Question

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

Answer 1

$\to \sin \frac{6\pi }{5}+i(1+\cos \frac{6\pi }{5})$

$\to 2\sin \frac{3\pi }{5}\cos \frac{3\pi }{5}+2i{\cos }^2\frac{3\pi }{5}$

$\to 2\cos \frac{3\pi }{5}(\sin \frac{3\pi }{5}+i\cos \frac{3\pi }{5})$

$\to 2\cos \frac{3\pi }{5}(\sin (\frac{\pi }{2}+\frac{\pi }{10})+i\cos (\frac{\pi }{2}+\frac{\pi }{10}))$

$2\cos \frac{3\pi }{5}(\cos \left(\frac{-\pi }{10}\right)+i\sin \left(\frac{-\pi }{10}\right))$

argument$=-\frac{\pi }{10}$