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A200100 Decimal expansion of greatest x satisfying x^2 - 4*cos(x) = sin(x).

%I A200100 #12 Feb 12 2025 09:54:26
%S A200100 1,3,5,4,5,7,5,5,5,8,2,1,5,8,5,7,8,4,4,9,0,8,9,0,7,7,0,1,6,4,6,4,6,3,
%T A200100 7,1,8,8,1,7,4,5,1,3,4,2,1,0,6,2,6,4,5,6,2,3,4,1,1,1,6,9,6,7,0,1,4,2,
%U A200100 1,3,1,9,1,6,3,0,2,2,8,8,3,3,1,9,0,4,0,2,9,8,1,8,3,5,3,7,7,0,2
%N A200100 Decimal expansion of greatest x satisfying x^2 - 4*cos(x) = sin(x).
%C A200100 See A199949 for a guide to related sequences. The Mathematica program includes a graph.
%H A200100 G. C. Greubel, <a href="/A200100/b200100.txt">Table of n, a(n) for n = 1..10000</a>
%H A200100 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%e A200100 least x:  -1.053352983600153733281110157999...
%e A200100 greatest x: 1.35457555821585784490890770164646...
%t A200100 a = 1; b = -4; c = 1;
%t A200100 f[x_] := a*x^2 + b*Cos[x]; g[x_] := c*Sin[x]
%t A200100 Plot[{f[x], g[x]}, {x, -3, 3}, {AxesOrigin -> {0, 0}}]
%t A200100 r = x /. FindRoot[f[x] == g[x], {x, -1.06, -1.05}, WorkingPrecision -> 110]
%t A200100 RealDigits[r]  (* A200099 *)
%t A200100 r = x /. FindRoot[f[x] == g[x], {x, 1.35, 1.36}, WorkingPrecision -> 110]
%t A200100 RealDigits[r]  (* A200100 *)
%o A200100 (PARI) a=1; b=-4; c=1; solve(x=1, 2, a*x^2 + b*cos(x) - c*sin(x)) \\ _G. C. Greubel_, Jun 24 2018
%Y A200100 Cf. A199949.
%K A200100 nonn,cons
%O A200100 1,2
%A A200100 _Clark Kimberling_, Nov 13 2011