A197002 -id:A197002 - cp's OEIS frontend

A197003 Decimal expansion of the slope of the line y=mx which meets the curve y=cos(x+Pi/4) orthogonally over the interval [0, 2*Pi] (as in A197002).

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

Offset: 1

Clark Kimberling, Oct 09 2011

See the Mathematica program for a graph.
xo=0.3695425666075803208276560438369...
yo=0.4039727532995172093189617400663...
m=1.09316974498501692208815321416057...
|OP|=0.54749949218543621432520415035...

Cf. A003957, A197002, A196996, A197000.

c = Pi/4;
xo = x /. FindRoot[x == Sin[x + c] Cos[x + c], {x, .8, 1.2}, WorkingPrecision -> 100]
RealDigits[xo] (* A197002 *)
m = 1/Sin[xo + c]
RealDigits[m]  (* A197003 *)
yo = m*xo
d = Sqrt[xo^2 + yo^2]
Show[Plot[{Cos[x + c], yo - (1/m) (x - xo)}, {x, -Pi/4, 1}], ContourPlot[{y == m*x}, {x, 0, Pi}, {y, 0, 1}], PlotRange -> All, AspectRatio -> Automatic, AxesOrigin -> Automatic]

my(d=solve(x=0,1,cos(x)-x)); sqrt(2-2*sqrt(1-d^2))/d \\ Gleb Koloskov, Jun 16 2021

Equals sqrt(2-2*sqrt(1-d^2))/d where d = A003957. - Gleb Koloskov, Jun 16 2021

A003957 The Dottie number: decimal expansion of root of cos(x) = x.

Original entry on oeis.org

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

Offset: 0

Leonid Broukhis

Let P be the point in quadrant I where the curve y=sin(x) meets the circle x^2+y^2=1. Let d be the Dottie number. Then P=(d,sin(d)), and d is the slope at P of the sine curve. - Clark Kimberling, Oct 07 2011
From Ben Branman, Dec 28 2011: (Start)
The name "Dottie" is of no fundamental mathematical significance since it refers to a particular French professor who--no doubt like many other calculator users before and after her--noticed that whenever she typed a number into her calculator and hit the cosine button repeatedly, the result always converged to this value.
The number is well-known, having appeared in numerous elementary works on algebra already by the late 1880s (e.g., Bertrand 1865, p. 285; Heis 1886, p. 468; Briot 1881, pp. 341-343), and probably much earlier as well. It is also known simply as the cosine constant, cosine superposition constant, iterated cosine constant, or cosine fixed point constant. Arakelian (1981, pp. 135-136; 1995) has used the Armenian small letter ayb (ա, the first letter in the Armenian alphabet) to denote this constant. (End)

			0.73908513321516064165531208767387340401341175890075746496568063577328...

H. Arakelian, The Fundamental Dimensionless Values (Their Role and Importance for the Methodology of Science). [In Russian.] Yerevan, Armenia: Armenian National Academy of Sciences, 1981.
A. Baker, Theorem 1.4 in Transcendental Number Theory. Cambridge, England: Cambridge University Press, 1975.

G. C. Greubel, Table of n, a(n) for n = 0..10000 (terms 0..499 from Ben Branman)
Hrant Arakelian, New Fundamental Mathematical Constant: History, Present State and Prospects, Nonlinear Science Letters B, Vol. 1, No. 4, pp. 183-193.
Mohammad K. Azarian, On the Fixed Points of a Function and the Fixed Points of its Composite Functions, International Journal of Pure and Applied Mathematics, Vol. 46, No. 1, 2008, pp. 37-44. Mathematical Reviews, MR2433713 (2009c:65129), March 2009. Zentralblatt MATH, Zbl 1160.65015.
Trefor Bazett, What is cos( cos( cos( cos( cos( cos( cos( cos( cos( cos( cos( cos(...?? // Banach Fixed Point Theorem, YouTube video, 2022.
J. Bertrand, Exercise III, in Traité d'algèbre, Vols. 1-2, 4th ed. Paris, France: Librairie Hachette et Cie, p. 285, 1865.
Steven Finch, Exercises in Iterational Asymptotics II, arXiv preprint (2025). arXiv:2501.06065 [math.NT]
Samuel R. Kaplan, The Dottie Number, Math. Magazine, 80 (No. 1, 2007), 73-74.
T. H. Miller, On the imaginary roots of cos x = x, Proc. Edinburgh Math. Soc. 21, 160-162, 1902.
V. Salov, Inevitable Dottie Number. Iterals of cosine and sine, arXiv preprint arXiv:1212.1027 [math.HO], 2012.
David R. Stoutemyer, Inverse spherical Bessel functions generalize Lambert W and solve similar equations containing trigonometric or hyperbolic subexpressions or their inverses, arXiv:2207.00707 [math.GM], 2022.
Eric Weisstein's World of Mathematics, Dottie Number
Index entries for transcendental numbers.

