If we don't want to use sqrt, then we have to approximate arcsin itself.
Here's the Taylor series of degree 7, expanded about the origin, and its error plot.
One might argue that, since half of the terms are zero, it might be more in the spirit of 7 d.f. to use an expansion with 7 nonzero terms:
Surprisingly, the max error is not much better.
It's generally better to expand a Taylor series about the center of the interval.
Unfortunately, here, again it's only a little better. Also here, adding extra terms wouldn't help much, because of the small radius of convergence.
Here is the, far better, Chebyshev approximation.
Here is the Pade quotient, converted from the 7 d.f. Taylor about the origin. It costs a division to execute, but is often more accurate, but not this time.
This is the Pade with 7 non-zero terms, converted from the Taylor about the origin.
The Pade approximation about the center of the interval is this:
Surprisingly, it's no better, altho at least it didn't fail this time.
This is the Chebyshev-Pade quotient, the formal transformation of the Chebyshev approximation to a, rational, expression.
This is the minimax polynomial.
Finally, rational minimax approximations would be expected to be even better than polynomials, especially when the function has non-polynomiallike things, such as singularities. However, in most cases, Maple and Mathematica fail to compute them, with the error message Error, (in numapprox/remez) error curve fails to oscillate sufficiently; try different degrees. These cases failed here: [3,3], [5,1], [4,2], [2,4]. Here is the [1,5] quotient.