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 A122775 The angle, in degrees, for which Ozanam's approximation is exact. 1
 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, 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, 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 (list; constant; graph; refs; listen; history; text; internal format)
 OFFSET 2,1 COMMENTS 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. "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) 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. 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. 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:   A (deg)     J   -------  -------      0     171.887      5     171.887     10     171.888     15     171.892     20     171.902     25     171.923     30     171.962     35     172.026     40     172.128     45     172.279 The degree of approximation may be shown by solving the triangle in which C=90 degrees, c=4156, a=2537. We find b = 3291.8, and, by Ozanam's formula, A = 2537*172/(3291.8 + 4156*2) = 436364/11603.8 = 37°36'18". 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 REFERENCES Rawdon Levett and Charles Davison, The Elements of Plane Trigonometry, Chapter XV, "Approximations and Errors", pp. 372-373, MacMillan and Co, London & NY, 1892. LINKS Robert G. Wilson v, Table of n, a(n) for n = 2..16385 R. A. Johnson, Determination of an angle of a right triangle, without tables, Amer. Math. Monthly, Vol 27, No. 10, Oct 1920, pp. 368, 369. Frank Swetz, Mathematical Treasure: Jacques Ozanam's Récréations, Convergence, August 2013. FORMULA If f(n) = 172n/(sqrt(1-n^2)+2) then A122775 is when f(sin(n*Pi/180)) = n. EXAMPLE 33.239565786056615600642100188324722742275831667577343680621765378873666... degrees =0.5801398648999450253504045320808762548459123764471181646784444551727458... radians. MATHEMATICA 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 *) CROSSREFS Sequence in context: A262212 A123676 A326814 * A086632 A038699 A249803 Adjacent sequences:  A122772 A122773 A122774 * A122776 A122777 A122778 KEYWORD nonn,cons AUTHOR Julian Havil (julian.havil(AT)ntlworld.com), Jun 25 2007 EXTENSIONS More terms from Robert G. Wilson v, Feb 22 2013 STATUS approved

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