Question

Find the arguments of each of the complex numbers.
1. $z=-1-i\sqrt{3}$
2. $z=-\sqrt{3}+i$

3. $z=1+i\sqrt{3}$

Answer 1

1. Argument of $z=-1-i\sqrt{3}$ $=-\pi +{\tan }^{-1}\left|\frac{-\sqrt{3}}{-1}\right|=-\pi +\frac{\pi }{3}=-\frac{2\pi }{3}$ [ Since the point represented by the complex number lies in the third quadrant]

2. Argument of $z=-\sqrt{3}+i$ $=\pi -{\tan }^{-1}\left|\frac{1}{-\sqrt{3}}\right|=\frac{5\pi }{6}$ [ Since the point represented by the complex number lies in the second quadrant]

3.Argument of $1+i\sqrt{3}$ $={\tan }^{-1}\left|\frac{\sqrt{3}}{1}\right|=\frac{\pi }{3}$ [ Since the point represented by the complex number lies in the first quadrant]