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  4. The argument of the complex number sin (6 π/5)+i(1+ cos (6 π/5)) is

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Q. The argument of the complex number $\sin \frac{6 \pi}{5}+i\left(1+\cos \frac{6 \pi}{5}\right)$ is

Complex Numbers and Quadratic Equations

Solution:

$\sin \frac{6 \pi}{5}+i\left(1+\cos \frac{6 \pi}{5}\right)$
$=-\sin \frac{\pi}{5}+i\left(1-\cos \frac{\pi}{5}\right)$
$=-2 \sin \frac{\pi}{10} \cos \frac{\pi}{10}+2 i \sin ^2 \frac{\pi}{10}$
$=2 \sin \frac{\pi}{10}\left(-\cos \frac{\pi}{10}+i \sin \frac{\pi}{10}\right)$
$=2 \sin \frac{\pi}{10}\left(\cos \frac{9 \pi}{10}+i \sin \frac{9 \pi}{10}\right)$
$\operatorname{Arg}(z)=\left(\frac{9 \pi}{10}\right)$