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A200278 Decimal expansion of greatest x satisfying 3*x^2 - 4*cos(x) = sin(x).

%I A200278 #15 Feb 12 2025 13:08:57
%S A200278 1,0,0,0,3,0,3,6,3,9,2,8,3,5,9,0,1,8,5,1,8,7,2,2,5,0,3,5,7,4,4,1,8,0,
%T A200278 0,1,2,2,5,8,5,6,5,6,1,8,5,2,9,2,0,2,6,4,2,6,3,2,2,5,7,4,9,5,4,6,0,9,
%U A200278 3,0,3,2,3,5,9,3,8,1,8,7,9,0,3,9,0,6,4,2,4,1,8,6,9,7,4,6,3,5,4
%N A200278 Decimal expansion of greatest x satisfying 3*x^2 - 4*cos(x) = sin(x).
%C A200278 See A199949 for a guide to related sequences. The Mathematica program includes a graph.
%H A200278 G. C. Greubel, <a href="/A200278/b200278.txt">Table of n, a(n) for n = 1..10000</a>
%H A200278 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%e A200278 least x: -0.81771521879230454511191454208365777...
%e A200278 greatest x: 1.000303639283590185187225035744180...
%t A200278 a = 3; b = -4; c = 1;
%t A200278 f[x_] := a*x^2 + b*Cos[x]; g[x_] := c*Sin[x]
%t A200278 Plot[{f[x], g[x]}, {x, -2, 2}, {AxesOrigin -> {0, 0}}]
%t A200278 r = x /.  FindRoot[f[x] == g[x], {x, -.82, -.81}, WorkingPrecision -> 110]
%t A200278 RealDigits[r]   (* A200277  *)
%t A200278 r = x /. FindRoot[f[x] == g[x], {x, 1.0, 1.1}, WorkingPrecision -> 110]
%t A200278 RealDigits[r]   (* A200278 *)
%o A200278 (PARI) a=3; b=-4; c=1; solve(x=1, 2, a*x^2 + b*cos(x) - c*sin(x)) \\ _G. C. Greubel_, Jul 05 2018
%Y A200278 Cf. A199949.
%K A200278 nonn,cons
%O A200278 1,5
%A A200278 _Clark Kimberling_, Nov 15 2011