Question

Amplitude of $\frac{1+\sqrt{3}i}{\sqrt{3}+1}$ is :

• A. $\frac{\pi }{6}$
• B. $\frac{\pi }{4}$
• C. $\frac{\pi }{3}$
• D. $\frac{\pi }{2}$

Answer 1

Answer: (C)

Let $r\left(cos\theta +isin\theta \right)$
$=\frac{1+i\sqrt{3}}{\sqrt{3}+1}=\frac{1}{\sqrt{3}+1}+i\frac{\sqrt{3}}{\sqrt{3}+1}$
$\Rightarrow rcos\theta =\frac{1}{\sqrt{3}+1};rsin\theta =\frac{\sqrt{3}}{\sqrt{3}+1}$
$\Rightarrow tan\theta =\sqrt{3}\Rightarrow \theta =\frac{\pi }{3}.$