Consider the polynomial $$ p(x) = -514-462 x+359 x^2+1129 x^3+165 x^4+490 x^5-418 x^6+497 x^7-227 x^8+60 x^9-10 x^{10}, $$ whose root $A\approx 3.14$ is very close to $\pi$: $$|A-\pi|=2.0746\times 10^{-33}.$$
Let's say I am given as input the eleven integer coefficients above, and nothing else. Is there a way to do calculate $\sin A$ that doesn't use multiple-precision floating-point arithmetic, just the regular double-precision arithmetic? (With multiple-precision arithmetic this is trivial, so I would like to know if there are any methods or ideas that could apply here with this restriction in place.)