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A200022 Decimal expansion of least x satisfying x^2 - 2*cos(x) = 3*sin(x) (negated).

%I A200022 #14 Feb 12 2025 04:58:35
%S A200022 5,1,4,5,1,4,8,3,0,4,7,6,4,5,8,6,8,6,5,6,5,6,3,8,9,4,5,6,7,5,3,7,1,8,
%T A200022 1,5,9,5,2,1,1,1,9,8,3,6,7,0,3,4,0,2,2,7,8,2,0,3,9,7,8,3,7,7,5,8,1,9,
%U A200022 3,2,4,4,5,9,9,6,6,4,1,8,7,7,8,9,1,5,6,2,9,3,8,0,2,0,5,0,3,4,2
%N A200022 Decimal expansion of least x satisfying x^2 - 2*cos(x) = 3*sin(x) (negated).
%C A200022 See A199949 for a guide to related sequences. The Mathematica program includes a graph.
%H A200022 G. C. Greubel, <a href="/A200022/b200022.txt">Table of n, a(n) for n = 0..10000</a>
%H A200022 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%e A200022 least x:  -0.5145148304764586865656389456753718159521...
%e A200022 greatest x: 1.669692169649763458252838305984917335937...
%t A200022 a = 1; b = -2; c = 3;
%t A200022 f[x_] := a*x^2 + b*Cos[x]; g[x_] := c*Sin[x]
%t A200022 Plot[{f[x], g[x]}, {x, -1, 2}, {AxesOrigin -> {0, 0}}]
%t A200022 r = x /. FindRoot[f[x] == g[x], {x, -.52, -.51}, WorkingPrecision -> 110]
%t A200022 RealDigits[r]  (* A200022 *)
%t A200022 r = x /. FindRoot[f[x] == g[x], {x, 1.66, 1.67}, WorkingPrecision -> 110]
%t A200022 RealDigits[r]  (* A200023 *)
%o A200022 (PARI) a=1; b=-2; c=3; solve(x=-1, 0, a*x^2 + b*cos(x) - c*sin(x)) \\ _G. C. Greubel_, Jun 24 2018
%Y A200022 Cf. A199949.
%K A200022 nonn,cons
%O A200022 0,1
%A A200022 _Clark Kimberling_, Nov 13 2011