A196611 Decimal expansion of the slope (negative) of the tangent line at the point of tangency of the curves y=c*cos(x) and y=1/x, where c is given by A196610.
1, 3, 5, 1, 0, 3, 3, 8, 8, 6, 8, 7, 8, 3, 7, 8, 6, 2, 4, 0, 0, 9, 1, 9, 2, 4, 7, 3, 5, 2, 8, 4, 3, 0, 2, 1, 7, 4, 8, 3, 4, 3, 7, 8, 0, 5, 9, 6, 3, 4, 7, 8, 1, 5, 9, 2, 3, 0, 1, 4, 5, 2, 3, 3, 6, 5, 4, 5, 9, 5, 8, 9, 8, 3, 5, 7, 6, 8, 7, 7, 2, 4, 9, 2, 4, 5, 3, 5, 7, 8, 7, 6, 5, 3, 0, 2, 9, 4, 9, 4
Offset: 1
A196603 Decimal expansion of the least x>0 satisfying sec(x)=2x.
6, 1, 0, 0, 3, 1, 2, 8, 4, 4, 6, 4, 1, 7, 5, 9, 7, 5, 3, 7, 0, 9, 6, 3, 0, 7, 3, 5, 1, 3, 4, 1, 0, 3, 2, 4, 6, 7, 3, 7, 2, 0, 9, 7, 9, 1, 1, 2, 1, 6, 9, 2, 3, 7, 8, 6, 3, 7, 5, 1, 6, 0, 7, 5, 3, 2, 8, 0, 9, 4, 8, 8, 6, 1, 0, 5, 1, 0, 6, 8, 8, 7, 8, 1, 4, 2, 4, 4, 1, 6, 0, 3, 4, 4, 4, 4, 1, 2, 4, 4
Offset: 0
Examples
0.61003128446417597537096307351341032...
Crossrefs
Cf. A196610.
Programs
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Mathematica
Plot[{1/x, Cos[x], 2 Cos[x], 3 Cos[x], 4 Cos[x]}, {x, 0, 2 Pi}] t = x /. FindRoot[1/x == Cos[x], {x, .1, 5}, WorkingPrecision -> 100] RealDigits[t] (* A133868 *) t = x /. FindRoot[1/x == 2 Cos[x], {x, .5, .7}, WorkingPrecision -> 100] RealDigits[t] (* A196603 *) t = x /. FindRoot[1/x == 3 Cos[x], {x, .3, .4}, WorkingPrecision -> 100] RealDigits[t] (* A196604 *) t = x /. FindRoot[1/x == 4 Cos[x], {x, .1, .3}, WorkingPrecision -> 100] RealDigits[t] (* A196605 *) t = x /. FindRoot[1/x == 5 Cos[x], {x, .15, .23}, WorkingPrecision -> 100] RealDigits[t] (* A196606 *) t = x /. FindRoot[1/x == 6 Cos[x], {x, .1, .2}, WorkingPrecision -> 100] RealDigits[t] (* A196607 *)
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
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...
Programs
Extensions
a(99) corrected by Georg Fischer, Jul 19 2021
Comments
Examples
Crossrefs
Programs
Mathematica
Plot[{1/x, (1.78222) Cos[x]}, {x, .7, 1}] xt = x /. FindRoot[x == Cot[x], {x, .8, 1}, WorkingPrecision -> 100] c = N[Csc[xt]/xt^2, 100] RealDigits[c] (* A196610 *) slope = -c*Sin[xt] RealDigits[slope] (* A196611 *)