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Equations Using DeMoivre's Theorem

Complex roots of an equation.

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Polar Theorems

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Theorem Overview

In your own words, describe each theorem as it relates to polar equations.
Theorem Description
Product Theorem ________________________________________________________________
Quotient Theorem ________________________________________________________________
DeMoivre's Theorem ________________________________________________________________

.

Complete the theorems:

Product Theorem

r1(cosθ1+isinθ1)r2(cosθ2+isinθ2)= ____________________________________

.

Quotient Theorem

r1(cosθ1+isinθ1)r2(cosθ2+isinθ2)= ____________________________________

.

DeMoivre's Theorem

zn=[r(cosθ+isinθ)]n= ____________________________________

Where z=r(cosθ+isinθ) and let n be a positive integer.

.

 The general rule for finding the nth roots of a complex number if z=r(cosθ+isinθ) is _____________________________________________ (Hint: start wth DeMoivre's Theorem).


Practice

Multiply each pair of complex numbers. If they are not in trigonometric form, change them before multiplying.

  1. 3(cos70+isin70)3(cos85+isin85)
  2. 7(cos85+isin85)2(cos40+isin40)
  3. (32i)(1+i)

Divide each pair of complex numbers. If they are not in trigonometric form, change them before dividing.
  1. 3(cos70+isin70)3(cos85+isin85)
  2. 7(cos85+isin85)2(cos40+isin40)
  3. (32i)(1+i)
Use DeMoivre's Theorem to evaluate each expression. Write your answer in standard form.
  1. [3(cosπ4+isinπ4)]5
  2. (25i)5
  3. (2+2i)4

Find the principal fifth roots of each complex number. Write your answers in standard form.

  1. 32(cosπ4+isinπ4)
  2. 2(cosπ3+isinπ3)
  3. 32i
Solve each equation.
  1. x3=343
  2. x7=128
  3. x4+5=86

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