Cf. A009442, A177413, A182503, A197002, A200309, A212112, A212113, A217066.

Cf. A330119 (degrees-based analog).

evalf(solve(cos(x)=x,x), 140);  # Alois P. Heinz, Feb 20 2024

RealDigits[ FindRoot[ Cos[x] == x, {x, {.7, 1} }, WorkingPrecision -> 120] [[1, 2] ]] [[1]]
FindRoot[Cos[x] == x, {x, {.7, 1}}, WorkingPrecision -> 500][[1, 2]][[1]] (* Ben Branman, Apr 12 2008 *)
N[NestList[Cos, 1, 100], 20] (* Clark Kimberling, Jul 01 2019 *)
RealDigits[Root[{# - Cos[#] &, 0.739085}], 10, 100][[1]] (* Eric W. Weisstein, Jul 15 2022 *)
RealDigits[Sqrt[1 - (2 InverseBetaRegularized[1/2, 1/2, 3/2] - 1)^2], 10, 100][[1]] (* Eric W. Weisstein, Jul 15 2022 *)

solve(x=0,1,cos(x)-x) \\ Charles R Greathouse IV, Dec 31 2011

from sympy import Symbol, nsolve, cos
x = Symbol("x")
a = list(map(int, str(nsolve(cos(x)-x, 1, prec=110))[2:-2]))
print(a) # Michael S. Branicky, Jul 15 2022

Equals twice A197002. - Hugo Pfoertner, Feb 20 2024

More terms from David W. Wilson

Additional references from Ben Branman, Dec 28 2011

A196996 Decimal expansion of xo, where P=(xo,yo) is the point nearest O=(0,0) at which a line y=m*x meets the curve y=cos(3x) orthogonally.

Original entry on oeis.org

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

Offset: 0

Clark Kimberling, Oct 09 2011

See the Mathematica program for a graph.
xo=0.9350272884749678361451944275323...
yo=0.3301955980451199836007253971727...
m=0.35314006565912096755666111412785...
|OP|=0.99161744799152518925689622748...

Cf. A196997, A197000, A197002.

c = 3;
xo = x /. FindRoot[0 == x + c*Sin[c*x] Cos[c*x], {x, .8, 1.2}, WorkingPrecision -> 100]
RealDigits[xo] (* A196996 *)
m = Sin[c*xo]/xo
RealDigits[m]  (* A196997 *)
yo = m*xo
d = Sqrt[xo^2 + yo^2]
Show[Plot[{Sin[c*x], yo - (1/m) (x - xo)}, {x, 0, Pi/c}], ContourPlot[{y == m*x}, {x, 0, 1.5}, {y, -.1, 1}], PlotRange -> All, AspectRatio -> Automatic]

A197000 Decimal expansion of xo, where P=(xo,yo) is the point nearest O=(0,0) in which a line y=mx meets the curve y=1+cos(x) orthogonally.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Oct 09 2011

See the Mathematica program for a graph.

			xo=1.2488014367215508560475125020128381535587614... <------ this constant
yo=1.3164595537507515212878992732671186100622603...
m=1.05417844265684217515747734305673483746142104...
|OP|=1.81454423617045980814297669595599066552030...

Cf. A197001, A196996, A197002.

c = 1;
xo = x /.
  FindRoot[x == Sin[x] (c + Cos[x]), {x, 1, 1.3}, WorkingPrecision -> 100]
RealDigits[xo] (* A197000 *)
m = 1/Sin[xo]
RealDigits[m]  (* A197001 *)
yo = m*xo
d = Sqrt[xo^2 + yo^2]
Show[Plot[{c + Cos[c*x], yo - (1/m) (x - xo)}, {x, 0, Pi}],  ContourPlot[{y == m*x}, {x, 0, Pi}, {y, 0, 2}], PlotRange -> All, AspectRatio -> Automatic, AxesOrigin -> Automatic]

solve(x=1,2, sin(x)*(cos(x)+1)-x) \\ Charles R Greathouse IV, Feb 03 2025

A196997 Decimal expansion of m, where y=mx is the line through (0,0) which meets the curve y=cos(3x) orthogonally at a point (x,y) satisfying 0 < x < 2*Pi.

