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Decimal expansion of the number c for which the curve y=1/(1+x^2) is tangent to the curve y=c*cos(x), and 0 < x < 2*Pi.
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%I #8 Apr 09 2021 22:46:33

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

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

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

%N Decimal expansion of the number c for which the curve y=1/(1+x^2) is tangent to the curve y=c*cos(x), and 0 < x < 2*Pi.

%F x=0.87446003662400926554814644840673725663073...

%e c=0.87446003662400926554814644840673725663073...

%t Plot[{1/(1 + x^2), 0.874*Cos[x]}, {x, .5, 1}]

%t t = x /. FindRoot[Tan[x] == 2 x/(1 + x^2), {x, .5, 1}, WorkingPrecision -> 100]

%t RealDigits[t] (* A196913 *)

%t c = N[Sqrt[t^4 + 6 t^2 + 1]/(t^4 + 2 t^2 + 1), 100]

%t RealDigits[c] (* A196914 *)

%t slope = N[-c*Sin[t], 100]

%t RealDigits[slope](* A196915 *)

%Y Cf. A196913, A196915.

%K nonn,cons

%O 0,1

%A _Clark Kimberling_, Oct 07 2011