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A200104 Decimal expansion of greatest x satisfying x^2 - 4*cos(x) = 3*sin(x).

%I A200104 #11 Feb 12 2025 09:54:29
%S A200104 1,6,4,3,5,5,6,5,6,7,5,2,0,1,7,1,6,5,6,9,0,6,5,2,4,7,6,1,6,3,4,8,8,8,
%T A200104 1,6,9,4,6,0,7,4,9,2,9,7,7,5,1,3,9,1,4,8,5,7,2,0,1,3,8,1,5,2,2,6,2,9,
%U A200104 4,3,3,5,9,0,4,2,1,6,1,2,9,4,5,5,9,5,8,9,8,4,0,0,3,6,4,1,3,8,4
%N A200104 Decimal expansion of greatest x satisfying x^2 - 4*cos(x) = 3*sin(x).
%C A200104 See A199949 for a guide to related sequences. The Mathematica program includes a graph.
%H A200104 G. C. Greubel, <a href="/A200104/b200104.txt">Table of n, a(n) for n = 1..10000</a>
%H A200104 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%e A200104 least x:  -0.79920081689509700594446006923211010...
%e A200104 greatest x: 1.643556567520171656906524761634888...
%t A200104 a = 1; b = -4; c = 3;
%t A200104 f[x_] := a*x^2 + b*Cos[x]; g[x_] := c*Sin[x]
%t A200104 Plot[{f[x], g[x]}, {x, -3, 3}, {AxesOrigin -> {0, 0}}]
%t A200104 r = x /. FindRoot[f[x] == g[x], {x, -.8, -.7}, WorkingPrecision -> 110]
%t A200104 RealDigits[r]  (* A200103 *)
%t A200104 r = x /. FindRoot[f[x] == g[x], {x, 1.64, 1.65}, WorkingPrecision -> 110]
%t A200104 RealDigits[r]  (* A200104 *)
%o A200104 (PARI) a=1; b=-4; c=3; solve(x=1, 2, a*x^2 + b*cos(x) - c*sin(x)) \\ _G. C. Greubel_, Jun 25 2018
%Y A200104 Cf. A199949.
%K A200104 nonn,cons
%O A200104 1,2
%A A200104 _Clark Kimberling_, Nov 13 2011