cp's OEIS Frontend

A197020 Decimal expansion of the radius of the circle tangent to the curve y=cos(2x) at points (x,y) and (-x,y), where 0

%I A197020 #7 Feb 04 2025 15:04:53
%S A197020 4,6,1,9,2,3,1,8,7,7,0,5,2,2,8,2,3,8,2,1,7,1,5,3,0,3,3,3,6,9,3,8,9,9,
%T A197020 9,9,6,2,0,4,3,4,7,2,6,7,0,5,6,8,8,6,5,7,9,7,6,7,0,6,4,3,0,3,7,9,5,1,
%U A197020 1,3,9,4,1,1,5,2,5,1,1,1,6,7,9,9,7,9,8,7,4,3,0,9,9,4,4,1,5,0,3,6
%N A197020 Decimal expansion of the radius of the circle tangent to the curve y=cos(2x) at points (x,y) and (-x,y), where 0<x<1.
%C A197020 Let (x,y) denote the point of tangency.  Then
%C A197020 x=0.371462711321448956555130330602759317162856415...
%C A197020 y=0.736492824477900896049098419167188850255855384...
%C A197020 slope=-1.3528907117613955482765053348775509428929...
%C A197020 (The Mathematica program includes a graph.)
%e A197020 0.461923187705228238217153033369389999620434726705688657976706430379511394115251....
%t A197020 r = .462; c = 2;
%t A197020 Show[Plot[Cos[c*x], {x, -2, 2}],
%t A197020  ContourPlot[x^2 + (y - r)^2 == r^2, {x, -1, 1}, {y, -1, 1}], PlotRange -> All, AspectRatio -> Automatic]
%t A197020 t = x /. FindRoot[c*Sin[c*x] Cos[c*x] - x ==
%t A197020     x*Sqrt[1 + (c*Sin[c*x])^2], {x, .25, .55}, WorkingPrecision -> 100]
%t A197020 RealDigits[t] (* x coordinate of tangency point *)
%t A197020 y = Cos[c*t]  (* y coordinate of tangency point *)
%t A197020 radius = Cos[c*t] - t/(c*Sin[c*t]) (* A197020 *)
%t A197020 RealDigits[radius]
%t A197020 slope = -c*Sin[c*t] (* slope at tangency point *)
%o A197020 (PARI) t=solve(x=.3,.4, 2*sin(2*x)*cos(2*x) - x*sqrt(1 + 4*sin(2*x)^2) - x); cos(2*t) - t/(2*sin(2*t)) \\ _Charles R Greathouse IV_, Feb 04 2025
%Y A197020 Cf. A197021, A196022, A196026, A197027, A197016.
%K A197020 nonn,cons
%O A197020 0,1
%A A197020 _Clark Kimberling_, Oct 08 2011