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Search: id:A131691
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| A131691 |
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Decimal expansion of convergence of iterated sine-cosine composite function. |
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+0
1
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| 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
(list; cons; graph; listen)
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OFFSET
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0,1
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COMMENT
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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 (mathar(AT)strw.leidenuniv.nl), Sep 28 2007
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FORMULA
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Let a(0) = some real number k (in radians); then a(n) = sin(cos(a(n-1))) which converges as n goes to infinity.
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EXAMPLE
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Let k = 0.5 radians; then a(0) = k = 0.5; a(1) = sin(cos(0.5)) = 0.76919...; a(2) = sin(cos(a(1))) = sin(cos(sin(cos(0.5)))) = 0.65823...; a(3) = 0.71110..., and so forth.
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MAPLE
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evalf( solve(sin(cos(x))=x, x)) ; - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Sep 28 2007
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CROSSREFS
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Sequence in context: A073240 A019853 A007332 this_sequence A021063 A110649 A037024
Adjacent sequences: A131688 A131689 A131690 this_sequence A131692 A131693 A131694
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KEYWORD
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cons,easy,nonn
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AUTHOR
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Alan Wessman (alanyst(AT)gmail.com), Sep 15 2007
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