Question

Write the argument of \[\left( 1 + i\sqrt{3} \right)\left( 1 + i \right)\left( \cos\theta + i\sin\theta \right)\].

Disclaimer: There is a misprinting in the question. It should be  \[\left( 1 + i\sqrt{3} \right)\]  instead of \[\left( 1 + \sqrt{3} \right)\].

Answer 1

Let the argument of \[\left( 1 + i\sqrt{3} \right)\] be α. Then,

\[\tan\alpha = \frac{\sqrt{3}}{1} = \tan\frac{\pi}{3}\]

\[ \Rightarrow \alpha = \frac{\pi}{3}\]

Let the argument of \[\left( 1 + i \right)\] be β. Then,

\[\text { tan }\beta = \frac{1}{1} = \tan\frac{\pi}{4}\]

\[ \Rightarrow \beta = \frac{\pi}{4}\]

Let the argument of \[\left( cos\theta + isin\theta \right)\] be γ. Then,

\[\text { tan }\gamma = \frac{sin\theta}{cos\theta} = \text { tan }\theta\]

\[ \Rightarrow \gamma = \theta\]

∴ The argument of 

\[\left( 1 + i\sqrt{3} \right)\left( 1 + i \right)\left( cos\theta + isin\theta \right) = \alpha + \beta + \gamma = \frac{\pi}{3} + \frac{\pi}{4} + \theta = \frac{7\pi}{12} + \theta\]

Hence, the argument of 

\[\left( 1 + i\sqrt{3} \right)\left( 1 + i \right)\left( cos\theta + isin\theta \right) is \frac{7\pi}{12} + \theta\]