A231985 Decimal expansion of the side length (in degrees) of the spherical square whose solid angle is exactly one deg^2.

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

Offset: 1

Author

Stanislav Sykora, Nov 17 2013

Keywords

• nonn
• cons
• easy

Comments

This answers the inverse problem of A231984 (not to be confused with its inverse value): what is the side arc-length of a spherical square required to subtend exactly 1 deg^2. Since the solid angle of a spherical square with side s (in rads) is Omega = 4*arcsin(sin(s/2)^2) (in sr), we have s = 2*arcsin(sqrt(Omega/4)). Converting Omega = 1 deg^2 into steradians (A231982), applying the formula, and converting the result from radians to degrees (A072097), one obtains the result.

Examples

			1.0000126923441633791606036333586617786396521852877666490350781364...
		

References

• G. V. Brummelen, Heavenly Mathematics: The Forgotten Art of Spherical Trigonometry, Princeton University Press, 2012, ISBN 978-0691148922.

Links

• Stanislav Sykora, Table of n, a(n) for n = 1..2000
• Wikipedia, Solid angle, Section 3.3 (Pyramid)
• Wikipedia, Square degree
• Wikipedia, Steradian

Crossrefs

Cf. A000796 (Pi), A072097 (rad/deg), A019685 (deg/rad), A231981 (sr/deg^2), A231982 (deg^2/sr), A231983, A231984 (inverse problem), A231986, A231985, A231987 (same problem for 1sr).

Programs

Mathematica

RealDigits[(360/Pi)*ArcSin[Sqrt[Sin[(Pi/360)^2]]],10,120][[1]] (* Harvey P. Dale, Jun 09 2021 *)

PARI

default(realprecision, 120);
(360/Pi)*asin(sqrt(sin((Pi/360)^2))) \\ or
(180/Pi)*solve(x = 0, 1, 4*asin(sin(x/2)^2) - (Pi/180)^2) \\ Rick L. Shepherd, Jan 29 2014

Formula

(360/Pi)*arcsin(sqrt(sin((Pi/360)^2))).