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. $z_{1}\left( \cos\frac{2\pi}{n} \pm i\sin\frac{2\pi}{n} \right)$
• B. $z_{1}\left( \cos\frac{\pi}{n} \pm i\sin\frac{\pi}{n} \right)$
• C. $z_{1}\left( \cos\frac{\pi}{2n} \pm i\sin\frac{\pi}{2n} \right)$
• D. None of these

Answer 1

Answer: (A)

$z_{1}\left( \cos\frac{2\pi}{n} \pm i\sin\frac{2\pi}{n} \right)$

Answer 2

Sol. 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

∴ $\frac{z_{2}}{z_{1}} = \left| \frac{z_{2}}{z_{1}} \right|e^{\frac{i2\pi}{2}}$ ⇒ $z_{2} = z_{1}e^{\frac{i2\pi}{2}}$ $(\because|z_{2}| = |z_{1}|)$

⇒ $z_{2} = z_{1}\left( \cos\frac{2\pi}{n} \pm i\sin\frac{2\pi}{n} \right)$