Question

if $\frac{1+2i}{2+i}=r(\cos \theta +i\sin \theta )$ then

• A. $r=1,\theta ={\tan }^{-1}\frac{3}{4}$
• B. $r=\sqrt{5},\theta ={\tan }^{-1}\frac{4}{3}$
• C. $r=1,\theta ={\tan }^{-1}\frac{4}{3}$
• D. None of these

Answer 1

The correct option is A $r=1,\theta ={\tan }^{-1}\frac{3}{4}$

$\frac{1+2i}{2+i}$
$=\frac{1}{5}\left[\right(1+2i\left)\right(2-i\left)\right]$
$=\frac{1}{5}[2-i+4i+2]$
$=\frac{1}{5}[4+3i]$
$=\frac{4}{5}+\frac{3}{5}i$
Therefore, $r=1$ and $\theta =ta{n}^{-1}\left(\frac{3}{4}\right)$
Hence, option 'A' is correct.