Search: a197384 -id:a197384
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A197133
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Decimal expansion of least x>0 having sin(x) = sin(2*x)^2.
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+10
94
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2, 7, 2, 9, 7, 1, 8, 4, 9, 2, 3, 6, 8, 2, 4, 9, 5, 0, 4, 0, 8, 6, 1, 6, 8, 0, 6, 0, 8, 3, 8, 6, 9, 8, 3, 1, 0, 4, 7, 4, 0, 6, 6, 5, 1, 9, 6, 6, 4, 4, 0, 1, 8, 2, 7, 6, 6, 8, 0, 0, 0, 1, 1, 4, 8, 4, 3, 3, 5, 9, 2, 7, 0, 1, 0, 2, 2, 0, 8, 9, 0, 4, 3, 5, 9, 2, 4, 4, 8, 6, 4, 3, 1, 9, 4, 0, 5, 6, 9, 0, 8
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OFFSET
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0,1
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COMMENTS
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The Mathematica program includes a graph.
Guide for least x>0 satisfying sin(b*x) = sin(c*x)^2 for selected numbers b and c:
b.....c.......x
4.....2.......A195693, x=arctan(1/(golden ratio))
Pi/6..Pi/4..........., x=1
Cf. A197476 for a similar table for sin(b*x) = sin(c*x)^2.
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LINKS
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FORMULA
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Equals arcsin(2*sin(arcsin(3*sqrt(3)/8)/3)/sqrt(3))
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EXAMPLE
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x=0.272971849236824950408616...
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MATHEMATICA
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b = 1; c = 2; f[x_] := Sin[x]
t = x /. FindRoot[f[b*x] == f[c*x]^2, {x, .1, .3}, WorkingPrecision -> 100]
Plot[{f[b*x], f[c*x]^2}, {x, 0, Pi}]
(* Second program: *)
RealDigits[ ArcSec[ Root[16 - 16 x^2 + x^6, 3]], 10, 100] // First (* Jean-François Alcover, Feb 19 2013 *)
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PROG
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(PARI) asin(2*sin(asin(3*sqrt(3)/8)/3)/sqrt(3)) \\ Gleb Koloskov, Sep 15 2021
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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