Question

Find the vertices of a regular polygon of n sides if its centre is located at z=0 and one of its vertices $z_1$ is known.

• A. $z_1(\cos 2\pi k/n-i\sin 2\pi k/n)$. $k=1,2,......n-1.$
• B. $z_1(\cos 2\pi k/n+i\sin 2\pi k/n)$. $k=1,2,......n-1.$
• C. $z_1(\sin 2\pi k/n+i\cos 2\pi k/n)$. $k=1,2,......n-1.$
• D. $z_1(\cos \pi k/n+i\sin \pi k/n)$. $k=1,2,......n-1.$

Answer 1

The correct option is B $z_1(\cos 2\pi k/n+i\sin 2\pi k/n)$. $k=1,2,......n-1.$

Let O be the origin and $A_1$ the vertex $z_1.$ Let the vertex adjacent to $A_1$ be $A_2.$

Then $z_2=z_1{e}^{2\pi i/n}$ since $\angle A_{2}O{A}_1=2\pi /n.$

Similarly if $z_3,z_4,\cdots z_n$ are the other vertices in order, then $z_3=z_1{e}^{4\pi i/n},z_4=z_1{e}^{6\pi i/n}$ etc.

Thus all the vertices are given by

$z_{k+1}=z_1{e}^{2\pi ki/n}=z_1(\cos 2\pi k/n+i\sin 2\pi k/n),$

$k=1,2,....n-1.$
Ans: B