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
Correct Answer: A
Solution :
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. \ \[\frac{{{z}_{2}}}{{{z}_{1}}}=\left| \frac{{{z}_{2}}}{{{z}_{1}}} \right|\,{{e}^{\pm \frac{i2\pi }{n}}}\] Þ \[{{z}_{2}}={{z}_{1}}{{e}^{\pm \frac{i2\pi }{n}}}\], \[\,(\because \,|{{z}_{2}}|\,=\,|{{z}_{1}}|)\] Þ \[{{z}_{2}}={{z}_{1}}\left( \cos \frac{2\pi }{n}\pm i\sin \frac{2\pi }{n} \right)\]You need to login to perform this action.
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