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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=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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