The argument of the complex number

Solution:

For $z=(-1+i \sqrt{3})(1+i)(\cos \theta+i \sin \theta)$

$\arg z=\arg (-1+i \sqrt{3})+\arg (1+i)+\arg (\cos \theta+i \sin \theta)$

$\arg (-1+i \sqrt{3}):-z_{1}=-1+i \sqrt{3}$

$\theta_{1}=\tan ^{-1}\left|\left(\frac{\sqrt{3}}{1}\right)\right|$

$\theta=\frac{\pi}{3}$

Since z1 lies in II quadrant

$\Rightarrow \arg z_{1}=\pi-\frac{\pi}{3}=\frac{2 \pi}{3}$

$\arg (1+i):-z_{2}=1+i$

$\theta_{2}=\tan ^{-1}\left|\frac{1}{1}\right|=\tan ^{-1} 1$

i. e $\theta_{2}=\frac{\pi}{4}$

Since z2 lies in I quadrant

$\Rightarrow \arg z_{2}=\frac{\pi}{4}$

$\arg z_{3}=\arg (\cos \theta+i \sin \theta)=\theta$

$\Rightarrow \arg z=\arg z_{1}+\arg z_{2}+\arg z_{3}$

$\arg z=\frac{2 \pi}{3}+\frac{\pi}{4}+\theta=\left|\frac{11 \pi}{12}+\theta\right|$