A131691 Real fixed point of the function sin(cos(x)) between x=0 and x=1.

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

Offset

0,1

Comments

This constant can be discovered by entering an arbitrary number in radians on a digital calculator and iteratively taking the cosine of the number and then the sine of that result, then the cosine of that result and so on, until it converges to two constants, one for when the sine is taken and the other for when the cosine is taken.

This is the solution to sin(cos(x))=x and to cos(cos(x))=sqrt(1-x^2). - R. J. Mathar, Sep 28 2007

The value A277077 is equal to the cosine of this value and this value is equal to the sine of A277077. - John W. Nicholson, Mar 16 2019

Links

G. C. Greubel, Table of n, a(n) for n = 0..5000

Formula

Let f(0) = some real number k (in radians); then f(n) = sin(cos(f(n-1))), which converges as n goes to infinity.

Example

Let k = 0.5 radians; then f(0) = k = 0.5; f(1) = sin(cos(0.5)) = 0.76919...; f(2) = sin(cos(f(1))) = sin(cos(sin(cos(0.5)))) = 0.65823...; f(3) = 0.71110... and so forth.
0.6948196907307875655784200727751937626855044467359379683700770954817215197...

Maple

evalf( solve(sin(cos(x))=x, x)) ; # R. J. Mathar, Sep 28 2007

Mathematica

RealDigits[x/.FindRoot[Sin[Cos[x]] -x, {x, 0, 1}, WorkingPrecision -> 105]][[1]] (* G. C. Greubel, Mar 16 2019 *)

Prog

(PARI) solve(x=0, 1, sin(cos(x))-x) \\ Michel Marcus, Oct 04 2016
(Sage) (sin(cos(x))==x).find_root(0, 1, x) # G. C. Greubel, Mar 16 2019

Crossrefs

Cf. A277077.

Sequence in context: A019853 A007332 A246041 * A258504 A273816 A350817

Adjacent sequences: A131688 A131689 A131690 * A131692 A131693 A131694

Keyword

cons,easy,nonn

Author

Alan Wessman (alanyst(AT)gmail.com), Sep 15 2007

Extensions

More terms from Michel Marcus, Oct 04 2016

Name clarified by Joerg Arndt, Oct 04 2016

Status

approved