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

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

2) What quadrant does \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?


Change to polar form

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