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 A131691 Real fixed point of the function sin(cos(x)) between x=0 and x=1. 2
 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 (list; constant; graph; refs; listen; history; text; internal format)
 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 A021063 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

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Last modified June 18 01:16 EDT 2021. Contains 345098 sequences. (Running on oeis4.)