Finding argument of a complex number

Question

How do you evaluate the following

$$\text{Arg}\{\sin\frac{8\pi}{5} + i(1 + \cos\frac{8\pi}{5})\}$$

Answer 1

$\displaystyle1+\cos\frac{8\pi}5=1-\cos\frac{3\pi}5>0$

and $\displaystyle\sin\frac{8\pi}5=\sin\left(\pi+\frac{3\pi}5\right)=-\sin\frac{3\pi}5<0$

So using the definition of atan2, $$\text{Arg}\left[\sin\frac{8\pi}{5} + i(1 + \cos\frac{8\pi}{5})\right]=\pi+\arctan\left(\frac{1-\cos\frac{3\pi}5}{-\sin\frac{3\pi}5}\right)$$

Now, $\displaystyle\frac{1-\cos2A}{-\sin2 A}=\frac{2\sin^2A}{-2\sin A\cos A}=-\tan A=\tan(-A)$

Here $\displaystyle2A=\frac{3\pi}5$

$$\implies \text{Arg}\{\sin\frac{8\pi}{5} + i(1 + \cos\frac{8\pi}{5})\}=\pi+\left(-\frac{3\pi}{10}\right)=\frac{7\pi}{10}$$