Question

The complex number Z= 1-(3 can also be written as?

Answer 1

Given the Complex Number:

$z=1-i\sqrt{3}$

You can identify that it is written in Rectangular Form:

$z=x+yi$

By definition, the coordinates in Rectangular Form are:

$(x,y)$

And in Polar Form:

$(r,\theta)$

In this case, you can identify that:

$\begin{gathered} x=1 \\ y=\sqrt{3} \end{gathered}$

In order to convert them to Polar Form, you need to use these formulas:

$\begin{gathered} x=r\cdot cos\theta \\ y=r\cdot sin\theta \end{gathered}$

Use this formula to find "r":

$r=\sqrt{x^2+y^2}$

Then, this is:

$r=\sqrt{1^2+(\sqrt{3})^2}=2$

You need to find the angle. You can use this formula:

$\theta=tan^{-1}(\frac{y}{x})$

Therefore the angle is:

$\theta=tan^{-1}(-\sqrt{3})+360\text{ \degree}$

$\theta=300°$

Then:

$z=2cos300\text{\degree}+2isin300\text{\degree}$

Hence, the answer is: Last option.