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

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, 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, 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

Offset: 0

Author

Clark Kimberling, Oct 08 2011

Keywords

• nonn
• cons

Comments

Let (x,y) denote the point of tangency. Then

x=0.371462711321448956555130330602759317162856415...

y=0.736492824477900896049098419167188850255855384...

slope=-1.3528907117613955482765053348775509428929...

(The Mathematica program includes a graph.)

Examples

			0.461923187705228238217153033369389999620434726705688657976706430379511394115251....
		

Crossrefs

Cf. A197021, A196022, A196026, A197027, A197016.

Programs

Mathematica

r = .462; c = 2;
Show[Plot[Cos[c*x], {x, -2, 2}],
 ContourPlot[x^2 + (y - r)^2 == r^2, {x, -1, 1}, {y, -1, 1}], PlotRange -> All, AspectRatio -> Automatic]
t = x /. FindRoot[c*Sin[c*x] Cos[c*x] - x ==
    x*Sqrt[1 + (c*Sin[c*x])^2], {x, .25, .55}, WorkingPrecision -> 100]
RealDigits[t] (* x coordinate of tangency point *)
y = Cos[c*t]  (* y coordinate of tangency point *)
radius = Cos[c*t] - t/(c*Sin[c*t]) (* A197020 *)
RealDigits[radius]
slope = -c*Sin[c*t] (* slope at tangency point *)

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