I want to consider the dimension of $\mathbb{Q}$ vector space generated by $\{sin(n \pi/2N):n\in \mathbb{Z}\}$ where $N$ is a fix integer. My intuition is that if $N$ is not divisible by 3 then all possible generators are independent and if $N$ is divisible by 3 then we lose one generator since $sin(\frac{\pi}{6})=\frac{1}{2}$.
I want to relate this to cyclotomic field extension. Thanks