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Polar Form of Complex Numbers

Main definition and formulas:

  • Complex numbers can be written in rectangular form z = x + yi, representing the rectangular coordinates of the point.
  • They can be also be written in polar form z = reiθ , representing the polar coordinates of the point.
  • r represents the distance from the origin and θ represents the angle it makes with the positive x-axis.
  • Conversions:
    r
    =

    √
     
    x2 + y2
     
    θ
    =
    arctan y
    x
    		 ,
    if x > 0 (Quadrants I and IV)
      
    π + arctan y
    x
    		 ,
    if x < 0 (Quadrants II and III)
    x
    =
    r cosθ
    y
    =
    r sinθ
  • Usually, r ≥ 0, but not necessarily.
  • Usually, 0 ≤ θ < 2π , but not necessarily.
  • Euler's formula:
    eiθ = cosθ + i sinθ
  • If z = r1eiθ 1 and w = r2eiθ 2 are two complex numbers given in polar form, then we can easily multiply them:
    zw
    =
    r1eiθ 1r2eiθ 2
    =
    r1r2 ei(θ 1+θ 2)

Example 1:

Convert the following complex numbers from rectangular form to polar form: z = − √ 3 + i, w = 6√ 2 − 6√ 2i

Example 2:

Convert the following complex numbers from polar form to rectangular form: z = 4e[−2π /3]i, w = 2e[3π/4]i

Example 3:

Perform the following multiplication by first converting each of the complex numbers to polar form. Check your answer by multiplying them directly in rectangular form.
(− 1+√ 3i)(− 2√ 3− 2i)

Example 4:

Convert z = − √ 2 − √ 2i from rectangular to polar form, and w = 6 e[5π /6]i from polar to rectangular form.

Example 5:

Simplify the expression (1+i)7 by converting to polar form, performing the exponentiation, and converting back to rectangular form.

Convert the following complex number from rectangular form to polar form: w = 3√{ 2} − 3√{2 i}

  • x = 3√2, y = − 3√2, r = √{x2 + y2}
  • θ = arctan[y/x] if x is positive or θ = arctan[y/x] + π if x is negative
  • Polar Form: reiθ
  • r = √{(3√2 )2 + ( − 3√2 )2} = √{36} = 6
  • θ = arctan[( − 3√2 )/(3√2 )] = − [(π)/4] + 2π = [(7π)/4]

Polar Form: w = 6e[(7π)/4]i

Convert the following complex number from rectangular form to polar form: w = − 2√2 + i

  • x = − 2√2, y = 1, r = √{x2 + y2}
  • θ = arctan[y/x] if x is positive or θ = arctan[y/x] + π if x is negative
  • Polar Form: reiθ
  • r = √{( − 2√2 )2 + (1)2} = √9 = 3
  • θ = arctan[1/( − 2√2 )] + π = 2.8

Polar Form: w = 3e2.8i

Convert the following complex number from rectangular form to polar form: w = − √3 − √3 i

  • x = − √3, y = − √3, r = √{x2 + y2}
  • θ = arctan[y/x] if x is positive or θ = arctan[y/x] + π if x is negative
  • Polar Form: reiθ
  • r = √{( − √3 )2 + ( − √3 )2} = √6
  • θ = arctan[( − √3 )/(√3 )] + π = [(π)/4] + π = [(5π)/4]

Polar Form: w = √6 e[(5π)/4]i

Convert the following complex number from polar form to rectangular form: w = 8e[(5π)/6]i

  • x = rcosθ, y = rsinθ
  • x = 8cos([(5π)/6])
  • x = 8( − [(√3 )/2])
  • x = − 4√3
  • y = 8sin([(5π)/6])
  • y = 8([1/2])
  • y = 4

w = − 4√3 + 4i

Convert the following complex number from polar form to rectangular form: w = 6e[( − 3π)/4]i

  • x = rcosθ, y = rsinθ
  • x = 6cos( − [(3π)/4])
  • x = 6( − [(√2 )/2])
  • x = − 3√2
  • y = 6sin([(5π)/6])
  • y = 6( − [(√2 )/2])
  • y = − 3√2

w = − 3√2 − 3√2 i

Convert the following complex number from polar form to rectangular form: w = 8e[(3π)/4]i

