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

%I A131691 #26 May 09 2019 04:35:17
%S A131691 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,
%T A131691 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,
%U A131691 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
%N A131691 Real fixed point of the function sin(cos(x)) between x=0 and x=1.
%C A131691 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.
%C A131691 This is the solution to sin(cos(x))=x and to cos(cos(x))=sqrt(1-x^2). - _R. J. Mathar_, Sep 28 2007
%C A131691 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
%H A131691 G. C. Greubel, <a href="/A131691/b131691.txt">Table of n, a(n) for n = 0..5000</a>
%F A131691 Let f(0) = some real number k (in radians); then f(n) = sin(cos(f(n-1))), which converges as n goes to infinity.
%e A131691 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.
%e A131691 0.6948196907307875655784200727751937626855044467359379683700770954817215197...
%p A131691 evalf( solve(sin(cos(x))=x,x)) ; # _R. J. Mathar_, Sep 28 2007
%t A131691 RealDigits[x/.FindRoot[Sin[Cos[x]] -x, {x, 0, 1}, WorkingPrecision -> 105]][[1]] (* _G. C. Greubel_, Mar 16 2019 *)
%o A131691 (PARI) solve(x=0, 1, sin(cos(x))-x) \\ _Michel Marcus_, Oct 04 2016
%o A131691 (Sage) (sin(cos(x))==x).find_root(0,1,x) # _G. C. Greubel_, Mar 16 2019
%Y A131691 Cf. A277077.
%K A131691 cons,easy,nonn
%O A131691 0,1
%A A131691 Alan Wessman (alanyst(AT)gmail.com), Sep 15 2007
%E A131691 More terms from _Michel Marcus_, Oct 04 2016
%E A131691 Name clarified by _Joerg Arndt_, Oct 04 2016