A277077 -id:A277077 - cp's OEIS frontend

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

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

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

			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...

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

Cf. A277077.

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

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

solve(x=0, 1, sin(cos(x))-x) \\ Michel Marcus, Oct 04 2016

(sin(cos(x))==x).find_root(0,1,x) # G. C. Greubel, Mar 16 2019

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

More terms from Michel Marcus, Oct 04 2016

Name clarified by Joerg Arndt, Oct 04 2016

cp's OEIS Frontend

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

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