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A231983
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Decimal expansion of the solid angle (in sr) of a spherical square having sides of one degree.
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5
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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, 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, 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
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OFFSET
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-3,1
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COMMENTS
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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.
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REFERENCES
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Glen Van Brummelen, Heavenly Mathematics: The Forgotten Art of Spherical Trigonometry, Princeton University Press, 2012, ISBN 978-0691148922.
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LINKS
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FORMULA
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Equals 4*arcsin(sin(R/2)sin(S/2)), where R = S = Pi/180.
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EXAMPLE
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0.0003046096875119366637825983210350747291625456181624489357027...
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MATHEMATICA
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RealDigits[4 * ArcSin[Sin[Pi/360]^2], 10, 120][[1]] (* Amiram Eldar, Jun 26 2023 *)
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PROG
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(PARI)
default(realprecision, 120);
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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