Question

Find the principal argument of the complex number $sin\frac{6\pi }{5}+i(1+cos\frac{6\pi }{5})$.

• A. $arg\left(z\right)=\frac{9\pi }{10},\left|z\right|=-2cos\frac{3\pi }{5}$
• B. $arg\left(z\right)=\frac{\pi }{10},\left|z\right|=-2cos\frac{3\pi }{5}$
• C. $arg\left(z\right)=\frac{9\pi }{10},\left|z\right|=2cos\frac{3\pi }{5}$
• D. $arg\left(z\right)=\frac{9\pi }{10},\left|z\right|=-2cos\frac{2\pi }{5}$

Answer 1

The correct option is A $arg\left(z\right)=\frac{9\pi }{10},\left|z\right|=-2cos\frac{3\pi }{5}$

$z=sin\frac{6\pi }{5}+i(1+cos\frac{6\pi }{5})$
$=2cos\frac{3\pi }{5}(sin\frac{3\pi }{5}+icos\frac{3\pi }{5})$
$=2cos\frac{3\pi }{5}(cos(\frac{\pi }{2}-\frac{3\pi }{5})+isin(\frac{\pi }{2}-\frac{3\pi }{5}))$
$=2cos\frac{3\pi }{5}[cos\left(\frac{-\pi }{10}\right)+isin\left(\frac{\pi }{10}\right)]$
$=-2cos\frac{3\pi }{5}[-cos\frac{\pi }{10}+isin\frac{\pi }{10}]$
$=-2cos\frac{3\pi }{5}[cos(\pi -\frac{\pi }{10})+isin(\pi -\frac{\pi }{10})]$
$=-2cos\frac{3\pi }{5}[cos\frac{9\pi }{10}+isin\frac{9\pi }{10}]$
$\therefore arg\left(Z\right)=\frac{9\pi }{10}$;
here $\left|z\right|=-2cos\frac{3\pi }{5}$
Ans: A