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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
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
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 March 29 09:59 EDT 2024. Contains 371268 sequences. (Running on oeis4.)