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)

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