Although I couldn’t understand it, I can see its still great.
Maybe because the video is 17 mins long and you commented 4 mins after the video was released lol
@@Ultiminati havent learn complex analysis, so…
@@Ultiminati I already don’t understand at the very start lol.
@@beautyofmath6821 You don't want to learn complex an4l lmao
A fast way to compute it is just to run x = cos(x) over and over, the fixed point will be at x=cos(x)
import math
x = 1
for i in range(100): x = math.cos(x)
for anyone who is more into JS (or just using the browser's console (Ctrl + Shift + K / F12)):
for(var x = 1, i = 0; i < 100; i++) x = Math.cos(x)
console.log(x)
@@guigazalu for(var i = 1; i < 100; i++) console.log(Math.cos(i))
@@HypeLevels But this... don't... alter the value of a variable, which should approximate the result of (cos x = x).
@@guigazalu ah true forgot that, I’m dumb sorry
@@HypeLevels No worries. It happens to everyone.
*wah is equal to the cosine of wah*
Yes
I calculated approximate value by Newton's method in Python
Python code
import math
b = True
i = 0
x0 = 0
while b:
x1 = x0
x0 = x0 - (x0-math.cos(x0))/(1+math.sin(x0))
i += 1
b = not(abs(x1-x0) < 1e-12)
For paper and pencil calculations x0=pi/4 would be probably better choice of initial guess
You can introduce variable for accuracy of calculations and change assign operator in second line of while loop
teacher:how is it x? me:COS(X)=X
Papa Flammy: *Have you checked out Flammy's Wood already?*
Me: *I haven't visited that part of the internet for a while*
Nice! But, personally, for computational purposes, I’d use Newton method
I think the secant method is actually slightly preferred here, since for Newton's method you'd need to calculate both the sine and the cosine, which would essentially lead to each step taking twice as long as a secant method step. So since phi^2 > 2 you end up with better convergence rate with the secant method when time is a factor.
That's not to say you're wrong with Newton's method, that would work perfectly fine, just wanted to spread some knowledge
Zach star: ahh yes, your slowly learning
cos(x) = x then -1 ≤ x ≤ 1. x cannot be in [-1, 0[ because for such values cos(x) is strictly positive. We conclude that x must be in [0, 1].
Let g(x) = cos(x) - x. This function is decreasing in [0, 1] and g(1) < 0 < g(0) so there exists a unique c such that g(c) = 0.
So, cos(x) = x has a unique real solution in [0, 1] which can be approximated by a dichotomic search (for example) with an error decreasing exponentially.
If you actually do the Taylor series for cosine (not the Maclaurin series) there is a y term. We just always use the Maclaurin and call it the Taylor series for some reason
Finally! The mathematical legends comes to life!!!!
cos(x) = x is easy to solve with fix point iteration, three lines of code in Python. This increases the velocity of convergence ;)
cos is lipschitz continuous with at most K=sin(1) on the interval [0,1], so you can also use the banach fixed point theorem and just do cos(cos(cos(1))) etc and get the answer too. tho banach is also overkill in this situation, the theorem you used looks like a real monster imo
Indeed complicated. What about to use the Intermediate Value Theorem (Bolzanos) from the start, to check if a solution exists. After that, things will be easy. f(n) = cos(n) - n = 0 and by the use of Bolzanos theorem it will be an easy numerical problem?
Dottie!
I'll see if you mentioned, but the analytic representation is a fractal expression basically,.
when did this become a physics channel?
cos of (death) = death
fml
dem!! dat dark
u draw ur summation operator with an underbite and it always makes me laugh
17:11
oh shi-.. it's spreading
Great video! I'm confused at why you refer to it being closed form, however. Being an infinite series, therefore having infinitely many operations, means it is not in closed form, correct?
He might have meant an analytic expression and just misspoken
@@stephendonovan9084 I thought as such. I just found it a bit abnormal from Papa who is usually very sound with his mathematical language, but we can all slip-up sometimes.
(I was also about 5% unsure that I might have had it wrong too!)
i just solved this problem in numerical analysis using Newton's method and fixed point method and got this in recommended. UA-cam is scary.
:'D
How do you just clean your board as clean as my search history?
watch my last video lol
new fundamental theorem just dropped
:D
New patch released
I love the "wa"
fun fact: Dottie's number is sometimes represented by the Armenian letter ա.
can use Banach fixed point theorem can't we ?
I tried doing it myself and was frustrated about how in the world you calculated the a_n sequence in a closed form. Watched the video and saw that you didn't 👍
Guys what's the weird spiral thing he writes after the infinite series???
What about the nontrivial solution for sin(x)=x ?
I don't think there are any non-trivial solutions. I can see this just by looking at the plot in Desmos for y = x and y = sin(x).
So, I think sin(0) = 0 is the fixed point.
The fixed point of x = sin(2x) would be more interesting to look at 😁.
Looks like there is one trivial solution and two non-trivial points. I guess if you look in a suitable domain around x = 0, a suitable domain around = 1 and a suitable domain around x= -1, then you can find the unique fixed points in each of these domain. The fixed points seems to be (to fourdecimal places)
x = 0,
x = 0.9477 and
x = -09477 respectively.
Could You introduce Lagrange's multipliers? Best regards
ua-cam.com/video/Uw_WUvWQEkg/v-deo.html
Yhis has bothered me for some time, I finally figured out why. The domain and range are disjoint sets, specifically, The argument of Cos is an angular measure, radians (if one is sane), and Cos(argument) is dimensionless.
