The On-Line Encyclopedia of Integer Sequences (OEIS)

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A122775

The angle, in degrees, for which Ozanam's approximation is exact.
(history; published version)

#47 by N. J. A. Sloane at Fri Dec 13 13:45:51 EST 2019

STATUS

proposed

approved

#46 by Jon E. Schoenfield at Wed Dec 11 22:12:13 EST 2019

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editing

proposed

Discussion

Thu Dec 12

00:38

Michel Marcus: you could ask to Robert G. Wilson v at Wed Jan 23 22:00:54 EST 2013

Fri Dec 13

13:45

N. J. A. Sloane: It seems to end in the last line of the COMMENTS section.

#45 by Jon E. Schoenfield at Wed Dec 11 22:10:34 EST 2019

COMMENTS

Ozanam's approximation states that in any right-angled 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.

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.

STATUS

approved

editing

Discussion

Wed Dec 11

22:12

Jon E. Schoenfield: Where does the quote that begins with "239. In any" end?

#44 by Joerg Arndt at Sun Dec 03 02:13:34 EST 2017

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reviewed

approved

#43 by Michel Marcus at Sun Dec 03 01:43:07 EST 2017

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proposed

reviewed

#42 by Jon E. Schoenfield at Sun Dec 03 01:12:59 EST 2017

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editing

proposed

#41 by Jon E. Schoenfield at Sun Dec 03 01:12:42 EST 2017

COMMENTS

The correct value of A is 37°37'17", so that the absolute error in this case is only 59". from [Levett and Davison. ] - Robert G. Wilson v, Jan 23 2013

Jacques Ozaman (1640-1718). - Robert G. Wilson v, Jul 19 2014

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proposed

editing

Discussion

Sun Dec 03

01:12

Jon E. Schoenfield: Yes, done -- thanks!

#40 by Jon E. Schoenfield at Sun Dec 03 01:03:16 EST 2017

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editing

proposed

Discussion

Sun Dec 03

01:06

Michel Marcus: so now we can remove Jacques Ozaman (1640-1718)  ??

#39 by Jon E. Schoenfield at Sun Dec 03 01:01:48 EST 2017

LINKS

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.

Discussion

Sun Dec 03

01:03

Jon E. Schoenfield: @Michel -- Thanks!  I had looked for a while, and that one seemed to be by far the most informative website I'd found on the subject.

I'm not sure how to format the Links entry.  Does this need any changes?  Thanks!

#38 by Jon E. Schoenfield at Sun Dec 03 01:00:23 EST 2017

LINKS

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>

STATUS

proposed

editing