Question

The amplitude of \[\frac{1 + i\sqrt{3}}{\sqrt{3} + i}\] is 

Options

• \[\frac{\pi}{3}\]

• \[- \frac{\pi}{3}\]

• \[\frac{\pi}{6}\]

• \[- \frac{\pi}{6}\]

Answer 1

\[\frac{\pi}{6}\]

\[\text { Let }z = \frac{1 + i\sqrt{3}}{\sqrt{3} + i}\]

\[ \Rightarrow z=\frac{1 + i\sqrt{3}}{\sqrt{3} + i}\times\frac{\sqrt{3} - i}{\sqrt{3} - i}\]

\[ \Rightarrow z=\frac{\sqrt{3} + 2i - \sqrt{3} i^2}{3 - i^2}\]

\[ \Rightarrow z=\frac{\sqrt{3} + \sqrt{3} + 2i}{4}\]

\[ \Rightarrow z = \frac{2\sqrt{3} + 2i}{4}\]

\[ \Rightarrow z = \frac{\sqrt{3}}{2} + \frac{1}{2}i\]

\[\tan \alpha = \left| \frac{Im(z)}{Re(z)} \right|\]

\[ = \frac{1}{\sqrt{3}}\]

\[ \Rightarrow \alpha = \frac{\pi}{6}\]

\[\text { Since, z lies in the first quadrant } . \]

\[\text{Therefore,} arg(z)=\tan^{- 1}\left( \frac{1}{\sqrt{3}} \right)=\frac{\pi}{6}\]