Question

Find the modulus and argument of the following complex number and hence express in the polar form:

\[\sqrt{3} + i\]

Answer 1

\[ z = \sqrt{3} + i\]

\[r = \left| z \right|\]

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

\[ = \sqrt{4}\]

\[ = 2\]

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

\[ \Rightarrow \tan \alpha = \left( \frac{1}{\sqrt{3}} \right)\]

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

\[\text { Since point } (\sqrt{3}, 1) \text { lies in the first quadrant, the argument of z is given by } \]

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

\[\text { Polar form } = r \left( \cos\theta + i\sin\theta \right) \]

                       \[ = 2 \left( \cos \frac{\pi}{6} + i\sin\frac{\pi}{6} \right)\]