cp's OEIS Frontend

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

%I A231985 #16 Jun 09 2021 13:22:06
%S A231985 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,
%T A231985 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,
%U A231985 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
%N A231985 Decimal expansion of the side length (in degrees) of the spherical square whose solid angle is exactly one deg^2.
%C A231985 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.
%D A231985 G. V. Brummelen, Heavenly Mathematics: The Forgotten Art of Spherical Trigonometry, Princeton University Press, 2012, ISBN 978-0691148922.
%H A231985 Stanislav Sykora, <a href="/A231985/b231985.txt">Table of n, a(n) for n = 1..2000</a>
%H A231985 Wikipedia, <a href="http://en.wikipedia.org/wiki/Solid_angle#Pyramid">Solid angle</a>, Section 3.3 (Pyramid)
%H A231985 Wikipedia, <a href="http://en.wikipedia.org/wiki/Square_degree">Square degree</a>
%H A231985 Wikipedia, <a href="http://en.wikipedia.org/wiki/Steradian">Steradian</a>
%F A231985 (360/Pi)*arcsin(sqrt(sin((Pi/360)^2))).
%e A231985 1.0000126923441633791606036333586617786396521852877666490350781364...
%t A231985 RealDigits[(360/Pi)*ArcSin[Sqrt[Sin[(Pi/360)^2]]],10,120][[1]] (* _Harvey P. Dale_, Jun 09 2021 *)
%o A231985 (PARI)
%o A231985 default(realprecision, 120);
%o A231985 (360/Pi)*asin(sqrt(sin((Pi/360)^2))) \\ or
%o A231985 (180/Pi)*solve(x = 0, 1, 4*asin(sin(x/2)^2) - (Pi/180)^2) \\ _Rick L. Shepherd_, Jan 29 2014
%Y A231985 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).
%K A231985 nonn,cons,easy
%O A231985 1,7
%A A231985 _Stanislav Sykora_, Nov 17 2013