

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
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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 rightangled 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 rightangled 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. 372373, 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(1n^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



