Write the argument of $(1+i \sqrt{3})(1+i)(\cos \theta+i \sin \theta)$.
Disclaimer: There is a misprinting in the question. It should be $(1+i \sqrt{3})$ instead of $(1+\sqrt{3})$.
Let the argument of $(1+i \sqrt{3})$ be $\alpha$. Then,
$\tan \alpha=\frac{\sqrt{3}}{1}=\tan \frac{\pi}{3}$
$\Rightarrow \alpha=\frac{\pi}{3}$
Let the argument of $(1+i)$ be $\beta$. Then,
$\tan \beta=\frac{1}{1}=\tan \frac{\pi}{4}$
$\Rightarrow \beta=\frac{\pi}{4}$
Let the argument of $(\cos \theta+i \sin \theta)$ be $y$. Then,
$\tan \gamma=\frac{\sin \theta}{\cos \theta}=\tan \theta$
$\Rightarrow \gamma=\theta$
$\therefore$ The argument of $(1+i \sqrt{3})(1+i)(\cos \theta+i \sin \theta)=\alpha+\beta+\gamma=\frac{\pi}{3}+\frac{\pi}{4}+\theta=\frac{7 \pi}{12}+\theta$
Hence, the argument of $(1+i \sqrt{3})(1+i)(\cos \theta+i \sin \theta)$ is $\frac{7 \pi}{12}+\theta$.
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