>>14124008In general it will be very difficult without a calculator (or a plotter).
But in specific cases like sin(x), the coefficients rapidly decay so the error is almost completely contained in the first dropped term (you can get nice bounds as well).
So if you use take the degree-5 polynomial, your first dropped term is x^7/5040. If you set that =10^-6, you get a=.469652. The exact result is a=.469857, so that works pretty well here.
But this can't work in general. For example, trying to approximate the function sin(x)+100000*x^11 with anything less than the degree-11 Taylor series will not work well at all, and just looking at the first few coefficients won't tell you that.
The usual error bounds you get from Lagrange etc. are almost always very loose.