A330119 -id:A330119 - cp's OEIS frontend

A003957 The Dottie number: decimal expansion of root of cos(x) = x.

7, 3, 9, 0, 8, 5, 1, 3, 3, 2, 1, 5, 1, 6, 0, 6, 4, 1, 6, 5, 5, 3, 1, 2, 0, 8, 7, 6, 7, 3, 8, 7, 3, 4, 0, 4, 0, 1, 3, 4, 1, 1, 7, 5, 8, 9, 0, 0, 7, 5, 7, 4, 6, 4, 9, 6, 5, 6, 8, 0, 6, 3, 5, 7, 7, 3, 2, 8, 4, 6, 5, 4, 8, 8, 3, 5, 4, 7, 5, 9, 4, 5, 9, 9, 3, 7, 6, 1, 0, 6, 9, 3, 1, 7, 6, 6, 5, 3, 1, 8, 4, 9, 8, 0, 1, 2, 4, 6

Offset: 0

Leonid Broukhis

Let P be the point in quadrant I where the curve y=sin(x) meets the circle x^2+y^2=1. Let d be the Dottie number. Then P=(d,sin(d)), and d is the slope at P of the sine curve. - Clark Kimberling, Oct 07 2011
From Ben Branman, Dec 28 2011: (Start)
The name "Dottie" is of no fundamental mathematical significance since it refers to a particular French professor who--no doubt like many other calculator users before and after her--noticed that whenever she typed a number into her calculator and hit the cosine button repeatedly, the result always converged to this value.
The number is well-known, having appeared in numerous elementary works on algebra already by the late 1880s (e.g., Bertrand 1865, p. 285; Heis 1886, p. 468; Briot 1881, pp. 341-343), and probably much earlier as well. It is also known simply as the cosine constant, cosine superposition constant, iterated cosine constant, or cosine fixed point constant. Arakelian (1981, pp. 135-136; 1995) has used the Armenian small letter ayb (ա, the first letter in the Armenian alphabet) to denote this constant. (End)

			0.73908513321516064165531208767387340401341175890075746496568063577328...

H. Arakelian, The Fundamental Dimensionless Values (Their Role and Importance for the Methodology of Science). [In Russian.] Yerevan, Armenia: Armenian National Academy of Sciences, 1981.
A. Baker, Theorem 1.4 in Transcendental Number Theory. Cambridge, England: Cambridge University Press, 1975.

G. C. Greubel, Table of n, a(n) for n = 0..10000 (terms 0..499 from Ben Branman)
Hrant Arakelian, New Fundamental Mathematical Constant: History, Present State and Prospects, Nonlinear Science Letters B, Vol. 1, No. 4, pp. 183-193.
Mohammad K. Azarian, On the Fixed Points of a Function and the Fixed Points of its Composite Functions, International Journal of Pure and Applied Mathematics, Vol. 46, No. 1, 2008, pp. 37-44. Mathematical Reviews, MR2433713 (2009c:65129), March 2009. Zentralblatt MATH, Zbl 1160.65015.
Trefor Bazett, What is cos( cos( cos( cos( cos( cos( cos( cos( cos( cos( cos( cos(...?? // Banach Fixed Point Theorem, YouTube video, 2022.
J. Bertrand, Exercise III, in Traité d'algèbre, Vols. 1-2, 4th ed. Paris, France: Librairie Hachette et Cie, p. 285, 1865.
Steven Finch, Exercises in Iterational Asymptotics II, arXiv preprint (2025). arXiv:2501.06065 [math.NT]
Samuel R. Kaplan, The Dottie Number, Math. Magazine, 80 (No. 1, 2007), 73-74.
T. H. Miller, On the imaginary roots of cos x = x, Proc. Edinburgh Math. Soc. 21, 160-162, 1902.
V. Salov, Inevitable Dottie Number. Iterals of cosine and sine, arXiv preprint arXiv:1212.1027 [math.HO], 2012.
David R. Stoutemyer, Inverse spherical Bessel functions generalize Lambert W and solve similar equations containing trigonometric or hyperbolic subexpressions or their inverses, arXiv:2207.00707 [math.GM], 2022.
Eric Weisstein's World of Mathematics, Dottie Number
Index entries for transcendental numbers.

