Advertisements
Advertisements
Question
Find the modulus and argument of the following complex number and hence express in the polar form:
sin 120° - i cos 120°
Advertisements
Solution
sin 120° - i cos 120°
\[ \frac{\sqrt{3}}{2} + \frac{i}{2}\]
\[r = \left| z \right|\]
\[ = \sqrt{\frac{3}{4} + \frac{1}{4}}\]
\[ = 1\]
\[\text { Let } \tan \alpha = \left| \frac{Im(z)}{Re(z)} \right|\]
\[\text { Then }, \tan \alpha = \left| \frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}} \right|\]
\[ = \frac{1}{\sqrt{3}}\]
\[ \Rightarrow \alpha = \frac{\pi}{6}\]
\[\text { Since point } \left( \frac{\sqrt{3}}{2}, \frac{1}{2} \right) \text { lies in the first quadrant, the argument is given by } \]
\[\theta = \alpha = \frac{\pi}{6}\]
\[\text { Polar form } = r\left( \cos\theta + i \sin\theta \right)\]
\[ = \cos \frac{\pi}{6} + i \sin \frac{\pi}{6}\]
