Solution
= Solution
Let $h=x-\pi/3$. At the expansion point, the function and its first three derivatives have values $1/2$, $-\sqrt3/2$, $-1/2$ and $\sqrt3/2$. <Taylor's theorem> therefore gives
$$
\boxed{\cos x=\frac12-\frac{\sqrt3}{2}h-\frac14h^2+\frac{\sqrt3}{12}h^3+O(h^4),\qquad h=x-\frac\pi3.}
$$
For real $x$, the magnitude of the omitted fourth-order remainder is at most $|h|^4/24$, because the fourth derivative is another cosine and has magnitude at most one.