Argand Plane and Polar Representation

If $\arg \left(z\right)<0,$ then $\arg \left(-z\right)-\arg \left(z\right)=$

Let $z$and $w$be two non-zero complex numbers such that  $\left|z\right|=\left|w\right|$  and  $\arg z+\arg w=\pi$  then $z$equals –

If $z=\frac{1+i\sqrt{3}}{1-i\sqrt{3}}$ where $i=\sqrt{-1}$, then what is the argument of $z$?

Euler form of $\frac{-1-i\sqrt{3}}{2}$

In a parallelogram $\mathrm{ABCD}$, $\mathrm{AC}=2\mathrm{BD}$ and the diagonals are at right angles. If the vertices $\mathrm{B}$ and $\mathrm{D}$ are $2+4i$ and $3-5i$ then the complex number representing $\mathrm{A}$ can be

Find the modulus and argument of $z=-i-i\sqrt{3}$

Find the modulus and argument of $\frac{1}{1+i}$.

Let $z$ be a complex number. Then the angle between vectors $z$ and $-iz$ is 

The principal value of amplitude of complex number $z=\frac{\left[\sin \left({\displaystyle \frac{8\mathrm{\pi }}{7}}\right)+i\cos \left({\displaystyle \frac{8\mathrm{\pi }}{7}}\right)\right]\left(1-i\right)}{\left(3+\sqrt{3}i\right)}$, where $i=\sqrt{-1}$ is equal to

Given complex numbers in modulus argument(in degrees) form, find the real part of $z=\frac{xy}{w}$, where $x=8 \arg \left(57°\right)$, $y=6 \arg \left(38°\right)$ and $w=5 \arg \left(33°\right)$.(Give answers to two decimal point)

Amplitude of $\frac{1+i}{1-i}$ is

Let $z_1$ and $z_2$ be two non-zero complex numbers. Then

The principal argument of the complex number $z=\frac{8+4i}{1+3i}$ is equal to

A complex number is such that it satisfies $\arg \left(\frac{z^3-1}{z^3+1}\right)=\frac{\mathrm{\pi }}{2}$. Find the arc length of $z$. 

Let $Z$ is a complex number such that $\left|Z\right|=1,$ then maximum value of $\left|Z+1\right|+\left|Z^2-Z+1\right|=?$

Find the argument of $\frac{1+i}{1-\sqrt{3}i}$

The value of ${\left(\frac{1+\cos \left({\displaystyle \frac{\mathrm{\pi }}{n}}\right)+i\sin \left({\displaystyle \frac{\mathrm{\pi }}{n}}\right)}{1+\cos \left(\frac{\mathrm{\pi }}{n}\right)-i\sin \left(\frac{\mathrm{\pi }}{n}\right)}\right)}^n$

If $x^2+x+1=0$ has one root '$\alpha$'. Then find ${\left(\alpha +\frac{1}{\alpha }\right)}^2+{\left({\alpha }^2+\frac{1}{{\alpha }^2}\right)}^2+{\left({\alpha }^3+\frac{1}{{\alpha }^3}\right)}^2...{\left({\alpha }^{27}+\frac{1}{{\alpha }^{27}}\right)}^2.$

Convert the complex number $z=\frac{i-1}{\cos \frac{\pi }{3}+i \sin \frac{\pi }{3}}$ in the polar form.

Find the modulus and argument of $\frac{1+i}{1-i}$.