A196774 -id:A196774 - cp's OEIS frontend

A196775 Decimal expansion of the slope (negative) at the point of tangency of the curves y=c+1/x and y=sin(x), where c is given by A196774.

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

Offset: 0

Clark Kimberling, Oct 06 2011

			x=-0.28810657283129896727398959945083925345500...

Cf. A196774.

Plot[{1/x + .42, Sin[x]}, {x, 0, 2 Pi}]
t = x /. FindRoot[-1 == (x^2) Cos[x], {x, 1.5, 2.5}, WorkingPrecision -> 100]
RealDigits[t]    (* A196773 *)
c = N[-1/t + Sin[t], 100]
RealDigits[c]    (* A196774 *)
slope = N[-1/t^2, 100]
RealDigits[slope](* A196775 *)

A196824 Decimal expansion of the slope (negative) at the point of tangency of the curves y=1/(1+x^2) and y=-c+cos(x), where c is given by A196774.

Original entry on oeis.org

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

Offset: 0

Clark Kimberling, Oct 06 2011

			slope=-0.6341649706958779561027498118640238055822484...

Cf. A196823.

Plot[{1/(1 + x^2), -.094 + Cos[x]}, {x, 0, 1}]
t = x /. FindRoot[2 x == ((1 + x^2)^2) Sin[x], {x, .5, 1}, WorkingPrecision -> 100]
RealDigits[t]     (* A196822 *)
c = N[-Cos[t] + 1/(1 + t^2), 100]
RealDigits[c]     (* A196823 *)
slope = N[-Sin[t], 100]
RealDigits[slope] (* A196824 *)

A196773 Decimal expansion of the least number x>0 satisfying -1=(x^2)*cos(x).

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Oct 06 2011

			x=1.863045308248699224069793204693635418...

Cf. A196774.

Plot[{1/x + .42, Sin[x]}, {x, 0, 2 Pi}]
t = x /. FindRoot[-1 == (x^2) Cos[x], {x, 1.5, 2.5}, WorkingPrecision -> 100]
RealDigits[t]    (* A196773 *)
c = N[-1/t + Sin[t], 100]
RealDigits[c]    (* A196774 *)
slope = N[-1/t^2, 100]
RealDigits[slope](* A196775 *)
RealDigits[x/.FindRoot[x^2 Cos[x]+1,{x,1/4},WorkingPrecision->120],10,120][[1]] (* Harvey P. Dale, Dec 14 2023 *)

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).

Original entry on oeis.org

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

Clark Kimberling, Oct 05 2011

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)

			c=0.60578021702155370914841756575969877104...

Cf. A139339, A196772.

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 *)

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

cp's OEIS Frontend

A196775 Decimal expansion of the slope (negative) at the point of tangency of the curves y=c+1/x and y=sin(x), where c is given by A196774.

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A196824 Decimal expansion of the slope (negative) at the point of tangency of the curves y=1/(1+x^2) and y=-c+cos(x), where c is given by A196774.

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A196773 Decimal expansion of the least number x>0 satisfying -1=(x^2)*cos(x).

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Mathematica

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).

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