Question

Express z1 33 - 9i in polar form.Express your answer in exact terms, using radians, where your angle is between 0 and 21 radians, inclusive.21 =

Answer 1

Answer:

$z_1=6\sqrt[]{3}\lbrack\cos (-\frac{\pi}{3})+i\sin (-\frac{\pi}{3})\rbrack$

Explanation:

The rectangular form of a complex number is generally given as;

$z=a+bi$

where;

$\begin{gathered} a=r\cos \theta \\ b=r\sin \theta \\ r=|z|=\sqrt[]{a^2+b^2} \\ \theta=\tan ^{-1}(\frac{b}{a})\text{ for a > 0} \end{gathered}$

Converting rectangular form to polar form, we'll have;

$z=r(\cos \theta+i\sin \theta)$

Given the below;

$z_1=3\sqrt[]{3}-9i$

We can see that;

$\begin{gathered} a=3\sqrt[]{3} \\ b=-9 \end{gathered}$

Let's go ahead and find r as shown below;

$\begin{gathered} r=\sqrt[]{(3\sqrt[]{3})^2+(-9)^2}=\sqrt[]{(9\times3)^{}+81}=\sqrt[]{27+81}=\sqrt[]{108} \\ r=\sqrt[]{36\times3}=\sqrt[]{36}\times\sqrt[]{3} \\ r=6\sqrt[]{3} \end{gathered}$

Let's now find theta,;

$\begin{gathered} \theta=\tan ^{-1}(\frac{-9}{3\sqrt[]{3}})=-\frac{\pi}{3} \\ \\ \end{gathered}$

If we go ahead and input the above values into our polar form equation, we'll have;

$z_1=6\sqrt[]{3}\lbrack\cos (-\frac{\pi}{3})+i\sin (-\frac{\pi}{3})\rbrack$