Answer
$\theta = 24.1^{\circ}$
Work Step by Step
We can write an expression for $L$ when $l = 5$:
$L = \sqrt{l(l+1)}~~\hbar$
$L = \sqrt{(5)(5+1)}~~\hbar$
$L = \sqrt{30}~~\hbar$
For any given value of $l$, the values of $m_l$ can be $~~m_l = 0, \pm 1, \pm 2,...,\pm l$
We can write an expression for $L_z$:
$L_z = m_l~\hbar$
To find the minimum possible value for the semi-classical angle $\theta$, we should maximize $L_z$
We can find the minimum possible value for $\theta$:
$cos~\theta = \frac{L_z}{L}$
$cos~\theta = \frac{5~\hbar}{\sqrt{30}~\hbar}$
$cos~\theta = \frac{5}{\sqrt{30}}$
$\theta = cos^{-1}~(\frac{5}{\sqrt{30}})$
$\theta = 24.1^{\circ}$
