If [ x ] denotes the greatest integer ≤ x, then the system of...

\left[sin \theta\right]x + \left[- cos \theta\right]y = 0$and \left[cos \theta\right]x + y = 0$ for infinite many solution
\left|$\begin{cases}\left[sin \theta\right] & \left[- cos \theta\right] \\ \left[cot \theta\right] & 1\end{cases}$\right| = 0$<br />i.e. \left[sin\theta\right] = \left[- cos\theta\right]\left[cot\theta\right] . . . \left(i\right)$
when $\theta \in \left(\dfrac{\pi}{2}, \dfrac{2\pi}{3}\right)$ then $sin\theta \in \left(0, \dfrac{1}{2}\right) ⇒ \left[sin\theta\right] = 0$ and $- cos\theta \in \left(0, \dfrac{1}{2}\right) ⇒ \left[- cos\theta\right] = 0$ and

$cot\theta \in \left(- \dfrac{1}{\sqrt{3}}, 0\right) ⇒ \left[cot\theta\right] = - 1$
When $\theta \in \left(\pi, \dfrac{7\pi}{6}\right)$ then $sin\theta \in \left(- \dfrac{1}{2}, 0\right) ⇒ \left[sin\theta\right] = - 1$ and $- cos\theta \in \left(\dfrac{\sqrt{3}}{2}, 1\right) ⇒ \left[- cos\theta\right] = 0$ and
$cot\theta \in \left(\sqrt{3}, \infty\right) ⇒ \left[cot\theta\right] \in \left\{2, 3, 4, . . . \infty\right\}$
When $\theta \in \left(\dfrac{\pi}{2}, \dfrac{2\pi}{3}\right)$ then the equation \left(i\right)$satisfied therefore infinitely many solution.<br />When$\theta \in \left(\pi, \dfrac{7\pi}{6}\right)$then the equation \left(i\right)$ not satisfied therefore unique solution.