Question

Write −1 + i \[\sqrt{3}\] in polar form .

Answer 1

\[\text{Let z }= - 1 + \sqrt{3}i . \text { Then } , \]

\[r = \left| z \right| = \sqrt{\left[ - 1 \right]^2 + \left[ \sqrt{3} \right]^2} = 2\]

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

\[ = \sqrt{3}\]

\[ \Rightarrow \alpha = \frac{\pi}{3}\]

\[\text { Since the point representing z lies in the second quadrant . Therefore, the argument of z is given by } \]

\[\theta = \pi - \alpha\]

\[ = \pi - \frac{\pi}{3}\]

\[ = \frac{2\pi}{3}\]

\[\text { So, the polar form is } r\left( \cos\theta + i\sin\theta \right)\]

\[ \therefore z = 2\left( \cos\frac{2\pi}{3} + i\sin\frac{2\pi}{3} \right)\]