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A200118 Decimal expansion of least x satisfying 2*x^2 - 2*cos(x) = 3*sin(x), negated.

%I A200118 #11 Feb 12 2025 11:31:30
%S A200118 4,6,6,8,2,3,6,0,7,5,7,0,9,8,6,7,9,9,5,8,4,1,3,4,1,5,4,4,3,1,5,8,4,0,
%T A200118 4,7,4,2,6,6,6,6,7,3,0,0,8,1,8,1,8,7,7,3,4,2,9,0,2,0,5,1,2,5,7,8,4,0,
%U A200118 2,8,8,6,8,6,8,7,4,3,9,5,5,4,5,2,5,8,6,5,8,5,4,5,5,4,8,1,6
%N A200118 Decimal expansion of least x satisfying 2*x^2 - 2*cos(x) = 3*sin(x), negated.
%C A200118 See A199949 for a guide to related sequences. The Mathematica program includes a graph.
%H A200118 G. C. Greubel, <a href="/A200118/b200118.txt">Table of n, a(n) for n = 0..10000</a>
%H A200118 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%e A200118 least x: -0.46682360757098679958413415443158404...
%e A200118 greatest x: 1.3071909920738130664046341866545604...
%t A200118 a = 2; b = -2; c = 3;
%t A200118 f[x_] := a*x^2 + b*Cos[x]; g[x_] := c*Sin[x]
%t A200118 Plot[{f[x], g[x]}, {x, -3, 3}, {AxesOrigin -> {0, 0}}]
%t A200118 r = x /. FindRoot[f[x] == g[x], {x, -.47, -.48}, WorkingPrecision -> 110]
%t A200118 RealDigits[r]  (* A200118 *)
%t A200118 r = x /. FindRoot[f[x] == g[x], {x, 1.3, 1.31}, WorkingPrecision -> 110]
%t A200118 RealDigits[r]  (* A200119 *)
%o A200118 (PARI) a=2; b=-2; c=3; solve(x=-1, 0, a*x^2 + b*cos(x) - c*sin(x)) \\ _G. C. Greubel_, Jun 29 2018
%Y A200118 Cf. A199949.
%K A200118 nonn,cons
%O A200118 0,1
%A A200118 _Clark Kimberling_, Nov 14 2011
%E A200118 Terms a(89) to a(96) corrected by _G. C. Greubel_, Jun 29 2018