I am having trouble understanding one of the inequalities involved in the proof of the Pólya-Vinogradov inequality, more precisely
$$ \left \vert{\frac{\sqrt p} p \sum_{a \mathop = 1}^{p-1} \frac{e^{\pi i a n / p} \sin \left({\pi a n / p}\right)} {e^{\pi i a / p} \sin \left({\pi a / p}\right)} }\right \vert \le \frac{\sqrt p} p \sum_{a \mathop = 1}^{p-1} \left \vert{\frac 1 {\sin \left({\pi \left \langle {a / p}\right \rangle}\right)} }\right \vert$$
where $\left\langle{x}\right\rangle$ denotes the absolute value of the difference between $x$ and the closest integer to $x$.
I think is using the triangle inequality, and as $\left\vert \frac{e^{\pi i a n / p}}{e^{\pi i a / p}} \right \vert =1$ it only remains to justify that
$$\left\vert \frac{sin( \pi an/p)}{sin( \pi a/p)} \right \vert \le \left\vert \frac{1}{\sin \left({\pi \left \langle {a / p}\right \rangle}\right)} \right \vert $$
but I can't see where the $\left \langle {a / p}\right \rangle$ comes up.
Thanks very much, any idea will be appreciated.