Question

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

• A. $ \cos \,(\pi /4)+i\,sin\,(\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) $