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A200121 Decimal expansion of greatest x satisfying 2*x^2 - 3*cos(x) = sin(x).

%I A200121 #11 Feb 12 2025 13:00:26
%S A200121 1,0,7,4,3,0,9,2,0,6,5,0,6,0,4,6,8,9,0,1,0,8,3,5,7,7,7,8,9,2,8,6,3,0,
%T A200121 6,3,4,2,8,6,1,7,0,7,8,6,8,2,3,6,6,6,0,5,3,6,8,9,9,5,0,4,9,9,8,3,8,8,
%U A200121 0,3,8,0,7,6,1,3,0,6,5,9,0,0,0,8,8,4,2,5,8,8,9,8,3,5,2,6,5,9,1
%N A200121 Decimal expansion of greatest x satisfying 2*x^2 - 3*cos(x) = sin(x).
%C A200121 See A199949 for a guide to related sequences. The Mathematica program includes a graph.
%H A200121 G. C. Greubel, <a href="/A200121/b200121.txt">Table of n, a(n) for n = 1..10000</a>
%H A200121 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%e A200121 least x: -0.815233223410514131205921200022220970300...
%e A200121 greatest x: 1.0743092065060468901083577789286306342...
%t A200121 a = 2; b = -3; c = 1;
%t A200121 f[x_] := a*x^2 + b*Cos[x]; g[x_] := c*Sin[x]
%t A200121 Plot[{f[x], g[x]}, {x, -3, 3}, {AxesOrigin -> {0, 0}}]
%t A200121 r = x /. FindRoot[f[x] == g[x], {x, -.82, -.81}, WorkingPrecision -> 110]
%t A200121 RealDigits[r]  (* A200120 *)
%t A200121 r = x /. FindRoot[f[x] == g[x], {x, 1.07, 1.08}, WorkingPrecision -> 110]
%t A200121 RealDigits[r]  (* A200121 *)
%o A200121 (PARI) a=2; b=-3; c=1; solve(x=1, 2, a*x^2 + b*cos(x) - c*sin(x)) \\ _G. C. Greubel_, Jun 29 2018
%Y A200121 Cf. A199949.
%K A200121 nonn,cons
%O A200121 1,3
%A A200121 _Clark Kimberling_, Nov 14 2011