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A200116 Decimal expansion of least x satisfying 2*x^2 - 2*cos(x) = sin(x), negated.

%I A200116 #11 Feb 12 2025 11:22:03
%S A200116 6,8,0,3,2,6,4,1,4,1,3,8,6,7,9,2,9,6,2,3,9,6,3,1,6,2,0,7,3,6,4,1,9,1,
%T A200116 7,6,8,6,5,5,3,0,2,5,8,0,2,1,0,8,1,4,5,3,5,6,0,8,0,7,7,9,5,9,8,9,2,6,
%U A200116 3,3,9,2,2,7,0,8,1,5,4,8,2,0,3,7,7,9,1,0,0,2,2,0,1,2,5,7,6,4,7
%N A200116 Decimal expansion of least x satisfying 2*x^2 - 2*cos(x) = sin(x), negated.
%C A200116 See A199949 for a guide to related sequences. The Mathematica program includes a graph.
%H A200116 G. C. Greubel, <a href="/A200116/b200116.txt">Table of n, a(n) for n = 0..10000</a>
%H A200116 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%e A200116 least x: -0.680326414138679296239631620736419...
%e A200116 greatest x: 0.9847126993630673524991380090748...
%t A200116 a = 2; b = -2; c = 1;
%t A200116 f[x_] := a*x^2 + b*Cos[x]; g[x_] := c*Sin[x]
%t A200116 Plot[{f[x], g[x]}, {x, -3, 3}, {AxesOrigin -> {0, 0}}]
%t A200116 r = x /. FindRoot[f[x] == g[x], {x, -.69, -.68}, WorkingPrecision -> 110]
%t A200116 RealDigits[r]  (* A200116 *)
%t A200116 r = x /. FindRoot[f[x] == g[x], {x, .98, .99}, WorkingPrecision -> 110]
%t A200116 RealDigits[r]  (* A200117 *)
%o A200116 (PARI) a=2; b=-2; c=1; solve(x=-1, 0, a*x^2 + b*cos(x) - c*sin(x)) \\ _G. C. Greubel_, Jun 29 2018
%Y A200116 Cf. A199949.
%K A200116 nonn,cons
%O A200116 0,1
%A A200116 _Clark Kimberling_, Nov 14 2011