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A243710
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Decimal expansion of the solid angle of an equilateral spherical triangle with a side length of 1 radian.
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2
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4, 9, 5, 5, 9, 4, 8, 9, 5, 7, 3, 3, 9, 6, 4, 7, 5, 0, 6, 9, 8, 8, 5, 7, 9, 1, 2, 9, 0, 8, 4, 0, 0, 2, 1, 1, 5, 6, 0, 3, 8, 0, 7, 9, 2, 1, 8, 8, 0, 4, 5, 1, 6, 8, 3, 7, 4, 7, 2, 7, 3, 0, 9, 0, 5, 8, 5, 8, 8, 6, 9, 2, 1, 6, 7, 4, 0, 4, 2, 8, 4, 7, 2, 0, 7, 5, 9, 0, 0, 4, 9, 7, 4, 3, 5, 0, 7, 2, 3, 3, 2, 5, 0, 1, 0
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OFFSET
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0,1
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COMMENTS
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Set theta_a = theta_b = theta_c = 1 in the formula below. The result is in steradians.
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LINKS
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Wikipedia, Solid angle, section 'Tetrahedron', L'Huillier's theorem.
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FORMULA
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For a spherical triangle with sides theta_a, theta_b, theta_c, the solid angle is 4*atan(sqrt(tan(theta/2)*tan((theta-theta_a)/2)*tan((theta-theta_b)/2)*tan((theta-theta_c)/2))), where theta = (theta_a+theta_b+theta_c)/2.
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EXAMPLE
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0.4955948957339647506988579129084002115603807921880... steradians.
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MATHEMATICA
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RealDigits[4(ArcTan[Sqrt[Tan[3/4]Tan[1/4]^3]]), 10, 120][[1]] (* Harvey P. Dale, Sep 13 2020 *)
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PROG
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(PARI) 4*atan(sqrt(tan(3/4)*tan(1/4)^3))
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CROSSREFS
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Cf. A243711 (fraction of full solid angle).
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KEYWORD
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AUTHOR
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STATUS
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approved
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