A196504 Decimal expansion of the least x > 0 satisfying x + tan(x) = 0.

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

Offset: 1

Author

Clark Kimberling, Oct 03 2011

Keywords

• nonn
• cons

Comments

Let L be the least x > 0 satisfying x + tan(x) = 0.

Then L is also the least x > 0 satisfying x = (sin(x))(sqrt(1+x^2)).

Consequently, for 0 < x < L, for all p > 0, 1/sqrt(1+x^2) - 1/x^p < sin(x) < 1/sqrt(1+x^2) for 0 < x < L.

See A196500-A196503 and A196505 for related constants and inequalities.

The number L also occurs in connection with Du Bois Reymond's constants; see the Finch reference.

For x = L the area of right triangle with vertices (0,0), (x,0) and (x,sin(x)), i.e., the one inscribed into the half-wave curve, is maximal. - Roman Witula, Feb 05 2015

Examples

			L = 2.02875783811043422357697112473471437610838002...
1/L = 0.4929124517549075741877801898222329769156970132...
		

References

• Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, page 239.

Links

• Index entries for transcendental numbers.

Crossrefs

Cf. A196505, A196500, A196502.

Programs

Mathematica

Plot[{Sin[x], x/Sqrt[1 + x^2]}, {x, 0, 9}]
t = x /.FindRoot[Sin[x] == x/Sqrt[1 + x^2], {x, .10, 3}, WorkingPrecision -> 100]
RealDigits[t]   (* A196504 *)
1/t
RealDigits[1/t] (* A196505 *)

PARI

solve(x=2,3, sin(x)-x/sqrt(1+x^2)) \\ Charles R Greathouse IV, Feb 11 2025