Question

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

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

Answer 1

Answer: (A), (C)

The correct options are
A $z_1(\cos \frac{2\pi }{n}+i\sin \frac{2\pi }{n})$
C $z_1(\cos \frac{2\pi }{n}-i\sin \frac{2\pi }{n})$

The angle subtended at the centre by two vertices is $2\pi /n$. Hence by rotating by an angle(either clockwise or anticlockwise) $2\pi /n$ we get the adjacent vertices. hence the nearest vertices are $z_1{e}^{i2\pi /n}$ and $z_1{e}^{-i2\pi /n}$.