Question

How do you find the conjugate of a complex number in polar form $?$

Answer 1

Finding the conjugate of a complex number in the polar form:

Let The complex number in the polar form with the coordinates $(r,\theta )$ is given by:

$z=(r\cos \theta )+i(r\sin \theta )$

Then the conjugate of this complex number in polar form is given as,

$\begin{array}{rcl}\overline{z}& =& (r\cos \theta )–i(r\sin \theta )\\ & =& r[\cos \theta –i\sin \theta ]\end{array}$

Conjugate of the polar form of a complex number can also be represented using exponential function as given below:

If $z=r{e}^{i\theta }$, then

$\overline{z}=r{e}^{-i\theta }$

Hence, conjugate of complex number in polar form is $\begin{array}{rcl}\overline{\mathbf{z}}\mathbf{}& \mathbf{=}& \mathbf{}\mathbf{(}\mathbf{r}\mathbf{}\mathbf{c}\mathbf{o}\mathbf{s}\mathbf{}\mathbf{\theta }\mathbf{)}\mathbf{}\mathbf{–}\mathbf{}\mathbf{i}\mathbf{(}\mathbf{r}\mathbf{}\mathbf{s}\mathbf{i}\mathbf{n}\mathbf{}\mathbf{\theta }\mathbf{)}\end{array}$ and $\overline{\mathbf{z}}\mathbf{}\mathbf{=}\mathbf{}\mathit{r}{\mathit{e}}^{\mathbf{-}\mathbf{i}\mathbf{\theta }}$