Question

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

 sin 120° - i cos 120° 

Answer 1

 sin 120° - i cos 120° 

\[ \frac{\sqrt{3}}{2} + \frac{i}{2}\]

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

\[ = \sqrt{\frac{3}{4} + \frac{1}{4}}\]

\[ = 1\]

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

\[\text { Then }, \tan \alpha = \left| \frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}} \right|\]

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

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

\[\text { Since point } \left( \frac{\sqrt{3}}{2}, \frac{1}{2} \right) \text { lies in the first quadrant, the argument is given by } \]

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

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

\[ = \cos \frac{\pi}{6} + i \sin \frac{\pi}{6}\]