OFFSET
1,7
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.
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
FORMULA
(360/Pi)*arcsin(sqrt(sin((Pi/360)^2))).
EXAMPLE
1.0000126923441633791606036333586617786396521852877666490350781364...
MATHEMATICA
RealDigits[(360/Pi)*ArcSin[Sqrt[Sin[(Pi/360)^2]]], 10, 120][[1]] (* Harvey P. Dale, Jun 09 2021 *)
PROG
(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
CROSSREFS
KEYWORD
AUTHOR
Stanislav Sykora, Nov 17 2013
STATUS
approved