Answer
$\theta = 30.0^{\circ}$
Work Step by Step
$L= \sqrt{12}~\hbar$
$L_z = 3~\hbar$
We can find the value for $\theta$:
$cos~\theta = \frac{L_z}{L}$
$cos~\theta = \frac{3~\hbar}{\sqrt{12}~\hbar}$
$cos~\theta = \frac{3}{\sqrt{12}}$
$\theta = cos^{-1}~(\frac{3}{\sqrt{12}})$
$\theta = 30.0^{\circ}$
