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A200131 Decimal expansion of greatest x satisfying 2*x^2 - 4*cos(x) = 3*sin(x).

%I A200131 #12 Feb 12 2025 13:01:20
%S A200131 1,3,6,8,1,4,9,1,1,2,0,4,2,0,6,7,6,6,7,9,9,7,6,9,9,1,0,8,8,9,0,6,9,2,
%T A200131 6,6,7,5,9,0,3,6,3,8,6,9,7,9,8,2,3,5,6,8,8,7,6,8,3,8,1,0,1,7,8,4,0,2,
%U A200131 0,0,9,5,7,2,9,5,6,3,9,7,4,8,1,6,1,3,1,7,9,1,3,4,2,8,1,1,0,1,7
%N A200131 Decimal expansion of greatest x satisfying 2*x^2 - 4*cos(x) = 3*sin(x).
%C A200131 See A199949 for a guide to related sequences. The Mathematica program includes a graph.
%H A200131 G. C. Greubel, <a href="/A200131/b200131.txt">Table of n, a(n) for n = 1..10000</a>
%H A200131 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%e A200131 least x: -0.719005064558842927859271780848179382...
%e A200131 greatest x: 1.368149112042067667997699108890...
%t A200131 a = 2; b = -4; c = 3;
%t A200131 f[x_] := a*x^2 + b*Cos[x]; g[x_] := c*Sin[x]
%t A200131 Plot[{f[x], g[x]}, {x, -3, 3}, {AxesOrigin -> {0, 0}}]
%t A200131 r = x /. FindRoot[f[x] == g[x], {x, -.72, -.71}, WorkingPrecision -> 110]
%t A200131 RealDigits[r]  (* A200130 *)
%t A200131 r = x /. FindRoot[f[x] == g[x], {x, 1.36, 1.37}, WorkingPrecision -> 110]
%t A200131 RealDigits[r]   (* A200131 *)
%o A200131 (PARI) a=2; b=-4; c=3; solve(x=1, 2, a*x^2 + b*cos(x) - c*sin(x)) \\ _G. C. Greubel_, Jul 01 2018
%Y A200131 Cf. A199949.
%K A200131 nonn,cons
%O A200131 1,2
%A A200131 _Clark Kimberling_, Nov 14 2011