Analysis and Approaches for IBDP Maths Ebook 2 | Page 277

2. You are asked to investigate the regular polygon inscribed in the unit circle on an Argand
diagram.
n
Let � be the roots of the equation z � 1, z � , k �0,1, 2, , n� 1.
(a)
k
Show that
2k�
arg( �k
) � , k �0,1, 2, , n� 1.
n
Let P
k
be the points on the Argand diagram representing the complex numbers �
k
,
k �0,1, 2, , n� 1. The points P
0
, P
1
, P
2
, …, P
n-1
are joined such that a regular polygon
inscribed in the unit circle is formed.
Let d
n
be the shortest distance of the side of the polygon P0 P1 P2 P
n-1
from the origin O ,
n � 3
�
, n� .
(b) Consider the case when n � 3.
[2]
20
(1) By considering the angle OPˆ
0P 1, find d
3
.
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