2. Given $\theta=\arctan \frac{5}{12}$, then the principal value of the argument of the complex number $z=\frac{\cos 2 \theta+i \sin 2 \theta}{239+i}$ is $\qquad$
$$
\frac{\pi}{4}
$$
2.【Analysis and Solution】 The principal value of the argument of $z$:
$$
\begin{aligned}
\arg z & =\arg \left[(12+5 i)^{2}(239-i)\right] \\
& =\arg [(119+120 i)(239-i)] \\
& =\arg [28561+28561 i] \\
& =\frac{\pi}{4} .
\end{aligned}
$$