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A197739 Decimal expansion of least x>0 having sin(2x)=3*sin(6x).

%I A197739 #19 Feb 22 2025 18:50:43
%S A197739 4,7,7,6,5,8,3,0,9,0,6,2,2,5,4,6,3,9,0,8,1,9,2,8,5,5,1,2,5,7,8,7,8,8,
%T A197739 7,7,1,2,1,7,0,7,3,4,7,5,0,5,0,0,2,7,4,5,4,7,9,8,4,9,0,6,4,6,6,0,9,5,
%U A197739 6,0,2,2,9,5,1,9,8,8,2,2,7,6,9,3,6,9,5,8,0,1,2,9,2,8,1,4,0,3,6
%N A197739 Decimal expansion of least x>0 having sin(2x)=3*sin(6x).
%C A197739 This constant is the least x>0 for which the function f(x)=(sin(x))^2+(cos(3x))^2 has its maximal value.  Least positive solutions of the equations f(x)=m/2, f(x)=m/3, f(x)=1, and f(x)=1/2 are given by sequences shown in the guide below.
%C A197739 In general, suppose that b and c are distinct positive real numbers.  Let f(x)=(sin(bx))^2+cos((cx))^2.  The extrema of f are the solutions of b*sin(2bx)=c*sin(2cx).
%C A197739 In the following guide, constants x given by the sequences (or explicit number) listed for each b,c are, in this order:
%C A197739 (1) least x>0 such that f(x)=(its maximum, m)
%C A197739 (2) m, the maximum of f
%C A197739 (3) least x>0 having f(x)=m/2
%C A197739 (4) least x>0 having f(x)=m/3
%C A197739 (5) least x>0 having f(x)=1
%C A197739 (6) least x>0 having f(x)=1/2
%C A197739 ...
%C A197739 (b,c)=(1,2):  A195700, x=25/16, A197589, A197591,
%C A197739   A019670, A197592
%C A197739 (b,c)=(1,3):  A197739, A197588, A197590, A197755,
%C A197739   A003881, A197488
%C A197739 (b,c)=(1,4):  A197758, A197759, A197760, A197761,
%C A197739   A019692 (x=pi/5), A003881
%C A197739 (b,c)=(1,pi): A197821, A197822, A197823, A197824,
%C A197739   A197726, A197826
%C A197739 (b,c)=(1,2*pi): A197827, A197828, A197829, A197830,
%C A197739   A197700, A197832
%C A197739 (b,c)=(1,3*pi): A197833, A197834, A197835, A197836,
%C A197739   A197837, A197838
%H A197739 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%e A197739 0.47765830906225463908192855125787887712170734750500...
%t A197739 b = 1; c = 3;
%t A197739 f[x_] := Cos[b*x]^2; g[x_] := Sin[c*x]^2; s[x_] := f[x] + g[x];
%t A197739 r = x /. FindRoot[b*Sin[2 b*x] == c*Sin[2 c*x], {x, .47, .48}, WorkingPrecision -> 110]
%t A197739 RealDigits[r]  (* A197739 *)
%t A197739 m = s[r]
%t A197739 RealDigits[m]  (* A197588 *)
%t A197739 Plot[{b*Sin[2 b*x], c*Sin[2 c*x]}, {x, 0, Pi}]
%t A197739 d = m/2; t = x /. FindRoot[s[x] == d, {x, 0.7, 0.8}, WorkingPrecision -> 110]
%t A197739 RealDigits[t]  (* A197590 *)
%t A197739 Plot[{s[x], d}, {x, 0, Pi}, AxesOrigin -> {0, 0}]
%t A197739 d = m/3; t = x /. FindRoot[s[x] == d, {x, 0.8, 0.9}, WorkingPrecision -> 110]
%t A197739 RealDigits[t]  (* A197755 *)
%t A197739 Plot[{s[x], d}, {x, 0, Pi}, AxesOrigin -> {0, 0}]
%t A197739 d = 1; t = x /. FindRoot[s[x] == d, {x, 0.7, 0.8}, WorkingPrecision -> 110]
%t A197739 RealDigits[t]  (* A003881 *)
%t A197739 Plot[{s[x], d}, {x, 0, Pi}, AxesOrigin -> {0, 0}]
%t A197739 d = 1/2; t = x /. FindRoot[s[x] == d, {x, .9, .93}, WorkingPrecision -> 110]
%t A197739 RealDigits[t]  (* A197488 *)
%t A197739 Plot[{s[x], d}, {x, 0, Pi}, AxesOrigin -> {0, 0}]
%t A197739 RealDigits[ ArcTan[ Sqrt[ 2-Sqrt[3] ] ], 10, 99] // First (* _Jean-François Alcover_, Feb 27 2013 *)
%Y A197739 Cf. A197739, A197588.
%K A197739 nonn,cons
%O A197739 0,1
%A A197739 _Clark Kimberling_, Oct 18 2011