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A197758 Decimal expansion of least x>0 having sin(2x)=4*sin(8x).

%I A197758 #12 Feb 27 2013 03:28:49
%S A197758 3,7,1,4,5,8,2,9,4,0,3,3,4,8,6,3,5,2,5,0,5,8,3,2,7,2,8,5,1,9,5,1,2,4,
%T A197758 0,9,8,0,8,9,6,8,2,6,0,7,3,9,5,7,5,3,9,0,7,2,3,4,4,5,2,9,1,0,6,3,6,6,
%U A197758 8,0,5,8,1,2,0,6,6,9,3,6,8,8,6,9,9,1,5,1,0,5,8,9,8,3,6,8,1,2,4
%N A197758 Decimal expansion of least x>0 having sin(2x)=4*sin(8x).
%C A197758 For a discussion and guide to related sequences, see A197739.
%e A197758 x=0.37145829403348635250583272851951240980...
%t A197758 b = 1; c = 4;
%t A197758 f[x_] := Cos[b*x]^2; g[x_] := Sin[c*x]^2; s[x_] := f[x] + g[x];
%t A197758 r = x /. FindRoot[b*Sin[2 b*x] == c*Sin[2 c*x], {x, .37, .38}, WorkingPrecision -> 110]
%t A197758 RealDigits[r]  (* A197758 *)
%t A197758 m = s[r]
%t A197758 RealDigits[m]  (* A197759 *)
%t A197758 Plot[{b*Sin[2 b*x], c*Sin[2 c*x]}, {x, 0, Pi}]
%t A197758 d = m/2; t = x /. FindRoot[s[x] == d, {x, 0.64, 0.65}, WorkingPrecision -> 110]
%t A197758 RealDigits[t]  (* A197760 *)
%t A197758 Plot[{s[x], d}, {x, 0, Pi}, AxesOrigin -> {0, 0}]
%t A197758 d = m/3; t = x /. FindRoot[s[x] == d, {x, 0.72, 0.73}, WorkingPrecision -> 110]
%t A197758 RealDigits[t]  (* A197761 *)
%t A197758 Plot[{s[x], d}, {x, 0, Pi}, AxesOrigin -> {0, 0}]
%t A197758 d = 1; t = x /. FindRoot[s[x] == d, {x, 0.6, 0.7}, WorkingPrecision -> 110]
%t A197758 RealDigits[t]  (* A019692, pi/5 *)
%t A197758 Plot[{s[x], d}, {x, 0, Pi}, AxesOrigin -> {0, 0}]
%t A197758 d = 1/2; t = x /. FindRoot[s[x] == d, {x, 0.6, 0.8}, WorkingPrecision -> 110]
%t A197758 RealDigits[t]   (* A003881 *)
%t A197758 Plot[{s[x], d}, {x, 0, 1}, AxesOrigin -> {0, 0}]
%t A197758 RealDigits[ ArcTan[ Sqrt[ Root[17#^3 - 109#^2 + 115# - 15&, 1] ] ], 10, 99] // First (* _Jean-François Alcover_, Feb 27 2013 *)
%Y A197758 Cf. A197739
%K A197758 nonn,cons
%O A197758 0,1
%A A197758 _Clark Kimberling_, Oct 19 2011