Original entry on oeis.org

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

Offset: 0

Clark Kimberling, Oct 09 2011

See the Mathematica program for a graph.
xo=0.9350272884749678361451944275323...
yo=0.3301955980451199836007253971727...
m=0.35314006565912096755666111412785...
|OP|=0.99161744799152518925689622748...

Cf. A196996, A197000, A197002.

c = 3;
xo = x /. FindRoot[0 == x + c*Sin[c*x] Cos[c*x], {x, .8, 1.2}, WorkingPrecision -> 100]
RealDigits[xo] (* A196996 *)
m = Sin[c*xo]/xo
RealDigits[m]  (* A196997 *)
yo = m*xo
d = Sqrt[xo^2 + yo^2]
Show[Plot[{Sin[c*x], yo - (1/m) (x - xo)}, {x, 0, Pi/c}], ContourPlot[{y == m*x}, {x, 0, 1.5}, {y, -.1, 1}], PlotRange -> All, AspectRatio -> Automatic]

A197006 Decimal expansion of xo, where P=(xo,yo) is the point nearest O=(0,0) in which a line y=mx meets the curve y=cos(x+Pi/6) orthogonally.

Original entry on oeis.org

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

Offset: 0

Clark Kimberling, Oct 10 2011

See the Mathematica program for a graph.
xo=0.460885580860965976987981282513698...
yo=0.553292712300593256734925495541442...
m=1.2004990723879979061250465124427113...
|OP|=0.7201030093885853693640956082816...

Cf. A197007, A197002, A196996, A197000.

c = Pi/6;
xo = x /.  FindRoot[x == Sin[x + c] Cos[x + c], {x, .8, 1.2}, WorkingPrecision -> 100]
RealDigits[xo] (* A197006 *)
m = 1/Sin[xo + c]
RealDigits[m] (* A197007 *)
yo = m*xo
d = Sqrt[xo^2 + yo^2]
Show[Plot[{Cos[x + c], yo - (1/m) (x - xo)}, {x, -Pi/4, Pi/2}], ContourPlot[{y == m*x}, {x, 0, Pi}, {y, 0, 1}], PlotRange -> All,
AspectRatio -> Automatic, AxesOrigin -> Automatic]

A196998 Decimal expansion of xo, where P=(xo,yo) is the point nearest O=(0,0) at which a line y=m*x meets the curve y=cos(5x/2) orthogonally.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Oct 09 2011

See the Mathematica program for a graph.
xo=1.055537135075475249854148417892290354122...
yo=0.481836913462240473673427172075977637742...
m=0.4564850420234501281397606474354137170643...
|OP|=1.1603126538559168441096914160911620183...

Cf. A196996, A197000, A197002.

c = 5/2;
xo = x /.  FindRoot[0 == x + c*Sin[c*x] Cos[c*x], {x, .8, 1.2}, WorkingPrecision -> 100]
RealDigits[xo] (* A196998 *)
m = Sin[c*xo]/xo
RealDigits[m]  (* A196999 *)
yo = m*xo
d = Sqrt[xo^2 + yo^2]
Show[Plot[{Sin[c*x], yo - (1/m) (x - xo)}, {x, 0, Pi/c}],
 ContourPlot[{y == m*x}, {x, 0, Pi/c}, {y, -.1, 1}], PlotRange -> All, AspectRatio -> Automatic]

A196999 Decimal expansion of slope of the line y=mx which meets the curve y=cos(5x/2) orthogonally (as in A196998).

Original entry on oeis.org

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

Offset: 0

Clark Kimberling, Oct 09 2011

See the Mathematica program for a graph.
xo=1.055537135075475249854148417892290354122...
yo=0.481836913462240473673427172075977637742...
m=0.4564850420234501281397606474354137170643...
|OP|=1.1603126538559168441096914160911620183...

