Worksheet: De Moivre's Theorem for Trigonometric Identities

In this worksheet, we will practice using de Moivre's theorem to obtain trigonometric identities.

Q1:

Using de Moivre’s theorem, find the exact value of ο„Έπœƒπœƒ.ο‘½οŽ‘οŠ¦οŠ­sind

  • A βˆ’ 1 6 3 5
  • B 5 1 2 3 5
  • C βˆ’ 1 , 0 2 4 3 5
  • D 1 6 3 5
  • E 1 , 0 2 4 3 5

Q2:

Which of the following is equivalent to cos3πœƒ?

  • A c o s c o s s i n   πœƒ βˆ’ 3 πœƒ πœƒ
  • B 1 βˆ’ πœƒ s i n 
  • C c o s c o s s i n   πœƒ βˆ’ 3 πœƒ πœƒ
  • D c o s c o s s i n   πœƒ + 3 πœƒ πœƒ
  • E c o s c o s s i n   πœƒ + 3 πœƒ πœƒ

Q3:

Use de Moivre’s theorem to find the exact value of ο„Έπœƒπœƒο‘½οŽ‘οŠ¦οŠ«cosd.

  • A 1 5 8
  • B1
  • C5
  • D 8 1 5
  • E0

Q4:

Express sincosοŠ¨οŠ©πœƒπœƒ in the form π‘Žπœƒ+𝑏3πœƒ+𝑐5πœƒcoscoscos, where π‘Ž,𝑏, and 𝑐 are constants to be found.

  • A 1 1 6 ( 2 πœƒ βˆ’ 5 πœƒ βˆ’ 3 πœƒ ) c o s c o s c o s
  • B 1 2 ( 2 πœƒ βˆ’ 5 πœƒ βˆ’ 3 πœƒ ) c o s c o s c o s
  • C 2 ( βˆ’ 2 πœƒ + 5 πœƒ + 3 πœƒ ) c o s c o s c o s
  • D 2 ( 2 πœƒ βˆ’ 5 πœƒ βˆ’ 3 πœƒ ) c o s c o s c o s
  • E βˆ’ 1 1 6 ( 2 πœƒ βˆ’ 5 πœƒ βˆ’ 3 πœƒ ) c o s c o s c o s

Hence, find all the solutions of coscos5πœƒ+3πœƒ=0 in the interval 0β‰€πœƒ<πœ‹. Give your answers in exact form.

  • A πœƒ = πœ‹ 8 , 3 πœ‹ 8 , πœ‹ 2 , 5 πœ‹ 8 , 7 πœ‹ 8
  • B πœƒ = πœ‹ 8 , 3 πœ‹ 8
  • C πœƒ = πœ‹ 8 , 3 πœ‹ 8 , 5 πœ‹ 8 , 7 πœ‹ 8
  • D πœƒ = πœ‹ 2
  • E πœƒ = πœ‹ 8 , 3 πœ‹ 8 , πœ‹ 2

Q5:

Use de Moivre’s theorem to express sin4πœƒ in terms of power of sinπœƒ and cosπœƒ.

  • A 4 πœƒ πœƒ + πœƒ πœƒ βˆ’ 4 πœƒ πœƒ s i n c o s s i n c o s s i n c o s    
  • B 4 πœƒ πœƒ βˆ’ πœƒ πœƒ βˆ’ 4 πœƒ πœƒ s i n c o s s i n c o s s i n c o s    
  • C s i n s i n c o s s i n c o s οŠͺ   πœƒ + 2 πœƒ πœƒ βˆ’ 4 πœƒ πœƒ
  • D 4 πœƒ πœƒ βˆ’ 4 πœƒ πœƒ s i n c o s s i n c o s  
  • E c o s s i n c o s s i n c o s οŠͺ   πœƒ + 2 πœƒ πœƒ βˆ’ 4 πœƒ πœƒ

Q6:

Use de Moivre’s theorem to express sin5πœƒ in terms of powers of sinπœƒ.

