Question

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

• A. $\frac{11\pi}{8},2\,\cos\left(\frac{\pi}{18}\right) $
• B. $-\frac{7\pi}{18},-2\,\cos\left(\frac{11\pi}{18}\right) $
• C. $\frac{2\pi}{9},2\,\cos\left(\frac{7\pi}{18}\right) $
• D. $-\frac{\pi}{9},-2\,\cos\left(\frac{\pi}{18}\right) $

Answer 1

Answer: (B)

$z=2\,cos^{2} \frac{11\pi}{18}+2i\,sin \frac{11\pi}{18}\, cos\, \frac{11\pi}{18}$
$=2\,cos\, \frac{11\pi}{18}\, cis \left(\frac{11\pi}{18}\right)$
But $\frac{11\pi}{18}$ is in the $Ilnd$ quadrant
$\therefore cos \frac{11\pi}{18} < 0$
$\therefore z=-2\,cos\left(\frac{11\pi}{18}\right)cis\left(\frac{11\pi}{18}-\pi\right)$
$= -2\,cos\left(\frac{11\pi}{18}\right)cis\left(-\frac{7\pi}{18}\right)$
$\therefore Arg z=-\frac{7\pi}{18}$
i.e., $\left|z\right|=-2\,cos\left(\frac{\pi}{18}\right)$