A069855 Decimal expansion of the root of x*tan(x)=1.

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

Offset: 0

Author

Benoit Cloitre, May 01 2002

Keywords

• cons
• easy
• nonn

Comments

Consider a lens-like shape S created by the curves cos(x) and -cos(x) for x in [-Pi/2,Pi/2] and the points A = (u, v), B = (-u, v), C = (-u, -v), D = (u, -v), K = (0, 2v), L = (-2u, 0), M = (0, -2v), N = (2u,0), where u is given by this sequence, and v = u/sqrt(1+u^2). Then ABCD is the rectangle of maximal area, inscribed in S, with sides parallel to the coordinate axes, and KLMN is the rhombus of minimal area, circumscribed around S, with vertices on the coordinate axes. Also, A,B,C,D are the tangent points where the sides of the rhombus touch S, see illustration in the links section. - Gleb Koloskov, Jul 05 2021

Examples

			0.860333589019379762483893424137662333411884363237653783...
		

Links

• Gleb Koloskov, Geometric illustration
• Eric Weisstein's World of Mathematics, Cotangent [From _Eric W. Weisstein_, Mar 03 2010]

Crossrefs

Cf. A009399, A085984, A346062.

Programs

Mathematica

N[Minimize[{(x+Cot[x])^2 Sin[x],{x>0,xGleb Koloskov, Jul 05 2021 *)
RealDigits[x/.FindRoot[x Tan[x]==1,{x,1},WorkingPrecision->120]][[1]] (* Harvey P. Dale, Dec 04 2021 *)

PARI

/* 300 significant digits */ s=0.1; for(n=1,500,s=s+sign(cotan(s)-s)/2^n; if(n>499, print(s*1.)))

Formula

Equals A346062 * sqrt(2 + 2*sqrt(1 + 256/A346062^2)) / 16. - Gleb Koloskov, Jul 05 2021