  • A s i n s i n s i n s i n 5 πœƒ = βˆ’ 4 πœƒ βˆ’ 2 0 πœƒ + 5 πœƒ  
  • B s i n s i n s i n s i n 5 πœƒ = 1 5 πœƒ βˆ’ 2 0 πœƒ + 5 πœƒ  
  • C s i n s i n s i n 5 πœƒ = 1 1 πœƒ βˆ’ 1 0 πœƒ  
  • D s i n s i n s i n 5 πœƒ = 1 6 πœƒ + 5 πœƒ 
  • E s i n s i n s i n s i n 5 πœƒ = 1 6 πœƒ βˆ’ 2 0 πœƒ + 5 πœƒ  

By considering the solutions of sin5πœƒ=0, find an exact representation for sinοŠ¨ο€»πœ‹5.

  • A s i n  ο€» πœ‹ 5  = 5 + √ 5 8
  • B s i n  ο€» πœ‹ 5  = 1 3
  • C s i n  ο€» πœ‹ 5  = 5 βˆ’ √ 5 8
  • D s i n  ο€» πœ‹ 5  = 1 0 1 1
  • E s i n  ο€» πœ‹ 5  = βˆ’ 5 + √ 3 0 2

Q7:

Express cosοŠ¬πœƒ in terms of cos6πœƒ, cos5πœƒ, cos4πœƒ, cos3πœƒ, cos2πœƒ, cosπœƒ, and any constant terms.

  • A c o s c o s c o s c o s  πœƒ = 1 3 2 6 πœƒ + 3 1 6 4 πœƒ + 1 5 3 2 2 πœƒ + 5 1 6
  • B c o s c o s c o s c o s  πœƒ = 1 6 4 6 πœƒ + 3 3 2 4 πœƒ + 1 5 6 4 2 πœƒ + 5 3 2
  • C c o s c o s c o s c o s  πœƒ = 2 6 πœƒ + 1 2 4 πœƒ + 3 0 2 πœƒ + 2 0
  • D c o s c o s c o s c o s  πœƒ = 6 πœƒ + 6 4 πœƒ + 1 5 2 πœƒ + 1 0
  • E c o s c o s c o s c o s  πœƒ = 1 1 6 6 πœƒ + 3 8 4 πœƒ + 1 5 1 6 2 πœƒ + 5 8

Q8:

Use de Moivre’s theorem to express cos5πœƒ in terms of power of cosπœƒ.

  • A 6 πœƒ βˆ’ 8 πœƒ βˆ’ 5 πœƒ c o s c o s c o s  
  • B 1 1 πœƒ βˆ’ 1 8 πœƒ βˆ’ 5 πœƒ c o s c o s c o s  
  • C 1 6 πœƒ βˆ’ 2 0 πœƒ + 5 πœƒ c o s c o s c o s  
  • D 1 6 πœƒ βˆ’ 1 2 πœƒ + 5 πœƒ c o s c o s c o s  
  • E 1 6 πœƒ + 2 0 πœƒ + 5 πœƒ c o s c o s c o s   

Q9:

Express sin6πœƒ in terms of powers of sinπœƒ and cosπœƒ.

  • A s i n c o s s i n c o s s i n c o s s i n 6 πœƒ = 6 πœƒ πœƒ βˆ’ 2 0 πœƒ πœƒ + 6 πœƒ πœƒ    
  • B s i n c o s c o s s i n c o s s i n s i n 6 πœƒ = πœƒ βˆ’ 1 5 πœƒ πœƒ + 1 5 πœƒ πœƒ βˆ’ πœƒ  οŠͺ   οŠͺ 
  • C s i n c o s s i n c o s s i n c o s s i n 6 πœƒ = βˆ’ 6 πœƒ πœƒ + 2 0 πœƒ πœƒ βˆ’ 6 πœƒ πœƒ    
  • D s i n c o s s i n c o s s i n c o s s i n 6 πœƒ = βˆ’ 6 πœƒ πœƒ βˆ’ 2 0 πœƒ πœƒ βˆ’ 6 πœƒ πœƒ    
  • E s i n c o s s i n c o s s i n c o s s i n 6 πœƒ = 6 πœƒ πœƒ + 2 0 πœƒ πœƒ + 6 πœƒ πœƒ    

Express cos6πœƒ in terms of powers of sinπœƒ and cosπœƒ.

