cos(x)=x

Today we calculate the transcendental number called "Dottie Number", which is the unique solution to the fixed point equation cos x=x . We use the Lagrange Inversion Theorem to get a perfectly disgusting series solution that is a numerical horror show. Enjoy! :D

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[garvett]

*You either die a mathematician, or prove long enough to see yourself become an engineer*

[kingbeauregard]

"I just solved sin(x) = x, how much harder could the cosine be?"

90 minutes and much blood later ...

[haider]

Cos(x) = x always holds true by the fundamental theorem of engineering.

[dylank]

Easy, since sin(x) = x it holds that cos(x) = (sin(x))' = (x)' = 1, so we just let x = 1

[nikhilnagaria]

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.

[sirtthetea]

Dude, you are slowly becoming a villain.

[TI]

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)

[shivam]

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:

[neilgerace]

3:31 Was waiting for that word, since you had already written it

[karabodibakoane]

"Let f be anal."

I've never laughed so much in a Math video.

[Bean-Time]

3:30 that is a sentence I never expected to hear.

[emilsriram]

I love you Flammable Maths! Even as a twelve year old, your content is extremely entertaining.

[jarogniewborkowski]

Yes, please make a video with derivation of that huge formula. Your explenations look so clear. You're the best on YT. Best regards

[raulpr]

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.

[leobrouk]

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).

[ЧингизНабиев-эг]

Nice! But, personally, for computational purposes, I’d use Newton method

[holyshit]

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 -
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

[beautyofmath]

Although I couldn’t understand it, I can see its still great.

[ulissemini]

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)

[HichemFrozenBlood]

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.