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A200117 Decimal expansion of greatest x satisfying 2*x^2 - 2*cos(x) = sin(x).

%I A200117 #11 Feb 12 2025 11:22:12
%S A200117 9,8,4,7,1,2,6,9,9,3,6,3,0,6,7,3,5,2,4,9,9,1,3,8,0,0,9,0,7,4,8,4,5,5,
%T A200117 2,4,3,2,3,5,0,7,8,9,3,1,1,5,1,0,5,9,5,6,0,4,9,2,5,3,6,5,5,6,9,1,3,4,
%U A200117 7,6,9,8,2,7,3,6,3,5,2,6,1,9,1,6,0,4,3,4,8,3,7,0,8,5,6,3,0,4,0
%N A200117 Decimal expansion of greatest x satisfying 2*x^2 - 2*cos(x) = sin(x).
%C A200117 See A199949 for a guide to related sequences. The Mathematica program includes a graph.
%H A200117 G. C. Greubel, <a href="/A200117/b200117.txt">Table of n, a(n) for n = 0..10000</a>
%H A200117 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%e A200117 least x: -0.680326414138679296239631620736419...
%e A200117 greatest x: 0.9847126993630673524991380090748...
%t A200117 a = 2; b = -2; c = 1;
%t A200117 f[x_] := a*x^2 + b*Cos[x]; g[x_] := c*Sin[x]
%t A200117 Plot[{f[x], g[x]}, {x, -3, 3}, {AxesOrigin -> {0, 0}}]
%t A200117 r = x /. FindRoot[f[x] == g[x], {x, -.69, -.68}, WorkingPrecision -> 110]
%t A200117 RealDigits[r]  (* A200116 *)
%t A200117 r = x /. FindRoot[f[x] == g[x], {x, .98, .99}, WorkingPrecision -> 110]
%t A200117 RealDigits[r]  (* A200117 *)
%o A200117 (PARI) a=2; b=-2; c=1; solve(x=0, 1, a*x^2 + b*cos(x) - c*sin(x)) \\ _G. C. Greubel_, Jun 29 2018
%Y A200117 Cf. A199949.
%K A200117 nonn,cons
%O A200117 0,1
%A A200117 _Clark Kimberling_, Nov 14 2011