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A197589 Decimal expansion of least x>0 satisfying f(x)=m/2, where m is the maximal value of the function f(x)=cos(x)^2+sin(2x)^2.

%I A197589 #6 Mar 30 2012 18:57:53
%S A197589 1,1,2,8,6,8,0,1,9,4,3,3,7,7,5,2,8,4,4,7,0,0,6,0,4,9,8,4,5,3,3,4,6,2,
%T A197589 9,4,7,2,6,0,9,5,3,6,4,3,8,6,6,8,3,8,6,0,6,0,5,8,6,9,2,8,2,5,2,1,7,5,
%U A197589 0,0,0,9,6,6,8,2,8,9,4,5,0,2,1,9,3,6,8,6,5,1,3,0,4,5,7,2,4,8,8
%N A197589 Decimal expansion of least x>0 satisfying f(x)=m/2, where m is the maximal value of the function f(x)=cos(x)^2+sin(2x)^2.
%C A197589 For a discussion and guide to related sequences, see A197739.
%e A197589 x=1.12868019433775284470060498453346294726...
%t A197589 b = 1; c = 2;
%t A197589 f[x_] := Cos[b*x]^2; g[x_] := Sin[c*x]^2; s[x_] := f[x] + g[x];
%t A197589 r = x /. FindRoot[b*Sin[2 b*x] == c*Sin[2 c*x], {x, .65, .66}, WorkingPrecision -> 110]
%t A197589 RealDigits[r]  (* A195700, arcsin(sqrt(3/8)) *)
%t A197589 m = s[r]
%t A197589 RealDigits[m]
%t A197589 Rationalize[{m, m/2, m/3, 2 m/3, m/4, 3 m/4}]
%t A197589 Plot[{b*Sin[2 b*x], c*Sin[2 c*x]}, {x, 0, Pi}]
%t A197589 d = m/2; t = x /. FindRoot[s[x] == d, {x, 1.1, 1.2}, WorkingPrecision -> 110]
%t A197589 RealDigits[t]  (* A197589 *)
%t A197589 Plot[{s[x], d}, {x, 0, Pi}, AxesOrigin -> {0, 0}]
%t A197589 d = m/3; t = x /. FindRoot[s[x] == d, {x, 1.1, 1.2}, WorkingPrecision -> 110]
%t A197589 RealDigits[t]  (* A197591 *)
%t A197589 Plot[{s[x], d}, {x, 0, Pi}, AxesOrigin -> {0, 0}]
%t A197589 d = 1; t = x /. FindRoot[s[x] == d, {x, 1.1, 1.2}, WorkingPrecision -> 110]
%t A197589 RealDigits[t] (* A019670, pi/3 *)
%t A197589 Plot[{s[x], d}, {x, 0, Pi}, AxesOrigin -> {0, 0}]
%t A197589 d = 1/2; t = x /. FindRoot[s[x] == d, {x, 1.1, 1.2}, WorkingPrecision -> 110]
%t A197589 RealDigits[t]  (* A197592 *)
%t A197589 Plot[{s[x], d}, {x, 0, Pi}, AxesOrigin -> {0, 0}]
%Y A197589 Cf. A197739, A195700.
%K A197589 nonn,cons
%O A197589 0,3
%A A197589 _Clark Kimberling_, Oct 18 2011