Is the set $\{\sin (p_{i})| p_{i} \,\,\mbox{is a prime number, for all}\,\,i\in \mathbb{N}\,\,\mbox{and}\,\, p_{i}\neq p_{j}\,\,\mbox{if}\,\,i\neq j\}$ is linearly independent when $\mathbb{R}$ is a vector space over $\mathbb{Q}$?
Thanks for any help.