Worksheet: Operations with Complex Numbers in Polar Form

In this worksheet, we will practice performing calculations with complex numbers in polar form.

Q1:

Given that 𝑧 = 2 0 ο€» πœ‹ 2 + 𝑖 πœ‹ 2  1 c o s s i n and 𝑧 = 4 ο€» πœ‹ 6 + 𝑖 πœ‹ 6  2 c o s s i n , find 𝑧 𝑧 1 2 in polar form.

  • A 5 ο€Ό 2 πœ‹ 3 + 𝑖 2 πœ‹ 3  c o s s i n
  • B 8 0 ο€» πœ‹ 3 + 𝑖 πœ‹ 3  c o s s i n
  • C 5 ο€» πœ‹ 2 + 𝑖 πœ‹ 2  c o s s i n
  • D 5 ο€» πœ‹ 3 + 𝑖 πœ‹ 3  c o s s i n
  • E 1 6 ο€» πœ‹ 3 + 𝑖 πœ‹ 3  c o s s i n

Q2:

What do we need to do to multiply two complex numbers in polar form?

  • Aadd their moduli together and multiply their arguments
  • Badd their moduli together and add their arguments
  • Cmultiply their moduli together and multiply their arguments
  • Dmultiply their moduli together and add their arguments
  • Emultiply their moduli together and subtract their arguments

Q3:

Given that 𝑧 = 2 ο€» πœ‹ 6 + 𝑖 πœ‹ 6   c o s s i n and 𝑧 = 1 √ 3 ο€» πœ‹ 3 + 𝑖 πœ‹ 3   c o s s i n , find 𝑧 𝑧   .

  • A 2 √ 3 3 ο€Ό 1 1 πœ‹ 6 + 𝑖 1 1 πœ‹ 6  c o s s i n
  • B ο€Ώ 2 + 1 √ 3  ο€» πœ‹ 2 + 𝑖 πœ‹ 2  c o s s i n
  • C ο€Ώ 2 + 1 √ 3  ο€Ό 1 1 πœ‹ 6 + 𝑖 1 1 πœ‹ 6  c o s s i n
  • D 2 √ 3 3 ο€» πœ‹ 2 + 𝑖 πœ‹ 2  c o s s i n
  • E 2 √ 3 3 ο€» πœ‹ 2 + 𝑖 πœ‹ 2  c o s s i n  

Q4:

What is the argument of the product of 𝑧 = π‘Ÿ ( πœƒ + 𝑖 πœƒ ) 1 c o s s i n and 𝑧 = 𝑠 ( πœ‘ + 𝑖 πœ‘ ) 2 c o s s i n ?

  • A π‘Ÿ 𝑠
  • B πœƒ Γ— πœ‘
  • C π‘Ÿ + 𝑠
  • D πœƒ + πœ‘
  • E π‘Ÿ πœƒ + 𝑠 πœ‘

Q5:

Consider the complex number 𝑧 = 1 + √ 3 𝑖 .

Find the modulus of 𝑧 .

Find the argument of 𝑧 .

  • A πœ‹ 3
  • B 2 πœ‹ 3
  • C √ 1 0
  • D πœ‹ 6
  • E2

Hence, use the properties of multiplication of complex numbers in polar form to find the modulus and argument of 𝑧 3 .

  • Amodulus = √ 3 , argument= πœ‹
  • Bmodulus = √ 1 0 , argument = πœ‹ 2
  • Cmodulus = 8, argument = πœ‹
  • Dmodulus = 8, argument = πœ‹ 2
  • Emodulus = √ 1 0 , argument = πœ‹

Hence, find the value of 𝑧 3 .

Q6:

Given that 𝑧 = 3 ο€Ό 1 1 πœ‹ 6 + 𝑖 1 1 πœ‹ 6  c o s s i n , find 1 𝑧 in exponential form.

  • A 1 𝑧 = 𝑒 ο‘½ οŽ₯ 
  • B 1 𝑧 = 1 3 𝑒   ο‘½ οŽ₯ 
  • C 1 𝑧 = 3 𝑒   ο‘½ οŽ₯ 
  • D 1 𝑧 = 1 3 𝑒 ο‘½ οŽ₯ 

Q7:

Given that 𝑧 = c o s ο€Ό 7 πœ‹ 6  + 𝑖 s i n ο€Ό 7 πœ‹ 6  , find 1 𝑧 .

