Question

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

$\sqrt{3}-i$

Short answer

The plot is

The polar form is$z=2\left(\cos 330°+i\sin 330°\right)$

Step-by-step solution

Step 1. Given information

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

Step 2.  Plotting point

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

Therefore the plot is

Step 3. conversion to polar form

The radius is

$$\begin{gathered} \left|z\right|=r=\sqrt{x^2+y^2} \\ r=\sqrt{(\sqrt{3}{)}^2+(-1{)}^2} \\ =\sqrt{4} \\ =2 \end{gathered}$$

and the argument is

$$\begin{gathered} \sin \theta =\frac{y}{r}=\frac{-1}{2} \\ \cos \theta =\frac{x}{r}=\frac{\sqrt{3}}{2} \\ \Rightarrow \theta ={330}^{\circ } \end{gathered}$$

The polar coordinate is $(2,{330}^{\circ })$

The polar form is$$\begin{gathered} z=r(\cos \theta +i\sin \theta ) \\ z=2\left(\cos 330°+i\sin 330°\right) \end{gathered}$$