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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: 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 ) 

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?


Change to polar form

  1. -3 -2i
  2.  2\sqrt{3}  - 2i
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 )
Click here for answers.

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