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$z=\sqrt{2}\left(\cos 135°+i\sin {135}^{\circ }\right)$

Step-by-step solution

Step 1. Given information

The complex number is given as$-1+i$

Step 2. Plotting point

Let $x=-1\&y=1$

Therefore the plot is

Step 3. conversion to polar form

The radius is

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

The argument is

$$\begin{gathered} \theta ={\tan }^{-1}\left(\frac{y}{x}\right) \\ ={\tan }^{-1}\left(\frac{1}{(-1)}\right) \\ \left.=135°\text{(as}0\le \theta <360°\right) \end{gathered}$$

The polar coordinate is $(\sqrt{2},{135}^{\circ })$

The polar form is$z=\sqrt{2}\left(\cos 135°+i\sin {135}^{\text{'}}\right)$