The Taylor series you expanded has six-place accuracy for |x| < .5
, five-place for |x| < .6
, four-place for |x| < .7
, three-place for |x| < .8
, and two-place for |x| < .9
.
Of course, there's no reason to think the Taylor polynomial is the best
polynomial of a given degree. Welcome to Numerical Analysis.
It takes too many terms to get a good estimation for |x| = 1
, because the derivative of arcsin(x) has a pole at x = 1
, so that its Taylor series converges very slowly. This means the Taylor expansion is not an efficient way of approximating arcsin(x) except for small x. If you printed each term of the expansion as you calculate it, you'd see it's extremely small to make the series converge in a reasonable amount of time.
For your aid, Milton Abramowitz and Irene Stegun in their book "Handbook of Mathematical Functions", p.81 derive this approximation formula:
arcsin(x) = pi/2 - sqrt(1 - x)(a0 + a1*x + a2*x^2 + a3*x^3)
where
a0 = 1.5707288
a1 = -0.2121144
a2 = 0.0742610
a3 = -0.0187293
which performs much better near 1.