Cf. A009442, A177413, A182503, A197002, A200309, A212112, A212113, A217066.

Cf. A330119 (degrees-based analog).

evalf(solve(cos(x)=x,x), 140);  # Alois P. Heinz, Feb 20 2024

RealDigits[ FindRoot[ Cos[x] == x, {x, {.7, 1} }, WorkingPrecision -> 120] [[1, 2] ]] [[1]]
FindRoot[Cos[x] == x, {x, {.7, 1}}, WorkingPrecision -> 500][[1, 2]][[1]] (* Ben Branman, Apr 12 2008 *)
N[NestList[Cos, 1, 100], 20] (* Clark Kimberling, Jul 01 2019 *)
RealDigits[Root[{# - Cos[#] &, 0.739085}], 10, 100][[1]] (* Eric W. Weisstein, Jul 15 2022 *)
RealDigits[Sqrt[1 - (2 InverseBetaRegularized[1/2, 1/2, 3/2] - 1)^2], 10, 100][[1]] (* Eric W. Weisstein, Jul 15 2022 *)

solve(x=0,1,cos(x)-x) \\ Charles R Greathouse IV, Dec 31 2011

from sympy import Symbol, nsolve, cos
x = Symbol("x")
a = list(map(int, str(nsolve(cos(x)-x, 1, prec=110))[2:-2]))
print(a) # Michael S. Branicky, Jul 15 2022

Equals twice A197002. - Hugo Pfoertner, Feb 20 2024

More terms from David W. Wilson

Additional references from Ben Branman, Dec 28 2011

A342204 Decimal expansion of the fixed point of the cosine function when measured in quadrants (1 quadrant = 90 degrees = Pi/2 radians).

Original entry on oeis.org

5, 9, 4, 6, 1, 1, 6, 4, 4, 0, 5, 6, 8, 3, 5, 5, 8, 2, 9, 8, 8, 4, 6, 1, 8, 8, 4, 6, 3, 4, 7, 7, 3, 9, 2, 4, 7, 8, 9, 9, 4, 9, 3, 7, 2, 7, 1, 7, 0, 2, 5, 1, 4, 5, 0, 3, 0, 1, 8, 5, 7, 4, 3, 0, 1, 4, 2, 9, 9, 0, 2, 7, 9, 6, 3, 9, 3, 8, 5, 8, 5, 2, 6, 2, 9, 7, 7, 0, 6, 2, 0, 7, 5, 7, 8, 2, 8, 6, 5, 2

Offset: 0

Michael Sheridan, Mar 04 2021

This is analogous to A003957, the fixed point of the cosine function in radians and A330119, the fixed point of the cosine function in degrees.  Each of the three are the unique real solutions to cos(x)-x=0, in their respective angular units.  The quadrant unit offers a nice symmetry, cos(0)=1 and cos(1)=0.  Unlike the previous two, the quadrant fixed point is not an attractor of its cosine function.  It cannot be found by iterative cosine application. Although not proven, iterative quadrant cosine can be seen empirically to diverge for all initial values.
A graphical solution can be demonstrated by plotting y = cos(x*Pi/2) - x, which shows a single zero near x=0.6.
The bisection method converges for the entire range of the cosine function (-1 to 1).  Newton's method also converges with reasonable initial estimate.

			0.594611644056835582988461884634773924789949372717025145030185743014299027963938...

Cf. A003957, A330119.

RealDigits[x /. FindRoot[Cos[Pi*x/2] == x, {x, 1}, WorkingPrecision -> 105], 10, 100][[1]] (* Amiram Eldar, Mar 05 2021 *)

fpprec :100;
bf_find_root(cos(x*%pi/2)-x,x,0,1) ;

PARI
```
\p 100
solve(x=-1,1,cos(Pi*x/2)-x)
```

cp's OEIS Frontend

A003957 The Dottie number: decimal expansion of root of cos(x) = x.

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