OT: Cosine curio « Next Oldest | Next Newest »

 ▼ Bob Patton Junior Member Posts: 33 Threads: 5 Joined: Jan 2009 12-11-2011, 05:17 PM ``` 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 Senior Member Posts: 306 Threads: 3 Joined: Sep 2009 12-11-2011, 09:23 PM It seems like the final 1 should be positive, not negative. Namir Posting Freak Posts: 2,247 Threads: 200 Joined: Jun 2005 12-12-2011, 06:49 PM 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 Senior Member Posts: 260 Threads: 0 Joined: Oct 2008 12-13-2011, 05:45 AM 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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