Question

If $arg\left(z\right)=\theta$ then $arg\left(\overline{z}\right)$ equal to

Answer 1

Find the value of :

Given $arg\left(z\right)=\theta$

Let $z=r\cos \theta +i\sin \theta$

then $\overline{z}=r\cos \theta -i\sin \theta$

$$\begin{gathered} =r\cos (-\theta )-i\sin (-\theta )(-1)\mathbf{}\mathbf{\left(}\mathbf{\because }\mathbf{}\mathbf{c}\mathbf{o}\mathbf{s}\mathbf{(}\mathbf{-}\mathbf{\theta }\mathbf{)}\mathbf{=}\mathbf{c}\mathbf{o}\mathbf{s}\mathbf{\theta }\mathbf{}\mathbf{;}\mathbf{}\mathbf{s}\mathbf{i}\mathbf{n}\mathbf{(}\mathbf{-}\mathbf{\theta }\mathbf{)}\mathbf{=}\mathbf{-}\mathbf{s}\mathbf{i}\mathbf{n}\mathbf{\theta }\mathbf{\right)} \\ =r\cos (-\theta )+i\sin (-\theta ) \end{gathered}$$

$\Rightarrow arg\left(\overline{z}\right)=-\theta$

Hence the$\mathit{a}\mathit{r}\mathit{g}\mathbf{\left(}\overline{\mathbf{z}}\mathbf{\right)}$ is $\mathbf{-}\mathit{\theta }$.