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.

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.

Another reference, possibly easier to get, is

Giroux A., Approximation of entire functions over bounded domains. J. Approx. Theory 28 (1980), no. 1, 45-53.

which extends Batyrev's result to $L^p$ norms, $2\leq p\leq\infty$.