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A200133 Decimal expansion of greatest x satisfying 3*x^2 - cos(x) = sin(x).

%I A200133 #16 Feb 12 2025 13:03:06
%S A200133 6,8,4,8,5,3,0,7,8,6,2,3,2,0,1,1,5,9,5,6,3,6,9,4,4,6,8,6,4,9,5,4,2,8,
%T A200133 8,8,4,5,1,8,4,2,6,1,0,3,1,8,2,0,2,6,7,1,9,2,8,2,6,1,9,9,7,6,4,6,0,2,
%U A200133 2,5,8,4,0,3,1,2,9,4,4,3,2,7,9,2,2,5,9,2,5,2,4,0,4,6,8,1,0,2,3
%N A200133 Decimal expansion of greatest x satisfying 3*x^2 - cos(x) = sin(x).
%C A200133 See A199949 for a guide to related sequences. The Mathematica program includes a graph.
%H A200133 G. C. Greubel, <a href="/A200133/b200133.txt">Table of n, a(n) for n = 0..10000</a>
%H A200133 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%e A200133 least x: -0.4137517591447739376844002798989...
%e A200133 greatest x: 0.684853078623201159563694468649...
%t A200133 a = 3; b = -1; c = 1;
%t A200133 f[x_] := a*x^2 + b*Cos[x]; g[x_] := c*Sin[x]
%t A200133 Plot[{f[x], g[x]}, {x, -3, 3}, {AxesOrigin -> {0, 0}}]
%t A200133 r = x /. FindRoot[f[x] == g[x], {x, -.42, -.41}, WorkingPrecision -> 110]
%t A200133 RealDigits[r]   (* A200132 *)
%t A200133 r = x /. FindRoot[f[x] == g[x], {x, .68, .69}, WorkingPrecision -> 110]
%t A200133 RealDigits[r]   (* A200133 *)
%o A200133 (PARI) a=3; b=-1; c=1; solve(x=0, 1, a*x^2 + b*cos(x) - c*sin(x)) \\ _G. C. Greubel_, Jul 05 2018
%Y A200133 Cf. A199949.
%K A200133 nonn,cons
%O A200133 0,1
%A A200133 _Clark Kimberling_, Nov 14 2011