Question

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

$4-4i$

Short answer

The plot is

The polar form is$z=4\sqrt{2}\left(\cos 315°+i\sin 315°\right)$

Step-by-step solution

Step 1. Given information

Complex number is given as$4-4i$

Step 2. Plotting point

Let $x=4\&y=-4$

Therefore the plot is

Step 3. Conversion to polar form

The radius is

$$\begin{gathered} \left|z\right|=r=\sqrt{x^2+y^2} \\ r=\sqrt{4^2+(-4{)}^2} \\ =\sqrt{16+16} \\ =\sqrt{32} \\ =4\sqrt{2} \end{gathered}$$

The angle is

$$\begin{gathered} \sin \theta =\frac{y}{r}=\frac{-4}{4\sqrt{2}}=\frac{-1}{\sqrt{2}} \\ \cos \theta =\frac{x}{r}=\frac{4}{4\sqrt{2}}=\frac{1}{\sqrt{2}} \\ \Rightarrow \theta ={135}^{\circ } \end{gathered}$$

The polar form is$$\begin{gathered} z=r(\cos \theta +i\sin \theta ) \\ z=4\sqrt{2}\left(\cos 315°+i\sin 315°\right) \end{gathered}$$