Polar Form and Principal Argument

Let $z$ and $w$ be complex numbers such that $\overline{z}+i\overline{w}=0$ and $\arg zw=\mathrm{\pi }$. Then $\arg z$ equals

If $\alpha , \beta$ are the roots of the equation $t^2-2t+2=0,$ a value of $x\in R$ so that $\frac{(x+\alpha {)}^n-(x+\beta {)}^n}{\alpha -\beta }=\frac{\sin n\theta }{{\sin }^n\theta }$ is

The value of $Arg\left[i\ln \left(\frac{a-ib}{a+ib}\right)\right],$ where $a$ and $b$ are real numbers, is

If $\arg \left({\overline{z}}_1\right)=\arg \left(z_2\right),$ then

The amplitude of $\sin \frac{\pi }{5}+i\left(1-\cos \frac{\pi }{5}\right)$ is

For $z=\sqrt{3}-i,$ find the principal value of $\arg \left(z\right)$

Let $z_1$ and $z_2$ be two complex numbers with $\alpha$ and $\beta$ as their principal arguments such that $\alpha +\beta >\pi ,$ then principal $arg\left(z_1{z}_2\right)$ is given by

If $z=r{e}^{i\theta },$ then $\left|e^{iz}\right|$ equals

The solutions of the equation $(1+i\sqrt{3}{)}^x-2^x=0$ form