Question

The centre of a regular polygon of n sides is located at the point z=0 and one of its vertex $z_1$ is known. If $z_2$ be the vertex adjacent to $z_1$, then $z_2$ is equal to

• A. $z_1(cos\frac{2\pi }{n}\pm isin\frac{2\pi }{n})$
• B. $z_1(cos\frac{\pi }{n}\pm isin\frac{\pi }{n})$
• C. $z_1(cos\frac{\pi }{2n}\pm isin\frac{\pi }{2n})$
• D. None of these

Answer 1

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

Let A be the vertex with affix $z_1$. There are two possibilities of $z_2$, i.e., $z_2$ can be obtained by rotating $z_1$ through $\frac{2\pi }{n}$ either in clockwise or in anticlockwise direction.


$\therefore \frac{z_2}{z_1}=\left|\begin{array}{c}\frac{z_2}{z_1}\end{array}\right|e^{\pm \frac{i2\pi }{n}}\Rightarrow z_2=z_1{e}^{\pm \frac{i2\pi }{n}},(\because |z_2|=|z_1|)$
$\Rightarrow z_2=z_1(cos\frac{2\pi }{n}\pm isin\frac{2\pi }{n})$