$\sum_{r=1}^{16}\left(\sin \frac{2 r \pi}{17}+i \cos \frac{2 r \pi}{17}\right)=$

If $1, \omega, \omega^{2}$ are three cube roots of unity,then $(1-\omega+\omega^{2})(1+\omega-\omega^{2})$ is

If $\alpha \neq 1$ is any $n^{th}$ root of unity,then $S = 1 + 3\alpha + 5\alpha^2 + \dots$ up to $n$ terms,is equal to

$\frac{(\cos \alpha + i\sin \alpha )^4}{(\sin \beta + i\cos \beta )^5} = $

Let $\omega (\neq 1)$ be a cubic root of unity. Then the minimum value of the set $\{|a+b\omega+c\omega^2|^2 : a, b, c \text{ are distinct non-zero integers}\}$ equals

If $1, \omega, \omega^2$ are the cube roots of unity,$k$ is a positive integer and $(1-\omega+\omega^2)^{3k} + (1-\omega^2+\omega)^{3k} = (1-\omega+\omega^2)^{3k+1} + (1+\omega-\omega^2)^{3k+1}$,then $k=$