Radians are also dimensionless. Proof: the distance traveled by a rotating circle equals the radius times the radians rotated. Radians times distance yields distance.
The whole underlying idea of radians is that they are the actual dimensionless measure of an angle, being the measure of the arc lengthU* divided by the radius. Two lengths divided by each other will be dimensionless.
*i.e. a portion of the circumference of a circle.
@@ZipplyZane Your statement makes good sense. I am out of my area of "expertise", to use the term loosely. Something doesn't taste quite right, but there are lots of things I don't understand. I do try to understand when to not argue, and hence, to not explicitly display my ignorance. Leave 'em in doubt, where ever possible, eh?
Papa Flammy, I think you are confusing the trigonometric functions. Oh sorry, I forgot that sin(x)=tan(x)=sin(x)/cos(x)=x, hence xcos(x)=sin(x)=tan(x)=sin(x)/cos(x)=x/cos(x)=tan(x)/cos(x)=1/sin(x), hence cos(x)=1/xsin(x)=1/x²=(cos²(x)+sin²(x))/x²=(cos(x)/x)²+1=1/tan²(x) +1=(1+x²)/x²=(x+1)²/x² for little values of 1 (every number is closer to 1 that to infinity, so this holds for every value of 1), hence cos(x)=(sin(x)+1)²/sin²(x)=(sin(x)+sin²(x)+cos²(x))²/sin²(x)=(1+x+1/x)²=((x²+x+1)/x)², but this means x²+x+1=x+1-> x²=0, so for little values of 1 we have x²=x-x+1-1=(x-1)+1-x=x+1, so cos(x)=((0+x²)/x)²=x²=x+1=x for little values of 1, Q.E.D.
Ah yes, Engineering
ua-cam.com/video/Uw_WUvWQEkg/v-deo.html
i beg pls do integrals again
If you interest for integrals, just see my channel..
for small x cosx ~1-x^2/2 . Solve for x. Find x = 0.73. Good enough
Who hasn't repeatedly pushed 'cos' on a calculator ?
It works in degree mode too, resulting in a different number but with *way* faster convergence. Why ?
quality content I love
Why did I expect to understand this video if I have just begun 11th grade?
Why not use 0 for x0?
Hey flammmy i wanna send you a meme, but i dont have instagram facebook or twitter (basically i live in a cave) do you have any other source from which i can send you that
piequals3@papaflammy.engineer
@@PapaFlammy69 sent hui hui hui hui
Could anyone please help me integrate 1/(x^2 * 2^x) ? I'm very stuck. Tried by parts which seems like it could be on the right track but its defeated me thus far.
Excellent video! I did a video involving the Dottie number last year, it was about eclipses :D
ohhhh, very nice! Gotta watch it :)
It reminds me of the FundamentalTheoremOfEngineering .
Its really rare for a guy who has an intrest in mathematics
wat?
Numerical Methods go brr.
Super string theory 😆
DOTTI BOI!!!
ua-cam.com/video/Uw_WUvWQEkg/v-deo.html
“Daddy’s Number”😏😏😏😏😏
ua-cam.com/video/Uw_WUvWQEkg/v-deo.html
Every functuon of x has to be equal to x for all x
This is the first video of yours that I watch, do you pronounce "y" as 'wah' on purpose? Your accent is really good so it would seem strange that you would have trouble pronouncing "y". It's pronounced like "why" FYI
Not scary? Lol, this is my favorite horror series on any media
I like how he says "wa!" instead of "y"
ua-cam.com/video/Uw_WUvWQEkg/v-deo.html
(meanwhile in the computer science department)
hehehe secant method go brrrrr
ua-cam.com/video/Uw_WUvWQEkg/v-deo.html
x = .739085 radians approximately
I have to pass on this one, as it's well beyond my pay grade ...
Silly flammy, we all know cos(x) = 1
Have you been skipping engineering class again?
yeye
cos(macaroni) = macaroni
ua-cam.com/video/Uw_WUvWQEkg/v-deo.html
cosine of WHAAA
Selam Olsun Okan Tekman'a
I come not for the video but for the comment section 😁
ua-cam.com/video/Uw_WUvWQEkg/v-deo.html
It's obviously transcendental from the Lindemann-Weierstrass theorem....duh!
What in engineering is this?
ua-cam.com/video/Uw_WUvWQEkg/v-deo.html
Dotty boi ❤️
Which one is better?
Like if cos
Comment if sin
People writing comments about math and smart stuff
Me: haha anal. Function
Newton raphson
I just wanna know the answer so someone who actually watched can tell me?
Y = waa 😼
Cosx was equal to 1
U seem much more serious in this video wts happening to u papaflammy
Nothing tbh, wasn't meant to be serious at all hehe^^
True engineers would use Newton’s method!
Elaborate.
Nah use the bisection algorithm lmao
Cos(x) = x? That’s easy just let cos(x) = sin(x + pi/2). Then obviously sin(x) = x and cos(x) = 1 by fundamental theorem of engineering. So we have 1 = x + pi/2 and so we solve for x to get x = 1 - pi/2
How is cos(x) = x if sin(x) = x?
Second condition implies x = 0, but cos(0) isn’t equal to 0 so there are no solutions
Isn't cos(x)=1?
Use numerical method
x=Cosx=1
а где 0.73
this is cursed
Solve Cos(x)=x ???
Newton's method !
ua-cam.com/video/Uw_WUvWQEkg/v-deo.html
Wah
👌