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A200020 Decimal expansion of least x satisfying x^2 - 2*cos(x) = 2*sin(x) (negated).

%I A200020 #14 Feb 12 2025 04:58:38
%S A200020 6,4,0,0,4,9,1,9,1,1,4,2,5,7,7,1,1,5,7,3,9,8,3,5,2,6,9,6,7,5,8,4,1,2,
%T A200020 0,1,2,9,6,6,3,4,6,8,4,7,7,3,3,5,0,9,7,1,1,0,6,6,7,1,3,8,2,1,5,6,7,3,
%U A200020 9,6,0,7,9,0,4,4,6,2,3,0,8,5,8,3,7,5,0,9,3,3,0,9,7,6,3,5,8,9,8
%N A200020 Decimal expansion of least x satisfying x^2 - 2*cos(x) = 2*sin(x) (negated).
%C A200020 See A199949 for a guide to related sequences. The Mathematica program includes a graph.
%H A200020 G. C. Greubel, <a href="/A200020/b200020.txt">Table of n, a(n) for n = 0..10000</a>
%H A200020 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%e A200020 least x:  -0.64004919114257711573983526967584120...
%e A200020 greatest x: 1.476368748380920391671696889789836416...
%t A200020 a = 1; b = -2; c = 2;
%t A200020 f[x_] := a*x^2 + b*Cos[x]; g[x_] := c*Sin[x]
%t A200020 Plot[{f[x], g[x]}, {x, -1, 2}, {AxesOrigin -> {0, 0}}]
%t A200020 r = x /. FindRoot[f[x] == g[x], {x, -.65, -.64}, WorkingPrecision -> 110]
%t A200020 RealDigits[r]  (* A200020 *)
%t A200020 r = x /. FindRoot[f[x] == g[x], {x, 1.47, 1.48}, WorkingPrecision -> 110]
%t A200020 RealDigits[r]  (* A200021 *)
%o A200020 (PARI) a=1; b=-2; c=2; solve(x=-1, 0, a*x^2 + b*cos(x) - c*sin(x)) \\ _G. C. Greubel_, Jun 24 2018
%Y A200020 Cf. A199949.
%K A200020 nonn,cons
%O A200020 0,1
%A A200020 _Clark Kimberling_, Nov 12 2011