Answer
$\theta = 150^{\circ}$
Work Step by Step
For each $l$, the values of $m_l$ can be $~~m_l = 0, \pm 1, \pm 2,...,\pm l$
When $l = 3,$ the smallest possible value of $m_l$ is $~~m_l = -3$
$L= \sqrt{12}~\hbar$
$L_z = m_l~\hbar = -3~\hbar$
We can find the value for $\theta$:
$cos~\theta = \frac{L_z}{L}$
$cos~\theta = \frac{-3~\hbar}{\sqrt{12}~\hbar}$
$cos~\theta = \frac{-3}{\sqrt{12}}$
$\theta = cos^{-1}~(\frac{-3}{\sqrt{12}})$
$\theta = 150^{\circ}$
