14. Let $z=\cos \theta+i\sin \theta(0<\theta<\pi)$, given that $\omega=\frac{1-(\bar{z})^{4}}{1+z^{4}},|\omega|=\frac{\sqrt{3}}{3}, \arg \omega<\frac{\pi}{2}$, find $\theta$.
14.
$$
\begin{aligned}
\omega & =\frac{1-(\cos \theta-\mathrm{i} \sin \theta)^{4}}{1+(\cos \theta+\mathrm{i} \sin \theta)^{4}}=\frac{1-\cos 4 \theta+\mathrm{i} \sin 4 \theta}{1+\cos 4 \theta+\mathrm{i} \sin 4 \theta} \\
& =\frac{2 \sin ^{2} 2 \theta+2 \mathrm{i} \sin 2 \theta \cos 2 \theta}{2 \cos ^{2} 2 \theta+2 \mathrm{i} \sin 2 \theta \cos 2 \theta}=\tan 2 \theta \frac{\sin 2 \theta+\mathrm{i} \cos 2 \theta}{\cos 2 \theta+\mathrm{i} \sin 2 \theta}=\tan 2 \theta(\sin 4 \theta+\mathrm{i} \cos 4 \theta)
\end{aligned}
$$

So \(|\tan 2 \theta|=\frac{\sqrt{3}}{3}\). Also, \(\arg \omega<\frac{\pi}{2}\),
so \(2 \theta=\frac{\pi}{6}\) or \(\frac{7}{6} \pi\), \(\theta=\frac{\pi}{12}\) or \(\frac{7}{12} \pi\).