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A200306 Decimal expansion of greatest x satisfying 4*x^2 - 4*cos(x) = sin(x).

%I A200306 #12 Feb 12 2025 14:37:17
%S A200306 9,0,3,0,5,9,6,1,3,9,0,4,8,6,4,2,5,0,9,1,0,2,2,7,4,6,4,5,5,2,6,1,6,5,
%T A200306 5,1,3,8,3,0,6,6,7,8,4,2,8,2,3,3,3,3,9,2,0,0,9,9,1,7,6,5,2,9,4,7,4,5,
%U A200306 4,3,7,5,5,2,1,0,1,4,2,6,7,5,1,7,6,7,4,5,5,7,4,9,6,8,2,7,4,7,0
%N A200306 Decimal expansion of greatest x satisfying 4*x^2 - 4*cos(x) = sin(x).
%C A200306 See A199949 for a guide to related sequences. The Mathematica program includes a graph.
%H A200306 G. C. Greubel, <a href="/A200306/b200306.txt">Table of n, a(n) for n = 0..10000</a>
%H A200306 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%e A200306 least x: -0.749483556259155223568363487735793...
%e A200306 greatest x: 0.9030596139048642509102274645526...
%t A200306 a = 4; b = -4; c = 1;
%t A200306 f[x_] := a*x^2 + b*Cos[x]; g[x_] := c*Sin[x]
%t A200306 Plot[{f[x], g[x]}, {x, -2, 2}, {AxesOrigin -> {0, 0}}]
%t A200306 r = x /. FindRoot[f[x] == g[x], {x, -.75, -.74}, WorkingPrecision -> 110]
%t A200306 RealDigits[r]   (* A200305 *)
%t A200306 r = x /. FindRoot[f[x] == g[x], {x, .90, .91}, WorkingPrecision -> 110]
%t A200306 RealDigits[r]    (* A200306 *)
%o A200306 (PARI) a=4; b=-4; c=1; solve(x=0, 1, a*x^2 + b*cos(x) - c*sin(x)) \\ _G. C. Greubel_, Jul 08 2018
%Y A200306 Cf. A199949.
%K A200306 nonn,cons
%O A200306 0,1
%A A200306 _Clark Kimberling_, Nov 16 2011