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A231985
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Decimal expansion of the side length (in degrees) of the spherical square whose solid angle is exactly one deg^2.
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5
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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
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OFFSET
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1,7
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COMMENTS
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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.
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REFERENCES
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G. V. 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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(360/Pi)*arcsin(sqrt(sin((Pi/360)^2))).
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EXAMPLE
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1.0000126923441633791606036333586617786396521852877666490350781364...
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MATHEMATICA
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RealDigits[(360/Pi)*ArcSin[Sqrt[Sin[(Pi/360)^2]]], 10, 120][[1]] (* Harvey P. Dale, Jun 09 2021 *)
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PROG
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(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
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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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