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A200129 Decimal expansion of greatest x satisfying 2*x^2 - 4*cos(x) = sin(x).

%I A200129 #12 Feb 12 2025 13:01:17
%S A200129 1,1,3,7,4,0,1,1,9,9,5,2,6,8,6,8,5,2,6,5,0,2,7,8,8,0,3,0,8,4,2,5,4,4,
%T A200129 8,8,0,5,3,0,2,1,1,9,6,5,1,5,2,5,1,3,6,5,2,7,2,9,1,7,5,8,7,9,5,2,0,9,
%U A200129 9,5,9,6,1,9,0,2,0,3,1,5,1,9,0,1,7,9,8,3,6,9,7,0,1,2,9,6,8,0,1
%N A200129 Decimal expansion of greatest x satisfying 2*x^2 - 4*cos(x) = sin(x).
%C A200129 See A199949 for a guide to related sequences. The Mathematica program includes a graph.
%H A200129 G. C. Greubel, <a href="/A200129/b200129.txt">Table of n, a(n) for n = 1..10000</a>
%H A200129 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%e A200129 least x: -0.91125136577248241254947318280293...
%e A200129 greatest x: 1.13740119952686852650278803084...
%t A200129 a = 2; b = -4; c = 1;
%t A200129 f[x_] := a*x^2 + b*Cos[x]; g[x_] := c*Sin[x]
%t A200129 Plot[{f[x], g[x]}, {x, -3, 3}, {AxesOrigin -> {0, 0}}]
%t A200129 r = x /. FindRoot[f[x] == g[x], {x, -.92, -.91}, WorkingPrecision -> 110]
%t A200129 RealDigits[r]  (* A200128 *)
%t A200129 r = x /. FindRoot[f[x] == g[x], {x, 1.13, 1.14}, WorkingPrecision -> 110]
%t A200129 RealDigits[r]  (* A200129 *)
%o A200129 (PARI) a=2; b=-4; c=1; solve(x=1, 2, a*x^2 + b*cos(x) - c*sin(x)) \\ _G. C. Greubel_, Jul 01 2018
%Y A200129 Cf. A199949.
%K A200129 nonn,cons
%O A200129 1,3
%A A200129 _Clark Kimberling_, Nov 14 2011