Cf. A196996, A196997, A197000, A197002.

c = 5/2;
xo = x /.  FindRoot[0 == x + c*Sin[c*x] Cos[c*x], {x, .8, 1.2}, WorkingPrecision -> 100]
RealDigits[xo] (* A196998 *)
m = Sin[c*xo]/xo
RealDigits[m]  (* A196999 *)
yo = m*xo
d = Sqrt[xo^2 + yo^2]
Show[Plot[{Sin[c*x], yo - (1/m) (x - xo)}, {x, 0, Pi/c}],
 ContourPlot[{y == m*x}, {x, 0, Pi/c}, {y, -.1, 1}], PlotRange -> All, AspectRatio -> Automatic]

A197001 Decimal expansion of the slope of the line y=mx which meets the curve y=1+cos(x) orthogonally over the interval [0, 2*Pi] (as in A197000).

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Oct 09 2011

See the Mathematica program for a graph.
xo=1.2488014367215508560475125020128381535587614...
yo=1.3164595537507515212878992732671186100622603...
m=1.05417844265684217515747734305673483746142104...
|OP|=1.81454423617045980814297669595599066552030...

Cf. A196700, A196996, A197002.

c = 1;
xo = x /.
  FindRoot[x == Sin[x] (c + Cos[x]), {x, 1, 1.3}, WorkingPrecision -> 100]
RealDigits[xo] (* A197000 *)
m = 1/Sin[xo]
RealDigits[m]  (* A197001 *)
yo = m*xo
d = Sqrt[xo^2 + yo^2]
Show[Plot[{c + Cos[c*x], yo - (1/m) (x - xo)}, {x, 0, Pi}],  ContourPlot[{y == m*x}, {x, 0, Pi}, {y, 0, 2}], PlotRange -> All, AspectRatio -> Automatic, AxesOrigin -> Automatic]

A197004 Decimal expansion of xo, where P=(xo,yo) is the point nearest O=(0,0) in which a line y=mx meets the curve y=cos(x+Pi/3) orthogonally.

Original entry on oeis.org

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

Offset: 0

Clark Kimberling, Oct 10 2011

See the Mathematica program for a graph.
xo=0.255465286103853596695882696613320272654788...
yo=0.264932084602776862434116494762571068650190...
m=1.0370570837365150046614795837584277605222343...
|OP|=0.3680373919265496189530095416155881110455...

Cf. A197002, A196996, A197000.

c = Pi/3;
xo = x /. FindRoot[x == Sin[x + c] Cos[x + c], {x, .8, 1.2}, WorkingPrecision -> 100]
RealDigits[xo] (* A197004 *)
m = 1/Sin[xo + c]
RealDigits[m]  (* A197005 *)
yo = m*xo
d = Sqrt[xo^2 + yo^2]
Show[Plot[{Cos[x + c], yo - (1/m) (x - xo)}, {x, -Pi/4, Pi/2}],
ContourPlot[{y == m*x}, {x, 0, Pi}, {y, 0, 1}], PlotRange -> All,
AspectRatio -> Automatic, AxesOrigin -> Automatic]

cp's OEIS Frontend

A197003 Decimal expansion of the slope of the line y=mx which meets the curve y=cos(x+Pi/4) orthogonally over the interval [0, 2*Pi] (as in A197002).

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

PARI

Formula

A003957 The Dottie number: decimal expansion of root of cos(x) = x.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Python

Formula

Extensions

A196996 Decimal expansion of xo, where P=(xo,yo) is the point nearest O=(0,0) at which a line y=m*x meets the curve y=cos(3x) orthogonally.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

A197000 Decimal expansion of xo, where P=(xo,yo) is the point nearest O=(0,0) in which a line y=mx meets the curve y=1+cos(x) orthogonally.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

A196997 Decimal expansion of m, where y=m*x is the line through (0,0) which meets the curve y=cos(3*x) orthogonally at a point (x,y) satisfying 0 < x < 2*Pi.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

A197006 Decimal expansion of xo, where P=(xo,yo) is the point nearest O=(0,0) in which a line y=mx meets the curve y=cos(x+Pi/6) orthogonally.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

A196998 Decimal expansion of xo, where P=(xo,yo) is the point nearest O=(0,0) at which a line y=m*x meets the curve y=cos(5x/2) orthogonally.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

A196999 Decimal expansion of slope of the line y=mx which meets the curve y=cos(5x/2) orthogonally (as in A196998).

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

A197001 Decimal expansion of the slope of the line y=mx which meets the curve y=1+cos(x) orthogonally over the interval [0, 2*Pi] (as in A197000).

Views

Author

A196997 Decimal expansion of m, where y=mx is the line through (0,0) which meets the curve y=cos(3x) orthogonally at a point (x,y) satisfying 0 < x < 2*Pi.