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A200230 Decimal expansion of greatest x satisfying 3*x^2 - 2*cos(x) = sin(x).

%I A200230 #11 Feb 12 2025 13:04:30
%S A200230 8,3,3,6,2,0,4,7,0,3,0,7,4,5,4,0,7,8,2,7,4,1,7,0,1,7,8,7,1,2,5,3,2,1,
%T A200230 2,3,7,6,7,9,6,5,3,2,7,8,9,8,2,4,5,2,5,4,1,1,9,4,2,2,6,0,7,2,6,4,5,0,
%U A200230 6,9,6,2,9,4,9,8,9,7,3,7,4,7,5,9,4,9,1,0,9,8,5,1,9,8,7,7,1,3,1
%N A200230 Decimal expansion of greatest x satisfying 3*x^2 - 2*cos(x) = sin(x).
%C A200230 See A199949 for a guide to related sequences. The Mathematica program includes a graph.
%H A200230 G. C. Greubel, <a href="/A200230/b200230.txt">Table of n, a(n) for n = 0..10000</a>
%H A200230 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%e A200230 least x: -0.6010846085447445780840915757937924370...
%e A200230 greatest x: 0.83362047030745407827417017871253212...
%t A200230 a = 3; b = -2; c = 1;
%t A200230 f[x_] := a*x^2 + b*Cos[x]; g[x_] := c*Sin[x]
%t A200230 Plot[{f[x], g[x]}, {x, -2, 2}, {AxesOrigin -> {0, 0}}]
%t A200230 r = x /. FindRoot[f[x] == g[x], {x, -.61, -.60}, WorkingPrecision -> 110]
%t A200230 RealDigits[r]   (* A200229 *)
%t A200230 r = x /. FindRoot[f[x] == g[x], {x, .83, .84}, WorkingPrecision -> 110]
%t A200230 RealDigits[r]   (* A200230 *)
%o A200230 (PARI) a=3; b=-2; c=1; solve(x=0, 1, a*x^2 + b*cos(x) - c*sin(x)) \\ _G. C. Greubel_, Jun 30 2018
%Y A200230 Cf. A199949.
%K A200230 nonn,cons
%O A200230 0,1
%A A200230 _Clark Kimberling_, Nov 14 2011