Question

The centre of regular polygon of $n$ sides is located at $z=0$ and one of its vertices is $z_1$. If $z_2$ is vertex adjacent to $z_1$, then $z_2=$

• A. $z_1(\cos \frac{2\pi }{n}\pm i\sin \frac{2\pi }{n})$
• B. $z_1(\cos \frac{\pi }{n}\pm i\sin \frac{\pi }{n})$
• C. $z_1(\cos \frac{\pi }{2n}\pm i\sin \frac{\pi }{2n})$
• D. $z_1(\cos \frac{\pi }{3n}\pm i\sin \frac{\pi }{3n})$

Answer 1

The correct option is A $z_1(\cos \frac{2\pi }{n}\pm i\sin \frac{2\pi }{n})$

Let $z_1$ represent a complex root of $x^n=a^n,a>0,a\in R.$
Then each vertex of a regular $n$ sided polygon of side length $a$ units will represent the $n$ roots of $x^n=a^n$
Also, the angle subtended by adjacent vertices at centre $=\frac{2\pi }{n}.$

Let $z_1=a{e}^{i\theta }$
$\Rightarrow$ Argument of the adjacent vertex will be $\theta \pm \frac{2\pi }{n}.$
$\therefore z_2=a{e}^{i(\theta \pm \frac{2\pi }{n})}$
$\Rightarrow z_2=a{e}^{i\theta }\cdot e^{\pm \frac{2\pi }{n}i}$
$\Rightarrow z_2=z_1(\cos \frac{2\pi }{n}\pm i\sin \frac{2\pi }{n})$