In most cases a Chebyshev approximation will just work. There is advice out there which says to use Clenshaw's summation formula to sum the Chebyshev polynomial

In most cases a Chebyshev approximation will just work. There is advice out there which says to use Clenshaw's summation formula to sum the Chebyshev polynomials, however I've found that expanding out the polynomial coefficients to get something like a5 x^5 + a4 x^4 + a3 x^3 to be more accurate in practice. Scaling the polynomial, as you suggest, would simply require reevaluating the coefficients. For example, a5 (2pix)^5 would become 32 a5 pi^5 x^5. In the cos case where regions of interest approach zero, divide by another polynomial with the same zeros to create limits. cos(x)/(pi^2/4 - x^2) -> 1/pi as x -> pi/2. Then approximate that function, and evaluate approx(x) \* (pi^2/4 - x^2), so you are multiplying the approximated value by something close to zero. This way you get precision in a way similar to sin(x) = x when x small.