Question

The amplitude of $(1+i{)}^5$ is

• A. $\frac{3\pi }{4}$
• B. $\frac{-3\pi }{4}$
• C. $\frac{-5\pi }{4}$
• D. $\frac{5\pi }{4}$

Answer 1

Answer: (D)

Given, ${\left(1+i\right)}^5$
$={\left(\sqrt{2}\right)}^5{\left(\frac{1}{\sqrt{2}}+\frac{i}{\sqrt{2}}\right)}^5$
$={\left(\sqrt{2}\right)}^5{\left(\cos \frac{\pi }{4}+i\sin \frac{\pi }{4}\right)}^5$
$={\left(\sqrt{2}\right)}^5\left(\cos \frac{5\pi }{4}+i\sin \frac{5\pi }{4}\right)$
[by De-Moivre’s theorem]
Now, amplitude $={\tan }^{-1}\left(\frac{y}{x}\right)$
$={\tan }^{-1}\left(\frac{\sin 5\pi /4}{\cos 5\pi /4}\right)=\frac{5\pi }{4}$