A196625 Decimal expansion of the number c for which the curve y=1/x is tangent to the curve y=cos(x-c), and 0 < x < 2*Pi; c = sqrt(r) - arccsc(r), where r = (1+sqrt(5))/2 (the golden ratio).

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

Offset: 0

Author

Clark Kimberling, Oct 05 2011

Keywords

• nonn
• cons

Comments

Let r=(1+sqrt(5))/2, the golden ratio. Let u=sqrt(r) and v=1/x. Let c=sqrt(r)-arccsc(r). The curve y=1/x is tangent to the curve y=cos(x-c) at (u,v), and the slope of the tangent line is r-1.

Guide to constants c associated with tangencies:

A196610: 1/x and c*cos(x)

A196619: 1/x - c and cos(x)

A196774: 1/x + c and sin(x)

A196625: 1/x and cos(c-x)

A196772: 1/x and sin(x+c)

A196758: 1/x and c*sin(x)

A196765: c/x and sin(x)

A196823: 1/(1+x^2) and -c+cos(x)

A196914: 1/(1+x^2) and c*cos(x)

A196832: 1/(1+x^2) and c*sin(x)

A197016: x=0, y=0, and cos(x)

Examples

			c=0.60578021702155370914841756575969877104...
		

Crossrefs

Cf. A139339, A196772.

Programs

Mathematica

Plot[{1/x, Cos[x - 0.60578]}, {x, 0, 2 Pi}]
r = GoldenRatio; xt = Sqrt[r];
x1 = N[xt, 100]
RealDigits[x1]     (* A139339 *)
c = Sqrt[r] - ArcCsc[r];
c1 = N[c, 100]
RealDigits[c1]     (* A196625 *)
slope = N[r - Sqrt[5], 100]
RealDigits[slope]  (* -1+A001622; -1+golden ratio *)

Extensions

a(99) corrected by Georg Fischer, Jul 19 2021