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)

$-\frac{7\pi}{18},-2\,\cos\left(\frac{11\pi}{18}\right)$

Answer 2

$z=2\,cos^{2} \frac{11\pi}{18}+2i\,sin \frac{11\pi}{18}\, cos\, \frac{11\pi}{18}$ $=2\,cos\, \frac{11\pi}{18}\, cis \left(\frac{11\pi}{18}\right)$ 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., $\left|z\right|=-2\,cos\left(\frac{\pi}{18}\right)$