De Moivre's Theorem

Let $z_1$  and  $z_2$ be nth roots of unity which subtend a right angle at the origin, then n must be of the form -

If $z=i^{\frac{1}{4}}$, Find the real and imaginary values.

Find the value of $k$ if ${\left[\frac{1+\sin \frac{\pi }{8}+i\cos \frac{\pi }{8}}{1+\sin \frac{\pi }{8}-i\cos \frac{\pi }{8}}\right]}^{\frac{8}{3}}=-k$.

The value of $\sqrt[4]{-1}$ is 

Find the value of ${\left(\frac{1+\sin \frac{\pi }{10}+i\cos \frac{\pi }{10}}{1+\sin \frac{\pi }{10}-i\cos \frac{\pi }{10}}\right)}^{10}$.

If $z={\left(\frac{\sqrt{3}}{2}+\frac{i}{2}\right)}^5+{\left(\frac{\sqrt{3}}{2}-\frac{i}{2}\right)}^5+{\left(1+i\right)}^2+{\left(1-i\right)}^2$, then  (where $i=\sqrt{-1}$)

Consider a regular $10-$gon with its vertices on the unit circle. With one vertex fixed, draw straight lines to the other $9$ vertices. Call them ${\mathrm{L}}_1, {\mathrm{L}}_2,\dots ,{\mathrm{L}}_9$ and denote their lengths by $l_1,l_2,\dots ,l_9$ respectively. Then the product $l_1{l}_2\dots l_9$ is

Let $a=\cos 1°$ and $b=\sin 1°$. We say that a real number is algebraic if it is a root of a polynomial with integer coefficients. Then,

On any given arc of positive length on the unit circle $\left|z\right|=1$ in the complex plane.

If $2\cos \frac{7\mathrm{\pi }}{5}$ is one of the values of $z^{\frac{1}{5}}$, then $\mathrm{z}=$

If the complex number $a$ is such that $\left|a\right|=1,$ and $\arg \left(a\right)=\theta$, then the roots of the equation ${\left(\frac{1+iz}{1-iz}\right)}^4=a$ are $z=$

The real part of each root of the equation ${\left(z+1\right)}^6+z^6=0$ is: (where $z$ is a complex number)

The number of solution(s) of the equation $z^2=4z+{\left|z\right|}^2\mathrm{ }+\frac{16}{{\left|z\right|}^3}$ is (where $z=x+iy, x,y\in R,i^2=–1 \mathrm{a}\mathrm{n}\mathrm{d} x\ne 2$)

If $n\in N$, then ${\left(-\sqrt{-1}\right)}^{4n+3}=$.

Let A₁, A₂, ……, A_n be the vertices of a regular polygon of n sides in a circle of radius unity and a = $|A_1{A}_2{|}^2+|A_1{A}_3{|}^2+.....|A_1{A}_n{|}^2$, b = $\left|A_1{A}_2\right|\left|A_1{A}_3\right|.....\left|A_1{A}_n\right|$, then $\frac{a}{b}$ is equal to

If $z=\cos \theta +i\sin \theta$, then imaginary part of $\sin 2°\sum _{k=1}^{15}{z}^{2k-1}$  at $\theta =2°$ is equal to $\lambda$. The value of $4\lambda$ is

If $1, \omega , {\omega }^2,\dots . {\omega }^{n-1}$ are $n, n^{th}$ roots of unity, then the then the value $\left(9-\omega \right)\left(9-{\omega }^2\right)\left(9-{\omega }^3\right)\dots . \left(9-{\omega }^{n-1}\right)$ is

If $1, z_1, z_2,\dots ,z_{n-1}$ are the $n^{th}$ roots of unity, then $\left(1-z_1\right) \left(1-z_2\right)\dots .\left(1-z_{n-1}\right)=$

If $\alpha , \beta$ are the non-real cube roots of $2$ then ${\alpha }^6+{\beta }^6=$