Question

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

$1+i$

Short answer

The plot is

The polar form is$z=\sqrt{2}\left(\cos 45°+i\sin 45°\right)$

Step-by-step solution

Step 1. Given information

Complex number is given as$1+i$

Step 2. Plotting graph

Let $x=1\&y=1$

Plot the point as,

Step 3. Conversion to polar form

The radius is

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

And the argument is

$$\begin{gathered} \sin \theta =\frac{y}{r}=\frac{1}{\sqrt{2}} \\ \cos \theta =\frac{x}{r}=\frac{1}{\sqrt{2}} \\ \Rightarrow \theta ={45}^{\circ } \end{gathered}$$

The polar coordinate is $(\sqrt{2},{45}^{\circ })$

The polar form is

$$\begin{gathered} z=r(\cos \theta +i\sin \theta ) \\ z=\sqrt{2}\left(\cos 45°+i\sin 45°\right) \end{gathered}$$