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A200013 Decimal expansion of greatest x satisfying x^2 - cos(x) = 2*sin(x).

%I A200013 #14 Feb 12 2025 04:51:45
%S A200013 1,4,5,0,9,3,8,4,4,9,6,3,4,9,7,4,4,3,1,1,2,8,2,8,5,5,7,6,6,9,0,3,5,7,
%T A200013 7,3,8,9,4,4,7,4,8,7,0,1,1,5,3,4,6,3,9,8,7,6,5,4,2,3,5,8,6,2,6,2,9,6,
%U A200013 1,9,2,8,4,3,0,8,1,3,5,0,5,9,1,7,2,0,8,4,3,0,4,9,7,0,9,4,6,6,0
%N A200013 Decimal expansion of greatest x satisfying x^2 - cos(x) = 2*sin(x).
%C A200013 See A199949 for a guide to related sequences. The Mathematica program includes a graph.
%H A200013 G. C. Greubel, <a href="/A200013/b200013.txt">Table of n, a(n) for n = 1..10000</a>
%H A200013 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%e A200013 least x:  -0.3941241928589759600997053993545900...
%e A200013 greatest x: 1.450938449634974431128285576690357...
%t A200013 a = 1; b = -1; c = 2;
%t A200013 f[x_] := a*x^2 + b*Cos[x]; g[x_] := c*Sin[x]
%t A200013 Plot[{f[x], g[x]}, {x, -1, 2}, {AxesOrigin -> {0, 0}}]
%t A200013 r = x /. FindRoot[f[x] == g[x], {x, -.4, -.3}, WorkingPrecision -> 110]
%t A200013 RealDigits[r]   (* A200012 *)
%t A200013 r = x /. FindRoot[f[x] == g[x], {x, 1.4, 1.5}, WorkingPrecision -> 110]
%t A200013 RealDigits[r]   (* A200013 *)
%o A200013 (PARI) a=1; b=-1; c=2; solve(x=1, 2, a*x^2 + b*cos(x) - c*sin(x)) \\ _G. C. Greubel_, Jun 23 2018
%Y A200013 Cf. A199949.
%K A200013 nonn,cons
%O A200013 1,2
%A A200013 _Clark Kimberling_, Nov 12 2011