  • A c o s c o s c o s s i n c o s s i n s i n 6 πœƒ = πœƒ βˆ’ 1 5 πœƒ πœƒ + 1 5 πœƒ πœƒ βˆ’ πœƒ  οŠͺ   οŠͺ 
  • B c o s c o s s i n c o s s i n c o s s i n 6 πœƒ = 6 πœƒ πœƒ βˆ’ 2 0 πœƒ πœƒ + 6 πœƒ πœƒ    
  • C c o s c o s c o s s i n c o s s i n s i n 6 πœƒ = πœƒ βˆ’ 1 5 πœƒ πœƒ + 1 5 πœƒ πœƒ + πœƒ  οŠͺ   οŠͺ 
  • D c o s c o s c o s s i n c o s s i n s i n 6 πœƒ = πœƒ + 1 5 πœƒ πœƒ + 1 5 πœƒ πœƒ + πœƒ  οŠͺ   οŠͺ 
  • E c o s c o s c o s s i n c o s s i n s i n 6 πœƒ = πœƒ βˆ’ 1 5 πœƒ πœƒ βˆ’ 1 5 πœƒ πœƒ βˆ’ πœƒ  οŠͺ   οŠͺ 

Hence, express tan6πœƒ in terms of powers of tanπœƒ.

  • A t a n t a n t a n t a n t a n t a n t a n 6 πœƒ = 6 πœƒ βˆ’ 2 0 πœƒ + 6 πœƒ 1 βˆ’ 1 5 πœƒ βˆ’ 1 5 πœƒ βˆ’ πœƒ    οŠͺ 
  • B t a n t a n t a n t a n t a n t a n t a n 6 πœƒ = 6 πœƒ + 2 0 πœƒ + 6 πœƒ 1 + 1 5 πœƒ + 1 5 πœƒ + πœƒ    οŠͺ 
  • C t a n t a n t a n t a n t a n t a n t a n 6 πœƒ = βˆ’ 6 πœƒ + 2 0 πœƒ βˆ’ 6 πœƒ 1 βˆ’ 1 5 πœƒ βˆ’ 1 5 πœƒ βˆ’ πœƒ    οŠͺ 
  • D t a n t a n t a n t a n t a n t a n t a n 6 πœƒ = 6 πœƒ βˆ’ 2 0 πœƒ + 6 πœƒ 1 βˆ’ 1 5 πœƒ + 1 5 πœƒ βˆ’ πœƒ    οŠͺ 
  • E t a n t a n t a n t a n t a n t a n t a n 6 πœƒ = 1 βˆ’ 1 5 πœƒ + 1 5 πœƒ βˆ’ πœƒ 6 πœƒ βˆ’ 2 0 πœƒ + 6 πœƒ  οŠͺ   

Q10:

Use de Moivre’s theorem to express tan5πœƒ in terms of power of tanπœƒ.

  • A 5 πœƒ βˆ’ 1 0 πœƒ + πœƒ 1 βˆ’ 1 0 πœƒ + 5 πœƒ t a n t a n t a n t a n t a n    οŠͺ
  • B t a n t a n t a n t a n t a n πœƒ βˆ’ πœƒ + πœƒ 1 βˆ’ πœƒ + πœƒ    οŠͺ
  • C 1 βˆ’ 1 0 πœƒ + 5 πœƒ 5 πœƒ βˆ’ 1 0 πœƒ + πœƒ t a n t a n t a n t a n t a n  οŠͺ  
  • D 5 πœƒ + 1 0 πœƒ βˆ’ πœƒ 1 + 1 0 πœƒ βˆ’ 5 πœƒ t a n t a n t a n t a n t a n    οŠͺ
  • E t a n t a n t a n t a n t a n πœƒ + πœƒ βˆ’ πœƒ 1 + πœƒ βˆ’ πœƒ    οŠͺ

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