A) \[\frac{3\pi }{4}\]
B) \[\frac{-3\pi }{4}\]
C) \[\frac{-5\pi }{4}\]
D) \[\frac{5\pi }{4}\]
Correct Answer: D
Solution :
Given, \[{{(1+i)}^{5}}\] \[={{(\sqrt{2})}^{5}}{{\left( \frac{1}{\sqrt{2}}+\frac{i}{\sqrt{2}} \right)}^{5}}\] \[={{(\sqrt{2})}^{5}}{{\left( \cos \frac{\pi }{4}+i\,\sin \frac{\pi }{4} \right)}^{5}}\] \[={{(\sqrt{2})}^{5}}\left( \cos \frac{5\pi }{4}+i\,\sin \frac{5\pi }{4} \right)\] [ by De-Moivre's theorem] Now, amplitude \[={{\tan }^{-1}}\left( \frac{y}{x} \right)\] \[={{\tan }^{-1}}\left( \frac{\sin \,\,5\pi /4}{\cos \,5\pi /4} \right)=\frac{5\pi }{4}\]
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