OT: Cosine curio

[Bob Patton]

      Laguerre cosine approximation

         Math history fun fact

 I came across this some 50 years ago,

 before calculators had trig keys.

 It's good to about 4 significant digits for first quadrant,

 and exact at 0, 45, 60, 90 degrees.

 m = angle/90 for degrees or angle/pi/2 for radians

                   -m^2

  ( --------------------------------- - 1)

    sqrt((-m^3 + 4m^2 -5m + 2)/3) + m

 This can be expressed in linear form as:

 -(((((((4-m)m-5)m+2)/3)^(1/2)+m)^(-1)mm)-1)

 which can be keyed in directly on a simple chain-logic memory

 calculator with a square root key and the ability to find the

 reciprocal using some combination of / and = keys. Handle the

 leading minus at the end, just mentally if necessary.

 To test it, on a 12C, for angles in degrees:

 90 / ENTER ENTER ENTER

 4 x<>y - X 5 - X 2 + 3 / g-sqrt + 1/x  X X 1 - CHS

<*** End of File ***>

[Crawl]

It seems like the final 1 should be positive, not negative.

[Namir]

Interesting approximation! Can we enhance it now that we have tools like Matlab and Excel?

Crawl it right in pointing out that the trailing -1 in the first two equations should be +1.

Namir

[C.Ret]

Really interesting approximation.

As Namir and Crawl have already point out, there is a typo in the developed formulae only.

The linear formulea and the RPN instructions are all correct.

Using my HP-41C, I just test a few points to observe error between cosine and the Laguerre approximation (express in the following table as ppm

 Angle(°)  m=a/90°    Laguerre(m)  Cos(a)      Error(ppm)

 -90       -1         0.0000000    0.0000000       0

 -45       -0.5       0.7124144    0.7071068    7506

   0        0         1.0000000    1.0000000       0

  10        1/9       0.9848786    0.9848078      72

  20        2/9       0.9398473    0.9396926     165

  30        1/3       0.8661695    0.8660254     166

  40        4/9       0.7660945    0.7660444      65

  45        0.5       0.7071068    0.7071068       0

  50        5/9       0.6427550    0.6427876     -50

  60        2/3       0.5000000    0.5000000       0

  70        7/9       0.3421859    0.3420201     484

  80        8/9       0.1739508    0.1736482    1742

  90        1         0.0000000    0.0000000       0

 120        4/3      -0.1927647   -0.5000000 -614471  !!!!!

 135        1.5      -0.3203263   -0.7071068    -546

 145        1.6111   -0.4175237   -0.8191520    -490

 180        2        -1.0000000   -1.0000000       0

 225        2.5       n.a.r.v.    -0.7071068     nan

As explain, in first quadrant, few error and exact value for same remarkable values are obtained.

No more approximation can be obtained after 180° due to the sign of the polynôme under the square root.



The following figure better illustrate accuracy and region of interest for this cosine approximation:



Note that in this graph, Lag(a) is the real part of the laguerre approximation. That’s why plot continue after the 180° limit.

As Namir point it out, we can enhance this approximation, but not only in accuracy, other way may be to make it usable on a larger domain.

Edited: 13 Dec 2011, 5:58 a.m.