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# Polar Form of Complex Numbers

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Polar Form of Complex Numbers

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### Vocabulary

##### Complete the table.
 Word Definition Reference angle ____________________________________________________________ rcisθ ____________________________________________________________

### Polar Form

There are multiple forms a number can take. These five bullets all represent the same number:

• complex number: $z = -1 - \sqrt{3}i$
• rectangular point $(-1, -\sqrt{3})$
• polar point:$\left (2, \frac{4\pi} {3}\right )$
•  $2 \left (\mbox{cos}\ \frac{4\pi} {3} + i\ \mbox{sin}\ \frac{4\pi} {3}\right )$
•  $2\ \mbox{cis}\ \left (\frac{4\pi} {3}\right )$
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1) Plot the complex number $z = 12 + 9i$

a) What is needed in order to plot this point on the polar plane?
b) How could the r-value be determined?
c) What is the r for this point?
d) How could $\theta$ be determined?
e) What is $\theta$ for this point?
f) What would $z = 12 + 9i$ look like on the polar plane?

2) What quadrant does $z = -3 + 2i$ occur in when graphed?

3) What are the coordinates of z = -3 + 2i in polar form and trigonometric form?

4) What would be the polar coordinates of the point graphed below?

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Change to polar form

1. $-3 -2i$
2. $2\sqrt{3} - 2i$
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Change to rectangular form
1. $15 (cos 120^o + i sin 120^o)$
2. $12 \left( cos \frac{\pi}{3} + i sin \frac{\pi}{3} \right)$
3. For the complex number in standard form $x + iy$ find: a) Polar Form b) Trigonometric Form (Hint: Recall that$x = r cos \theta$ and $y = r sin \theta$ )
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