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September 26

Approximation of the cosine function

In the Cosine article on Mathworld, an interesting approximation is mentioned:

A close approximation to cos(pix/2) for x in [0,1] is

${\begin{aligned}H(x)&=1-{\frac {x^{2}}{x+(1-x){\sqrt {\frac {2-x}{3}}}}}&\quad (8)\\&\approx \cos \left({\frac {\pi }{2}}x\right)&\quad (9)\end{aligned}}$

(Hardy 1959), where the difference between cos(pix/2) and Hardy's approximation is plotted above.

However, all it mentions is Hardy's work, and the work cited does not contain any other information about this approximation! The derivation and/or potential applications for this would be interesting, but no other information about this can be found on the internet. Do you know any other sources/information about this? Llightex (talk) 21:05, 26 September 2014 (UTC)[reply]

Here is a way you might reverse engineer the formula, though I have no idea how Hardy derived it. Let C(x) = cos(πx/2). We know from the Taylor series that C(x) = 1 - x²/constant + other terms. Rewrite this as C(x) = 1 - x²/K(x) where K is to be determined. We also know C(1) = 0 from which K(1)=1. Expand K at x=1 to get K(x)=1+constant⋅(x-1)+higher terms. Again, collecting the the constant and higher terms into a single function, write K(x)=1+(x-1)L(x). At this point you can get a fairly good approximation for C by plugging in a linear approximation for L. But we also know C(1/2)=√2/2 which would imply (after some computation) L(1/2) = 1 - √(1/2). So perhaps a better approximation of L would be L(x)≈ 1 - √(1/2+m(x-1/2)) for some constant m. If you plug in C(2/3)=1/2 you get m=-1/3 which produces the approximation given, but other values of m might work just as well or better. I found m=-.337 gives the lowest mean square error on the interval. Note that there are points in the derivation where different choices could be made, for example you could write C(x) = 1 - x²⋅K(x) or K(x)=1+(x-1)/L(x). It might be fun to explore these variations to see how they compare with the one given. --RDBury (talk) 00:01, 27 September 2014 (UTC)[reply]