Use Taylor's Theorem to answer the following:

a$)$ Find a polynomial approximation for $f(x)=sin(x)$ centered at $0,$ accurate to within $0\mathrm{.}009$ for values of $x$ in the interval $[-\frac{}{2},\frac{}{2}].$ Your series should not include more terms than are necessary.

b$)$ What is the actual bound on the error in your approximation given by Taylor's Theorem? Give your answer as a decimal number accurate to at least $4$ decimal places.