Question

The principal value of the $\arg (z)$ and $|z|$ of the complex number $z=1+\cos \left(\frac{11 \pi}{9}\right)+i \sin \left(\frac{11 \pi}{9}\right)$ are respectively:

• A. $\frac{11 \pi}{18}, 2 \cos \frac{\pi}{18}$
• B. $-\frac{7 \pi}{18}, 2 \cos \frac{7 \pi}{18}$
• C. $\frac{2 \pi}{9}, 2 \cos \frac{7 \pi}{18}$
• D. $-\frac{\pi}{9},-2 \cos \frac{\pi}{18}$

Answer 1

Answer: (B)

$z=1+\cos \frac{11 \pi}{9}+i \sin \frac{11 \pi}{9}$
$z=1-\cos \frac{2 \pi}{9}-i \sin \frac{2 \pi}{9}$
$z=2 \sin ^2 \frac{\pi}{9}-2 i \sin \frac{\pi}{9} \cos \frac{\pi}{9}$
$z=2 \sin \frac{\pi}{9}\left(\sin \frac{\pi}{9}-i \cos \frac{\pi}{9}\right)$
$=2 \cos \left(\frac{7 \pi}{18}\right)\left(\cos \frac{7 \pi}{18}-i \sin \frac{7 \pi}{18}\right)$
$=2 \cos \left(\frac{7 \pi}{18}\right)\left(\cos \left(-\frac{7 \pi}{18}\right)+i \sin \left(\frac{-7 \pi}{18}\right)\right)$
$\arg (z)=\frac{-7 \pi}{18}$
$|z|=2 \cos \frac{7 \pi}{18}$