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A200108 Decimal expansion of greatest x satisfying 2*x^2 - cos(x) = sin(x).

%I A200108 #11 Feb 12 2025 09:55:32
%S A200108 8,4,0,2,6,3,5,1,7,7,1,5,7,6,7,8,9,9,3,4,7,9,7,3,4,9,9,6,4,8,3,5,5,7,
%T A200108 9,7,3,6,5,0,2,5,3,9,0,5,3,5,1,5,2,6,6,1,1,7,3,5,4,3,6,3,9,2,5,1,7,4,
%U A200108 5,5,5,6,5,3,6,2,5,0,2,1,5,6,7,8,0,3,5,1,8,3,7,2,4,6,3,0,2,7,7
%N A200108 Decimal expansion of greatest x satisfying 2*x^2 - cos(x) = sin(x).
%C A200108 See A199949 for a guide to related sequences. The Mathematica program includes a graph.
%H A200108 G. C. Greubel, <a href="/A200108/b200108.txt">Table of n, a(n) for n = 0..10000</a>
%H A200108 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%e A200108 least x: -0.4690323711198093057335493058025105005500...
%e A200108 greatest x: 0.840263517715767899347973499648355797365...
%t A200108 a = 2; b = -1; c = 1;
%t A200108 f[x_] := a*x^2 + b*Cos[x]; g[x_] := c*Sin[x]
%t A200108 Plot[{f[x], g[x]}, {x, -3, 3}, {AxesOrigin -> {0, 0}}]
%t A200108 r = x /.
%t A200108   FindRoot[f[x] == g[x], {x, -.47, -.46}, WorkingPrecision -> 110]
%t A200108 RealDigits[r]  (* A200107 *)
%t A200108 r = x /. FindRoot[f[x] == g[x], {x, .84, .85}, WorkingPrecision -> 110]
%t A200108 RealDigits[r]  (* A200108 *)
%o A200108 (PARI) a=2; b=-1; c=1; solve(x=0, 1, a*x^2 + b*cos(x) - c*sin(x)) \\ _G. C. Greubel_, Jun 25 2018
%Y A200108 Cf. A199949.
%K A200108 nonn,cons
%O A200108 0,1
%A A200108 _Clark Kimberling_, Nov 13 2011