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

%I #15 Feb 12 2025 13:05:44

%S 6,3,7,6,6,1,1,5,7,9,4,6,0,7,3,1,3,4,1,1,9,8,9,5,4,5,6,5,8,8,1,9,6,2,

%T 0,1,3,7,3,3,9,9,2,2,8,0,7,2,7,3,3,8,6,9,5,5,6,1,0,6,2,3,0,9,0,1,0,1,

%U 4,0,5,4,5,4,4,3,3,0,7,9,6,8,0,0,1,4,3,8,4,6,5,1,0,7,1,4,4,7,8

%N Decimal expansion of least x satisfying 3*x^2-3*cos(x)=2*sin(x).

%C See A199949 for a guide to related sequences. The Mathematica program includes a graph.

%H G. C. Greubel, <a href="/A200239/b200239.txt">Table of n, a(n) for n = 0..10000</a>

%H <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.

%e least x: -0.63766115794607313411989545658819620...

%e greatest x: 1.039829693324607596071793532120387...

%t a = 3; b = -3; c = 2;

%t f[x_] := a*x^2 + b*Cos[x]; g[x_] := c*Sin[x]

%t Plot[{f[x], g[x]}, {x, -2, 2}, {AxesOrigin -> {0, 0}}]

%t r = x /. FindRoot[f[x] == g[x], {x, -.64, -.63}, WorkingPrecision -> 110]

%t RealDigits[r] (* A200239 *)

%t r = x /. FindRoot[f[x] == g[x], {x, 1.0, 1.1}, WorkingPrecision -> 110]

%t RealDigits[r] (* A200240 *)

%o (PARI) a=3; b=-3; c=2; solve(x=-.64, -.63, a*x^2 + b*cos(x) - c*sin(x)) \\ _G. C. Greubel_, Jun 22 2018

%Y Cf. A199949.

%K nonn,cons,changed

%O 0,1

%A _Clark Kimberling_, Nov 15 2011