Question

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

$9\sqrt{3}+9i$

Short answer

The plot is

the polar form is$z=18\left(\cos 30°+i\sin 30°\right)$

Step-by-step solution

Given information

A complex number is given as$9\sqrt{3}+9i$

Step 2. Plotting point

Let $x=9\sqrt{3}\&y=9i$

Therefore the plot is

Step 3. Conversion into polar coordinates

The radius is

$$\begin{gathered} r=\sqrt{x^2+y^2} \\ =\sqrt{(9\sqrt{3}{)}^2+(9{)}^2} \\ =18 \end{gathered}$$

And the argument is $$\begin{gathered} \theta ={\tan }^{-1}\left(\frac{y}{x}\right) \\ ={\tan }^{-1}\left(\frac{9}{9\sqrt{3}}\right) \\ =30°\left(\text{as}0\le \theta <360°\right) \end{gathered}$$

Therefore the polar coordinate is $(18,{30}^{\circ })$

The polar form is$z=18\left(\cos 30°+i\sin 30°\right)$