Question

If \[z = \frac{- 2}{1 + i\sqrt{3}}\],then the value of arg (z) is

विकल्प

• π

• \[\frac{\pi}{3}\]

• \[\frac{2\pi}{3}\]

• \[\frac{\pi}{4}\]

Answer 1

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

z =\[\frac{- 2}{1 + i\sqrt{3}}\]

Rationalising z, we get,

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

\[ \Rightarrow z = \frac{- 2 + i2\sqrt{3}}{1 + 3}\]

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

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

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

\[ = \sqrt{3}\]

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

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

\[\text { Therefore,}\arg (z) = \pi - \frac{\pi}{3}\]

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