Calculate the Taylor polynomials $T_2(x)$ and $T_3(x)$ centered at $x = \frac{\pi }{2}$ for $f(x) = \sin\!\left(x\right)$.

$T_2(x)$ must be of the form $$A + B(x-\frac{\pi }{2}) + C(x-\frac{\pi }{2})^2$$ where
$A$ equals:
$B$ equals: and
$C$ equals:

$T_3(x)$ must be of the form $$D + E(x-\frac{\pi }{2}) + F(x-\frac{\pi }{2})^2 + G(x-\frac{\pi }{2})^3$$ where
$D$ equals:
$E$ equals:
$F$ equals: and
$G$ equals: