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A200224 Decimal expansion of greatest x satisfying 3*x^2 - cos(x) = 2*sin(x).

%I A200224 #11 Feb 12 2025 13:03:07
%S A200224 8,4,8,9,0,5,1,8,8,3,2,9,5,2,2,3,6,1,7,3,4,5,6,3,8,1,6,2,6,6,1,3,2,6,
%T A200224 7,6,8,3,4,5,9,5,3,3,8,7,9,9,7,8,6,6,5,2,5,8,0,4,1,9,1,3,6,6,3,5,7,5,
%U A200224 7,3,3,0,4,0,5,9,1,3,7,2,4,9,5,3,8,3,2,3,5,4,5,6,3,8,6,2,5,8,6
%N A200224 Decimal expansion of greatest x satisfying 3*x^2 - cos(x) = 2*sin(x).
%C A200224 See A199949 for a guide to related sequences. The Mathematica program includes a graph.
%H A200224 G. C. Greubel, <a href="/A200224/b200224.txt">Table of n, a(n) for n = 0..10000</a>
%H A200224 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%e A200224 least x: -0.3231143338862262845356370492311714...
%e A200224 greatest x: 0.84890518832952236173456381626613...
%t A200224 a = 3; b = -1; c = 2;
%t A200224 f[x_] := a*x^2 + b*Cos[x]; g[x_] := c*Sin[x]
%t A200224 Plot[{f[x], g[x]}, {x, -3, 3}, {AxesOrigin -> {0, 0}}]
%t A200224 r = x /. FindRoot[f[x] == g[x], {x, -.33, -.32}, WorkingPrecision -> 110]
%t A200224 RealDigits[r]   (* A200223 *)
%t A200224 r = x /. FindRoot[f[x] == g[x], {x, .84, .85}, WorkingPrecision -> 110]
%t A200224 RealDigits[r]   (* A200224 *)
%o A200224 (PARI) a=3; b=-1; c=2; solve(x=0, 1, a*x^2 + b*cos(x) - c*sin(x)) \\ _G. C. Greubel_, Jun 30 2018
%Y A200224 Cf. A199949.
%K A200224 nonn,cons
%O A200224 0,1
%A A200224 _Clark Kimberling_, Nov 14 2011