  • x = rcosθ
  • x = 8cos([(3π)/4]) ⇒ x = 8( − [(√2 )/2]) ⇒ x = − 4√2
  • y = rsinθ
  • y = 8sin([(3π)/4]) ⇒ y = 8([(√2 )/2]) ⇒ y = 4√2

w = − 4√2 + 4√2 i

Simplify the expression (1 + i)6 by converting to polar form, performing the exponentiation, and converting back to rectangular form

  • r = √{x2 + y2}
  • θ = arctan[y/x] if x is positive or θ = arctan[y/x] + π
  • Polar Form: reiθ
  • eiθ = cosθ+ isinθ
  • r = √{12 + 12} = √2
  • θ = arctan1 = [(π)/4]
  • Polar Form: √2 e[(π)/4]i
  • Exponentiation Form: (1 + i)6 = (√2 e[(π)/4]i)6 (1 + i)6 = (√2 )6e[(6π)/4]i (1 + i)6 = 8e[(3π)/2]i

Rectangular Form: 8(cos[(3π)/2] + isin[(3π)/2]) = 8(0 + i( − 1)) = − 8i

Simplify the expression (1 − i)5 by converting to polar form, performing the exponentiation, and converting back to rectangular form

  • r = √{x2 + y2}
  • θ = arctan[y/x] if x is positive or θ = arctan[y/x] + π
  • Polar Form: reiθ
  • eiθ = cosθ+ isinθ
  • r = √{12 + ( − 1)2} = √2
  • θ = arctan( − 1) = − [(π)/4] + 2π = [(7π)/4]
  • Polar Form: √2 e[(7π)/4]i
  • Exponentiation Form: (1 − i)5 = (√2 e[(7π)/4]i)5 ⇒ (1 − i)5 = (√2 )5e[(35π)/4]i ⇒ (1 − i)5 = 4√2 e[(3π)/4]i

Rectangular Form: 4√2 (cos[(3π)/4] + isin[(3π)/4]) = 4√2 ( − [(√2 )/2] + i([(√2 )/2])) = − 4 + 4i

Perform the following multiplication by first converting each of the complex numbers to polar form. Do not multiply them directly in rectangular form: ( − √3 − i)( − 2 + 2√3 )

  • r = √{x2 + y2}
  • θ = arctan[y/x] if x is positive or θ = arctan[y/x] + π
  • Polar Form: reiθ
  • eiθ = \cosq + isinθ
  • r = √{( − √3 )2 + ( − 1)2} = 2
  • r = √{( − 2)2 + (2√3 )2} = 4
  • θ = arctan([( − 1)/( − √3 )]) + π = [(π)/6] + π = [(7π)/6]
  • θ = arctan([(2√3 )/( − 2)]) + π = − [(π)/3] + π = [(2π)/3]
  • Polar Form: (2e[(7π)/6]i)(4e[(2π)/3]i) = 8e[(7π)/6]i + [(2π)/3]i = 8e[(11π)/6]i
  • 8e[11p/6]i = 8(cos[(11π)/6] + isin[(11π)/6]) = 8([(√3 )/2] + i( − [1/2]))

4√3 − 4i

Perform the following multiplication by first converting each of the complex numbers to polar form. Do not multiply them directly in rectangular form: (2 + 2i)(3 − 3i)

  • r = √{x2 + y2}
  • θ = arctan[y/x] if x is positive or θ = arctan[y/x] + π
  • Polar Form: reiθ
  • eiθ = cosθ+ isinθ
  • r = √{(2)2 + (2)2} = 2√2, r = √{(3)2 + ( − 3)2} = 3√2
  • θ = arctan([2/2]) = [p/4], θ =  arctan([( − 3)/3]) + π = − [(π)/4] + 2p = [(7π)/4]
  • Polar Form: (2√2 e[(π)/4]i)(3√2 e[(7π)/4]i) = 12e[(π)/4]i + [(7π)/4]i = 12e2πi
  • 12e2πi = 12(cos2π+ isin2π) = 12(1 + i(0))

12

*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.

Answer

Polar Form of Complex Numbers

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Mathematics: Trigonometry

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