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A200019 Decimal expansion of greatest x satisfying x^2 - 2*cos(x) = sin(x).

%I A200019 #12 Feb 12 2025 04:52:51
%S A200019 1,2,5,4,1,8,7,9,6,2,4,7,7,9,1,9,5,5,3,3,6,3,9,1,2,3,2,6,3,2,1,8,0,1,
%T A200019 3,7,4,2,6,0,6,2,4,1,3,8,6,9,8,6,9,2,8,5,1,5,7,7,7,5,1,6,0,3,0,8,2,5,
%U A200019 6,2,2,5,8,3,8,4,0,8,2,4,8,4,9,2,4,2,6,4,5,9,4,9,4,6,1,3,9,5,5
%N A200019 Decimal expansion of greatest x satisfying x^2 - 2*cos(x) = sin(x).
%C A200019 See A199949 for a guide to related sequences. The Mathematica program includes a graph.
%H A200019 G. C. Greubel, <a href="/A200019/b200019.txt">Table of n, a(n) for n = 1..10000</a>
%H A200019 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%e A200019 least x:  -0.8096299991295524131861096984840271321...
%e A200019 greatest x: 1.254187962477919553363912326321801374...
%t A200019 a = 1; b = -2; c = 1;
%t A200019 f[x_] := a*x^2 + b*Cos[x]; g[x_] := c*Sin[x]
%t A200019 Plot[{f[x], g[x]}, {x, -1, 2}, {AxesOrigin -> {0, 0}}]
%t A200019 r = x /. FindRoot[f[x] == g[x], {x, -.9, -.8}, WorkingPrecision -> 110]
%t A200019 RealDigits[r]  (* A200018 *)
%t A200019 r = x /. FindRoot[f[x] == g[x], {x, 1.2, 1.3}, WorkingPrecision -> 110]
%t A200019 RealDigits[r]  (* A200019 *)
%o A200019 (PARI) a=1; b=-2; c=1; solve(x=1, 2, a*x^2 + b*cos(x) - c*sin(x)) \\ _G. C. Greubel_, Jun 24 2018
%Y A200019 Cf. A199949.
%K A200019 nonn,cons
%O A200019 1,2
%A A200019 _Clark Kimberling_, Nov 12 2011