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A231983 Decimal expansion of the solid angle (in sr) of a spherical square having sides of one degree. 5

%I #23 Jun 26 2023 02:21:13

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

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

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

%N Decimal expansion of the solid angle (in sr) of a spherical square having sides of one degree.

%C Arc-lengths are planar angles, expressed either in radians (rad) or in degrees (deg), while solid angles are areas subtended on a unit sphere and expressed in steradians (sr) or degree-squares (deg^2). Consequently, the solid angle Omega of a spherical rectangle with sides r, s expressed in degrees, cannot be computed as the product of its sides and then converted into steradians by applying the deg^2/sr conversion factor A231982. Rather, one must use the general formula Omega = 4*arcsin(sin(R/2)sin(S/2)), where R=(Pi/180)r, S=(Pi/180)s are the sides expressed in radians. Due to spherical excess, the result differs slightly, but significantly, from A231982.

%D Glen Van Brummelen, Heavenly Mathematics: The Forgotten Art of Spherical Trigonometry, Princeton University Press, 2012, ISBN 978-0691148922.

%H Stanislav Sykora, <a href="/A231983/b231983.txt">Table of n, a(n) for n = -3..2000</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Solid_angle#Pyramid">Solid angle</a>, Section 3.3 (Pyramid).

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Square_degree">Square degree</a>.

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Steradian">Steradian</a>.

%F Equals 4*arcsin(sin(R/2)sin(S/2)), where R = S = Pi/180.

%e 0.0003046096875119366637825983210350747291625456181624489357027...

%t RealDigits[4 * ArcSin[Sin[Pi/360]^2], 10, 120][[1]] (* _Amiram Eldar_, Jun 26 2023 *)

%o (PARI)

%o default(realprecision, 120);

%o 4*asin(sin(Pi/360)^2) \\ _Rick L. Shepherd_, Jan 28 2014

%Y Cf. A000796 (Pi), A072097 (rad/deg), A019685 (deg/rad), A231981 (sr/deg^2), A231982 (deg^2/sr), A231984 (this constant in deg^2), A231986 (same for square with 1 rad side), A231985, A231987.

%K nonn,cons,easy

%O -3,1

%A _Stanislav Sykora_, Nov 17 2013

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