Write the argument of
Question:

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 $(1+\sqrt{3})$.

Solution:

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 $\frac{7 \pi}{12}+\theta$.