OT: Cosine curio



#5

      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 ***>

#6

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

#7

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

#8

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.


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