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

Author

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

Keywords

• cons
• easy
• nonn

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

Examples

			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...
		

Links

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

Crossrefs

Cf. A277077.

Programs

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 *)

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

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.

Extensions

More terms from Michel Marcus, Oct 04 2016

Name clarified by Joerg Arndt, Oct 04 2016