Worksheet: De Moivre's Theorem

In this worksheet, we will practice finding powers and roots of complex numbers and using de Moivre's theorem to simplify calculations of powers and roots.

Q1:

What is ( 1 βˆ’ 2 𝑖 ) 4 ?

  • A 5 + 1 0 𝑖
  • B βˆ’ 3 βˆ’ 4 𝑖
  • C 4 βˆ’ 8 𝑖
  • D βˆ’ 7 + 2 4 𝑖
  • E 1 βˆ’ 2 𝑖

Q2:

Use De Moivre’s theorem to find the two square roots of 9 ο€Ό 2 πœ‹ 3 + 𝑖 2 πœ‹ 3  c o s s i n .

  • A { βˆ’ 3 , 3 }
  • B  3 2 + 3 √ 3 2 𝑖 , βˆ’ 3 2 + 3 √ 3 2 𝑖 
  • C  √ 3 2 βˆ’ 1 2 𝑖 , βˆ’ √ 3 2 + 1 2 𝑖 
  • D  3 2 + 3 √ 3 2 𝑖 , βˆ’ 3 2 βˆ’ 3 √ 3 2 𝑖 
  • E  βˆ’ 1 2 βˆ’ 1 2 𝑖 , 1 2 + 1 2 𝑖 

Q3:

If 𝑍 = π‘Ÿ ( πœƒ + 𝑖 πœƒ ) c o s s i n , what is 𝑍 𝑛 ?

  • A π‘Ÿ ( 𝑛 πœƒ + 𝑖 𝑛 πœƒ ) c o s s i n
  • B π‘Ÿ ( πœƒ + 𝑖 πœƒ ) 𝑛 c o s s i n
  • C π‘Ÿ ο€½ πœƒ 𝑛 + 𝑖 πœƒ 𝑛  c o s s i n
  • D π‘Ÿ ( 𝑛 πœƒ + 𝑖 𝑛 πœƒ ) 𝑛 c o s s i n

Q4:

What is ( 1 + 𝑖 ) 1 0 ?

  • A2
  • B 1 0 𝑖
  • C 2 + 2 𝑖
  • D 3 2 𝑖
  • E 1 + 𝑖

Q5:

What is ( βˆ’ 1 βˆ’ 3 𝑖 ) 4 ?

  • A βˆ’ 1 0 + 3 0 𝑖
  • B βˆ’ 8 + 6 𝑖
  • C βˆ’ 4 βˆ’ 1 2 𝑖
  • D 2 8 βˆ’ 9 6 𝑖
  • E βˆ’ 1 βˆ’ 3 𝑖

Q6:

Use De Moivre’s theorem to find the two square roots of c o s s i n πœ‹ 3 + 𝑖 πœ‹ 3 .

  • A { 𝑖 , βˆ’ 𝑖 }
  • B  √ 3 2 + 1 2 𝑖 , 𝑖 
  • C  1 2 βˆ’ √ 3 2 𝑖 , βˆ’ 1 2 + √ 3 2 𝑖 
  • D  √ 3 2 + 1 2 𝑖 , βˆ’ √ 3 2 βˆ’ 1 2 𝑖 
  • E  1 2 + √ 2 𝑖 , βˆ’ 1 2 βˆ’ √ 2 𝑖 

Q7:

Use De Moivre’s theorem to find the two square roots of 9 ο€» πœ‹ 3 + 𝑖 πœ‹ 3  c o s s i n .

  • A { 3 𝑖 , βˆ’ 3 𝑖 }
  • B  3 √ 3 2 + 3 2 𝑖 , 3 𝑖 
  • C { βˆ’ 1 , 1 }
  • D  3 √ 3 2 + 3 2 𝑖 , βˆ’ 3 √ 3 2 βˆ’ 3 2 𝑖 
  • E  1 2 + √ 2 𝑖 , βˆ’ 1 2 βˆ’ √ 2 𝑖 

Q8:

What is ( βˆ’ 1 + 𝑖 ) 8 ?

  • A2
  • B βˆ’ 8 𝑖
  • C βˆ’ 8 + 8 𝑖
  • D16
  • E βˆ’ 1 + 𝑖

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