Question

In Problems $11–22$,plot each complex number in the complex plane and write it in polar form. Express the argument in degrees.

$1-\sqrt{3}i$

Short answer

The plot is

The polar form is$z=2\left(\cos 300°+i\sin 300°\right)$

Step-by-step solution

Step 1. Given information

Complex number is given as$1-\sqrt{3}i$

Step 2. Plotting point

Let $x=1\&y=-\sqrt{3}$

Therefore the plot is

Step 3. Conversion to polar form

The radius is

$$\begin{gathered} r=\sqrt{x^2+y^2} \\ =\sqrt{1^2+(-\sqrt{3}{)}^2} \\ =2 \end{gathered}$$

The argument is

$$\begin{gathered} \theta ={\tan }^{-1}\left(\frac{y}{x}\right) \\ ={\tan }^{-1}\left(\frac{(-\sqrt{3})}{1}\right) \\ =300°\left(\text{as}0\le \theta <360°\right) \end{gathered}$$

The polar coordinate is $(2,{300}^{\circ })$

The polar form is$z=2\left(\cos 300°+i\sin 300°\right)$