Quote:

Hi All,
In your opinion, among the sin, cos, and tan functions (as you get any two trig functions from the value of the third trig function), which one is easier to calculate using polynomial approximations or special approximation? If you are referring to a special approximation can you please state it or offer a link to that information?

Thanks,

Namir

Namir,

I tend to lean towards sine. However, I find the convergence for the Taylor Series for sine to be super slow. The approximations listed are stated to be good for eight-digit accuracy for the interval [0 to pi/2].

From "Scientific Analysis on the Pocket Calculator" by Jon M. Smith (published 1975):

Sine and Cosine: error < 2 x 10^-9 with 0 <= x <= pi/2
sin x = x*(1 + x^2*(a2 + x^2*(a4 + x^2*(a6 + x^2*(a8 + a10*x^2)))))

a2 = -0.16666 66664

a4 = 0.00833 33315

a6 = -0.00019 84090

a8 = 0.00000 27526

a10 = -0.00000 00239

cos x = 1 + x^2*(a2 + x^2*(a4 + x^2*(a6 + x^2*(a8 + a10*x^2)))))

a2 = -0.49999 99963

a4 = 0.04166 66418

a6 = -0.00138 88397

a8 = 0.00002 47609

a10 = -0.00000 26050

The "compacted" approximations are said to give three-digit accuracy for [0, pi/2].

Sine and Cosine: Error = 2 * 10^-4 and 2 * 10^-9, respectively, for 0 <= x <= pi/2
sin x = x * (1 + x^2*(a2 + a4*x^2))

a2 = -0.16605

a4 = 0.00761

cos x = 1 + x^2*(a2 + a4*x^2))

a2 = -0.49670

a4 = 0.03705

There was also a thread of calculating trig functions with the HP 12C, that was a while ago.

Hope this helps,

Eddie