Answer
$\frac{\pi}{6}$
Work Step by Step
Step 1. Identify the parameters; we have
$A=7, B=-6\sqrt 3, C=13$
Step 2. To eliminate the $x'y'$ term, the rotation angle $\phi$ needs to satisfy
$cot2\phi =\frac{A-C}{B}=\frac{7-13}{-6\sqrt 3}=\frac{\sqrt 3}{3}$
Thus $2\phi =\frac{\pi}{3}$ and $\phi =\frac{\pi}{6}$
