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A200007 Decimal expansion of greatest x satisfying 3*x^2 + cos(x) = 4*sin(x).

%I A200007 #14 Feb 12 2025 04:50:17
%S A200007 9,3,5,7,8,1,9,5,4,5,6,0,2,0,1,6,9,0,6,4,7,6,9,0,3,5,6,7,4,8,3,5,0,6,
%T A200007 5,5,1,8,6,4,5,4,2,4,0,5,5,1,0,1,8,8,3,7,9,3,0,4,1,7,2,9,1,9,1,7,6,6,
%U A200007 5,6,9,6,0,6,1,3,3,0,9,3,0,3,4,5,2,2,9,7,6,4,4,8,2,1,4,6,2,6,7
%N A200007 Decimal expansion of greatest x satisfying 3*x^2 + cos(x) = 4*sin(x).
%C A200007 See A199949 for a guide to related sequences. The Mathematica program includes a graph.
%H A200007 G. C. Greubel, <a href="/A200007/b200007.txt">Table of n, a(n) for n = 0..10000</a>
%H A200007 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%e A200007 least x:  0.31916558449395611450944828046123878...
%e A200007 greatest x: 0.935781954560201690647690356748350...
%t A200007 a = 3; b = 1; c = 4;
%t A200007 f[x_] := a*x^2 + b*Cos[x]; g[x_] := c*Sin[x]
%t A200007 Plot[{f[x], g[x]}, {x, -1, 2}, {AxesOrigin -> {0, 0}}]
%t A200007 r = x /. FindRoot[f[x] == g[x], {x, .31, .32}, WorkingPrecision -> 110]
%t A200007 RealDigits[r]  (* A200006 *)
%t A200007 r = x /. FindRoot[f[x] == g[x], {x, .93, .94}, WorkingPrecision -> 110]
%t A200007 RealDigits[r]  (* A200007 *)
%o A200007 (PARI) a=3; b=1; c=4; solve(x=.5, 1, a*x^2 + b*cos(x) - c*sin(x)) \\ _G. C. Greubel_, Jun 23 2018
%Y A200007 Cf. A199949.
%K A200007 nonn,cons
%O A200007 0,1
%A A200007 _Clark Kimberling_, Nov 12 2011
%E A200007 a(89)-a(98) corrected by _G. C. Greubel_, Jun 23 2018