Question

The center of 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 $z_1$ and $z_2$ is equal to

• 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. None of these

Answer 1

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

Let $A$ be the vertex with affix $z_1$.

There are two possibilities and can be obtain by rotating

$z_1$ though $\frac{2\pi }{n}$ either clockwise or in anti clockwise direction.

$z_2=z_1{e}^{\pm \frac{2\pi }{n}}\{\because \left|z_2\right|=\left|z_1\right|\}$

$=z_1(\cos \frac{2\pi }{n}\pm \sin \frac{2\pi }{n})$