Question

Convert $\frac{(i+1)}{(\cos (\frac{\pi }{4})-i\sin (\frac{\pi }{4})}$ in polar form?

• A. $\cos (\pi /4)+isin(\pi /4)$
• B. $\cos (\pi /2)+i\sin (\pi /2)$
• C. $\sqrt{2}(\cos (\pi /4)+i\sin (\pi /4))$
• D. $\sqrt{2}(\cos (\pi /2)+i\sin (\pi /2))$

Answer 1

Answer: (D)

$\frac{i+1}{\cos \frac{\pi }{4}-i\sin \frac{\pi }{4}}=\frac{i+1}{\frac{1}{\sqrt{2}(1-i)}}\times \frac{1+i}{1+i}$
$=\frac{\sqrt{2}{(1+i)}^2}{1-i^2}$
$=\frac{\sqrt{2}(1-1+2i)}{1+1}$
$=\frac{\sqrt{2}.2i}{2}=\sqrt{2}i$
$=\sqrt{2}\left(\cos \frac{\pi }{2}+i\sin \frac{\pi }{2}\right)$