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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  
  • rectangular point (1,3)  
  • polar point:(2,4π3) 
  •  2(cos 4π3+i sin 4π3) 
  •  2 cis (4π3) 
.

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 θ be determined?
e) What is θ 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. 32i
  2. 232i
.
Change to rectangular form
  1. 15(cos120o+isin120o)
  2. 12(cosπ3+isinπ3)
  3. For the complex number in standard form x+iy find: a) Polar Form b) Trigonometric Form (Hint: Recall thatx=rcosθ and y=rsinθ )
.
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