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# Polar Form of Complex Numbers

## Conversion of a + bi to (a, b), (r, theta), and rcistheta.

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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=13i\begin{align*}z = -1 - \sqrt{3}i\end{align*}
• rectangular point (1,3)\begin{align*}(-1, -\sqrt{3})\end{align*}
• polar point:(2,4π3)\begin{align*}\left (2, \frac{4\pi} {3}\right )\end{align*}
•  2(cos 4π3+i sin 4π3)\begin{align*}2 \left (\mbox{cos}\ \frac{4\pi} {3} + i\ \mbox{sin}\ \frac{4\pi} {3}\right )\end{align*}
•  2 cis (4π3)\begin{align*}2\ \mbox{cis}\ \left (\frac{4\pi} {3}\right )\end{align*}
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1) Plot the complex number z=12+9i\begin{align*}z = 12 + 9i\end{align*}

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 θ\begin{align*}\theta\end{align*} be determined?
e) What is θ\begin{align*}\theta\end{align*} for this point?
f) What would z=12+9i\begin{align*} z = 12 + 9i\end{align*} look like on the polar plane?

2) What quadrant does z=3+2i\begin{align*}z = -3 + 2i\end{align*} 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. 32i\begin{align*}-3 -2i\end{align*}
2. 232i\begin{align*} 2\sqrt{3} - 2i\end{align*}
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Change to rectangular form
1. 15(cos120o+isin120o)\begin{align*}15 (cos 120^o + i sin 120^o)\end{align*}
2. 12(cosπ3+isinπ3)\begin{align*}12 \left( cos \frac{\pi}{3} + i sin \frac{\pi}{3} \right)\end{align*}
3. For the complex number in standard form x+iy\begin{align*}x + iy\end{align*} find: a) Polar Form b) Trigonometric Form (Hint: Recall thatx=rcosθ\begin{align*} x = r cos \theta\end{align*} and y=rsinθ\begin{align*}y = r sin \theta\end{align*} )
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