The rate of polynomial approximation 
$$E_{n}(f,\mathcal{K}):=\inf\{\max_{z\in \mathcal{K}}|f(z)-P_{n}(z)|,~\text{deg}~P_{n}\leq n\}$$
to an entire function $f$ on a compact set $\mathcal{K}$ of the complex plane of positive capacity was derived by A. Batyrev in

> Batyrev, A. V., On the best approximation of analytic functions by
> polynomials.  Doklady Akad. Nauk SSSR (N.S.) 76, (1951) 173--175.

His theorem states that
$$
\operatorname { limsup } _ { n \rightarrow \infty } n \left[ E _ { n } ( f , \mathcal{K} ) \right] ^ { \rho / n } = e \rho \tau ^ { \rho } \sigma.
$$
Here $\tau$ is the capacity of $\mathcal{K}$, $\rho$ and $\sigma$ are the order and type of $f$ respectively.

Since the segment $\mathcal{K}=[-K,K]$ has capacity $K/2$, and the sine function is of order 1 and type 1, one gets that $E_{n}(\sin,\mathcal{K})$ decreases like 
$$\left(\frac{eK}{2n}\right)^{n},$$
as $n$ tends to infinity.