Example 9 (1999 National High School League Question) Given $\theta=\arctan \frac{5}{12}$, then the principal value of the argument of the complex number $z=\frac{\cos 2 \theta+\mathrm{i} \sin 2 \theta}{239+\mathrm{i}}$ is
Solution: Fill in $\frac{\pi}{4}$. Reason: the principal value of the argument of $z$, $\arg z=\arg \left[(12+5 \mathrm{i})^{2}(239-\mathrm{i})\right]=\arg [(119+$ $120 \mathrm{i})(239-\mathrm{i})]=\arg (28561+28561 \mathrm{i})=\frac{\pi}{4}$.