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A200018 Decimal expansion of least x satisfying x^2 - 2*cos(x) = sin(x) (negated).

%I A200018 #14 Feb 12 2025 04:52:43
%S A200018 8,0,9,6,2,9,9,9,9,1,2,9,5,5,2,4,1,3,1,8,6,1,0,9,6,9,8,4,8,4,0,2,7,1,
%T A200018 3,2,1,9,6,6,9,1,7,6,0,3,5,6,5,4,5,3,7,8,5,3,8,9,4,7,8,4,6,3,2,9,8,3,
%U A200018 6,9,0,8,2,3,2,1,6,6,6,8,3,9,6,9,0,1,7,5,9,9,6,0,8,6,5,1,2,5,2
%N A200018 Decimal expansion of least x satisfying x^2 - 2*cos(x) = sin(x) (negated).
%C A200018 See A199949 for a guide to related sequences. The Mathematica program includes a graph.
%H A200018 G. C. Greubel, <a href="/A200018/b200018.txt">Table of n, a(n) for n = 0..10000</a>
%H A200018 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%e A200018 least x:  -0.8096299991295524131861096984840271321...
%e A200018 greatest x: 1.254187962477919553363912326321801374...
%t A200018 a = 1; b = -2; c = 1;
%t A200018 f[x_] := a*x^2 + b*Cos[x]; g[x_] := c*Sin[x]
%t A200018 Plot[{f[x], g[x]}, {x, -1, 2}, {AxesOrigin -> {0, 0}}]
%t A200018 r = x /. FindRoot[f[x] == g[x], {x, -.9, -.8}, WorkingPrecision -> 110]
%t A200018 RealDigits[r]  (* A200018 *)
%t A200018 r = x /. FindRoot[f[x] == g[x], {x, 1.2, 1.3}, WorkingPrecision -> 110]
%t A200018 RealDigits[r]  (* A200019 *)
%o A200018 (PARI) a=1; b=-2; c=1; solve(x=-1, 0, a*x^2 + b*cos(x) - c*sin(x)) \\ _G. C. Greubel_, Jun 24 2018
%Y A200018 Cf. A199949.
%K A200018 nonn,cons
%O A200018 0,1
%A A200018 _Clark Kimberling_, Nov 12 2011