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A200300 Decimal expansion of greatest x satisfying 4*x^2 - 3*cos(x) = sin(x).

%I A200300 #12 Feb 12 2025 14:36:09
%S A200300 8,3,0,8,5,0,3,2,7,6,6,0,5,4,7,4,0,2,7,6,6,6,2,0,9,9,3,5,6,6,5,9,7,2,
%T A200300 8,9,7,8,5,3,0,3,0,1,5,7,3,0,2,8,1,4,8,0,7,4,7,1,6,5,1,2,1,8,3,5,0,0,
%U A200300 1,8,5,4,8,1,3,3,8,1,5,2,2,3,2,5,4,0,6,8,6,3,2,0,8,3,6,2,8,0,6
%N A200300 Decimal expansion of greatest x satisfying 4*x^2 - 3*cos(x) = sin(x).
%C A200300 See A199949 for a guide to related sequences. The Mathematica program includes a graph.
%H A200300 G. C. Greubel, <a href="/A200300/b200300.txt">Table of n, a(n) for n = 0..10000</a>
%H A200300 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%e A200300 least x: -0.661826141188850993743026123357094...
%e A200300 greatest x: 0.8308503276605474027666209935665...
%t A200300 a = 4; b = -3; c = 1;
%t A200300 f[x_] := a*x^2 + b*Cos[x]; g[x_] := c*Sin[x]
%t A200300 Plot[{f[x], g[x]}, {x, -1, 1}, {AxesOrigin -> {0, 0}}]
%t A200300 r = x /. FindRoot[f[x] == g[x], {x, -.67, -.66}, WorkingPrecision -> 110]
%t A200300 RealDigits[r]   (* A200299 *)
%t A200300 r = x /. FindRoot[f[x] == g[x], {x, .83, .84}, WorkingPrecision -> 110]
%t A200300 RealDigits[r]   (* A200300 *)
%o A200300 (PARI) a=4; b=-3; c=1; solve(x=0, 1, a*x^2 + b*cos(x) - c*sin(x)) \\ _G. C. Greubel_, Jul 08 2018
%Y A200300 Cf. A199949.
%K A200300 nonn,cons
%O A200300 0,1
%A A200300 _Clark Kimberling_, Nov 15 2011