Q.If the Arg $\bar{z}_1$ and Arg $z_2$ are $\frac{\pi}{5}$ and $\frac{\pi}{3}$ respectively, then find (Arg $z_1$ + Arg $z_2$).

Use $\text{Arg}(\bar z)=-\text{Arg}(z)$ to get $\text{Arg}\,z_1$ from $\text{Arg}\,\bar z_1$, then add.

For any complex number $z=r\,\text{cis}\,\theta$, its conjugate is $\bar z=r\,\text{cis}(-\theta)$, so

$\text{Arg}(\bar z)=-\text{Arg}(z).$

Given $\text{Arg}\,\bar z_1=\dfrac{\pi}{5}$, this means $-\text{Arg}\,z_1=\dfrac{\pi}{5}$, i.e. $\text{Arg}\,z_1=-\dfrac{\pi}{5}$.

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