A199949 Decimal expansion of least x satisfying x^2 + cos(x) = 2*sin(x).

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

Offset: 0

Author

Clark Kimberling, Nov 12 2011

Keywords

• nonn
• cons

Comments

For many choices of a,b,c, there are exactly two numbers x>0 satisfying a*x^2+b*cos(x)=c*sin(x).

Guide to related sequences, with graphs included in Mathematica programs:

a.... b.... c.... least x, greatest x

1.... 1.... 2.... A199949, A199950

1.... 1.... 3.... A199951, A199952

1.... 1.... 4.... A199953, A199954

1.... 2.... 3.... A199955, A199956

1.... 2.... 4.... A199957, A199958

1.... 3.... 3.... A199959, A199960

1.... 3.... 4.... A199961, A199962

1.... 4.... 3.... A199963, A199964

1.... 4.... 4.... A199965, A199966

2.... 1.... 3.... A199967, A200003

2.... 1.... 4.... A200004, A200005

3.... 1.... 4.... A200006, A200007

4.... 1.... 4.... A200008, A200009

1... -1.... 1.... A200010, A200011

1... -1.... 2.... A200012, A200013

1... -1.... 3.... A200014, A200015

1... -1.... 4.... A200016, A200017

1... -2.... 1.... A200018, A200019

1... -2.... 2.... A200020, A200021

1... -2.... 3.... A200022, A200023

1... -2.... 4.... A200024, A200025

1... -3.... 1.... A200026, A200027

1... -3.... 2.... A200093, A200094

1... -3.... 3.... A200095, A200096

1... -3.... 4.... A200097, A200098

1... -4.... 1.... A200099, A200100

1... -4.... 2.... A200101, A200102

1... -4.... 3.... A200103, A200104

1... -4.... 4.... A200105, A200106

2... -1.... 1.... A200107, A200108

2... -1.... 2.... A200109, A200110

2... -1.... 3.... A200111, A200112

2... -1.... 4.... A200114, A200115

2... -2.... 1.... A200116, A200117

2... -2.... 3.... A200118, A200119

2... -3.... 1.... A200120, A200121

2... -3.... 2.... A200122, A200123

2... -3.... 3.... A200124, A200125

2... -3.... 4.... A200126, A200127

2... -4.... 1.... A200128, A200129

2... -4.... 3.... A200130, A200131

3... -1.... 1.... A200132, A200133

3... -1.... 2.... A200223, A200224

3... -1.... 3.... A200225, A200226

3... -1.... 4.... A200227, A200228

3... -2.... 1.... A200229, A200230

3... -2.... 2.... A200231, A200232

3... -2.... 3.... A200233, A200234

3... -2.... 4.... A200235, A200236

3... -3.... 1.... A200237, A200238

3... -3.... 2.... A200239, A200240

3... -3.... 4.... A200241, A200242

3... -4.... 1.... A200277, A200278

3... -4.... 2.... A200279, A200280

3... -4.... 3.... A200281, A200282

3... -4.... 4.... A200283, A200284

4... -1.... 1.... A200285, A200286

4... -1.... 2.... A200287, A200288

4... -1.... 3.... A200289, A200290

4... -1.... 4.... A200291, A200292

4... -2.... 1.... A200293, A200294

4... -2.... 3.... A200295, A200296

4... -3.... 1.... A200299, A200300

4... -3.... 2.... A200297, A200298

4... -3.... 3.... A200301, A200302

4... -3.... 4.... A200303, A200304

4... -4.... 1.... A200305, A200306

4... -4.... 3.... A200307, A200308

Suppose that f(x,u,v) is a function of three real variables and that g(u,v) is a function defined implicitly by f(g(u,v),u,v)=0. We call the graph of z=g(u,v) an implicit surface of f.

For an example related to A199949, take f(x,u,v)=x^2+u*cos(x)-v*sin(x) and g(u,v) = a nonzero solution x of f(x,u,v)=0. If there is more than one nonzero solution, care must be taken to ensure that the resulting function g(u,v) is single-valued and continuous. A portion of an implicit surface is plotted by Program 2 in the Mathematica section.

Examples

			least x:  0.659266045766946074537348579563067611...
greatest x: 1.2710268008159460640047188480978502...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..10000
• Index entries for transcendental numbers.

Crossrefs

Cf. A199950.

Programs

Mathematica

(* Program 1:  A199949 *)
a = 1; b = 1; c = 2;
f[x_] := a*x^2 + b*Cos[x]; g[x_] := c*Sin[x]
Plot[{f[x], g[x]}, {x, -1, 2}, {AxesOrigin -> {0, 0}}]
r = x /. FindRoot[f[x] == g[x], {x, .65, .66}, WorkingPrecision -> 110]
RealDigits[r]   (* A199949 *)
r = x /. FindRoot[f[x] == g[x], {x, 1.27, 1.28}, WorkingPrecision -> 110]
RealDigits[r]   (* A199950 *)
(* Program 2: implicit surface of x^2+u*cos(x)=v*sin(x) *)
f[{x_, u_, v_}] := x^2 + u*Cos[x] - v*Sin[x];
t = Table[{u, v, x /. FindRoot[f[{x, u, v}] == 0, {x, 0, 1}]}, {u, -5, 0}, {v, 0, 1}];
ListPlot3D[Flatten[t, 1]]  (* for A199949 *)

PARI

a=1; b=1; c=2; solve(x=0, 1, a*x^2 + b*cos(x) - c*sin(x)) \\ G. C. Greubel, Jul 05 2018

Extensions

A-number corrected by Jaroslav Krizek, Nov 27 2011