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%I #47 Dec 13 2019 13:45:51
%S 3,3,2,3,9,5,6,5,7,8,6,0,5,6,6,1,5,6,0,0,6,4,2,1,0,0,1,8,8,3,2,4,7,2,
%T 2,7,4,2,2,7,5,8,3,1,6,6,7,5,7,7,3,4,3,6,8,0,6,2,1,7,6,5,3,7,8,8,7,3,
%U 6,6,6,7,2,1,3,0,7,3,0,1,7,8,6,3,5,3,9,5,7,2,5,5,7,2,3,3,8,2,3,7,3,6,5,3,6,1,9,3,8,8,8,2,9,4
%N The angle, in degrees, for which Ozanam's approximation is exact.
%C Ozanam's approximation states that in any right triangle the number of degrees in the smallest angle is very nearly equal to the smallest side times 172 divided by the other side plus twice the hypotenuse. The approximation is remarkably accurate and for the angle 33.239565... degrees the approximation is exact.
%C "239. In any right-angled triangle the number of degrees in the smallest angle divided by 172 is very nearly equal to the smallest side divided by the sum of the other side and twice the hypotenuse. (Ozanam's Formula)
%C In the right-angled triangle ABC, let C be the right angle, and A the smallest angle; Let A be the number of degrees, and a the number of radians in this angle, so that a = Pi*A/180 = 3A/172, approximately.
%C Now, a/(b+2c) = c*sin A/(2c+c cos A) = sin a/(2+cos a) = (a - a^3/6)/(3 - a^2/2), approximately, a/3 = A/172, approximately.
%C This proves Ozanam's formula, when A is not large. Writing J for the fraction A*(2 + cos A)/sin A we see then that, for small values of A, J does not differ greatly from 172. In the following table, the value of J is given to three places of decimals for every five degrees 0 degrees to 45 degrees:
%C A (deg) J
%C ------- -------
%C 0 171.887
%C 5 171.887
%C 10 171.888
%C 15 171.892
%C 20 171.902
%C 25 171.923
%C 30 171.962
%C 35 172.026
%C 40 172.128
%C 45 172.279
%C The degree of approximation may be shown by solving the triangle in which C=90 degrees, c=4156, a=2537.
%C We find b = 3291.8, and, by Ozanam's formula,
%C A = 2537*172/(3291.8 + 4156*2) = 436364/11603.8 = 37°36'18".
%C The correct value of A is 37°37'17", so that the absolute error in this case is only 59". [Levett and Davison] - _Robert G. Wilson v_, Jan 23 2013
%D Rawdon Levett and Charles Davison, The Elements of Plane Trigonometry, Chapter XV, "Approximations and Errors", pp. 372-373, MacMillan and Co, London & NY, 1892.
%H Robert G. Wilson v, <a href="/A122775/b122775.txt">Table of n, a(n) for n = 2..16385</a>
%H R. A. Johnson, <a href="http://www.jstor.org/stable/2972556">Determination of an angle of a right triangle, without tables</a>, Amer. Math. Monthly, Vol 27, No. 10, Oct 1920, pp. 368, 369.
%H Frank Swetz, <a href="https://www.maa.org/press/periodicals/convergence/mathematical-treasure-jacques-ozanam-s-r-cr-ations">Mathematical Treasure: Jacques Ozanam's Récréations</a>, Convergence, August 2013.
%F If f(n) = 172n/(sqrt(1-n^2)+2) then A122775 is when f(sin(n*Pi/180)) = n.
%e 33.239565786056615600642100188324722742275831667577343680621765378873666... degrees
%e =0.5801398648999450253504045320808762548459123764471181646784444551727458... radians.
%t f[n_] := 172n/(2 + Sqrt[1 - n^2]); FindRoot[ f[ Sin[ t*Pi/180]] == t, {t, 30}, AccuracyGoal -> Infinity, WorkingPrecision -> 2^7, PrecisionGoal -> 2^7][[1, 2]] (* _Robert G. Wilson v_, Feb 22 2013 *)
%K nonn,cons
%O 2,1
%A Julian Havil (julian.havil(AT)ntlworld.com), Jun 25 2007
%E More terms from _Robert G. Wilson v_, Feb 22 2013
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