$\mathrm{Arg}(\sin \frac{6\pi }{5}+i(1+\cos \frac{6\pi }{5}))=$

$=\mathrm{Arg}(\sin \frac{6\pi }{5}+i(1+\cos \frac{6\pi }{5}))$

$=-(\sin (\pi /5)+i(1-\cos \pi /5\left)\right)$

$arg\left(z\right)=\pi -{\tan }^{-1}|y/x|$

$=\pi -{\tan }^{-1}\left|\frac{1-\cos (\pi /5)}{\sin \left(\pi /5\right)}\right|$

$=\pi -{\tan }^{-1}\left|\frac{2{\sin }^2\left(\pi /10\right)}{2\sin \left(\pi /10\right)\cos \left(\pi /10\right)}\right|$

$=\pi -{\tan }^{-1}\left|\tan \left(\pi /10\right)\right|$

$=\frac{9\pi }{10}$