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)

$\frac {5 \pi}{4}$

Answer 2

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 $= \tan^{-1} \left(\frac{y}{x}\right)$
$= \tan^{-1} \left(\frac{\sin 5\pi /4}{\cos 5 \pi /4}\right) = \frac{5\pi}{4}$