  • A s i n ο€Ό 5 πœ‹ 6  + 𝑖 c o s ο€Ό 5 πœ‹ 6 
  • B c o s ο€Ό 7 πœ‹ 6  + 𝑖 s i n ο€Ό 7 πœ‹ 6 
  • C c o s ο€» πœ‹ 6  + 𝑖 s i n ο€» πœ‹ 6 
  • D c o s ο€Ό 5 πœ‹ 6  + 𝑖 s i n ο€Ό 5 πœ‹ 6 

Q8:

Given that 𝑧 = 1 6 ( 4 5 + 𝑖 4 5 )  ∘ ∘  c o s s i n and 𝑧 = 2 ( βˆ’ 2 8 5 βˆ’ 𝑖 2 8 5 )  ∘ ∘  s i n c o s , find 𝑧 𝑧   .

  • A 3 2 ( 1 2 0 + 𝑖 1 2 0 ) c o s s i n ∘ ∘
  • B 8 ( 6 0 + 𝑖 6 0 ) c o s s i n ∘ ∘
  • C 3 2 ( 6 0 + 𝑖 6 0 ) c o s s i n ∘ ∘
  • D 8 ( 1 2 0 + 𝑖 1 2 0 ) c o s s i n ∘ ∘
  • E 8 ( 0 + 𝑖 0 ) c o s s i n ∘ ∘

Q9:

Given that , , , and , find .

  • A
  • B
  • C
  • D

Q10:

Given that , , and , where , find .

  • A
  • B
  • C
  • D
  • E

Q11:

Given that 𝑧 = 4 ( 4 5 + 𝑖 4 5 )  ∘ ∘ c o s s i n and 𝑧 = 6 ( 9 0 + 𝑖 9 0 )  ∘ ∘ c o s s i n , find the exponential form of 𝑧 𝑧   .

  • A 2 3 𝑒 ο‘½  
  • B 2 3 𝑒  ο‘½  
  • C 4 𝑒  ο‘½  
  • D 3 2 𝑒 ο‘½  
  • E 6 𝑒  ο‘½  

Q12:

Given that 𝑧 = 1 2 βˆ’ √ 3 2 𝑖  and 𝑧 = 2 √ 3 + 2 𝑖  , find 𝑧 𝑧   , giving your answer in exponential form.

  • A 𝑧 𝑧 = 4 𝑒     ο‘½ 
  • B 𝑧 𝑧 = 1 4 𝑒     ο‘½ 
  • C 𝑧 𝑧 = 𝑒     ο‘½ 
  • D 𝑧 𝑧 = 1 4 𝑒     ο‘½ 
  • E 𝑧 𝑧 = 1 4 𝑒    ο‘½ οŽ₯

Q13:

If 𝑍 = 7 ( πœƒ + 𝑖 πœƒ )    c o s s i n , 𝑍 = 1 6 ( πœƒ + 𝑖 πœƒ )    c o s s i n , and πœƒ + πœƒ = πœ‹   , then what is 𝑍 𝑍   ?

  • A 1 1 2 𝑖
  • B112
  • C βˆ’ 1 1 2 𝑖
  • D βˆ’ 1 1 2

Q14:

Simplify 4 ( c o s 9 0 ∘ + 𝑖 s i n 9 0 ∘ ) Γ— 5 ( c o s 8 0 ∘ + 𝑖 s i n 8 0 ∘ ) Γ— 4 ( c o s 4 5 ∘ + 𝑖 s i n 4 5 ∘ ) , giving your answer in trigonometric form.

  • A 8 0 ( c o s 1 2 5 ∘ + 𝑖 s i n 1 2 5 ∘ )
  • B 1 3 ( c o s 2 1 5 ∘ + 𝑖 s i n 2 1 5 ∘ )
  • C 8 0 ( s i n 2 1 5 ∘ + 𝑖 c o s 2 1 5 ∘ )
  • D 8 0 ( c o s 2 1 5 ∘ + 𝑖 s i n 2 1 5 ∘ )

Q15:

Given that and , find .

  • A
  • B
  • C
  • D
  • E

Q16:

Given that and , where , determine the trigonometric form of .

  • A
  • B
  • C
  • D
  • E

Q17:

Given that 𝑧 = 2 ( 5 π‘Ž + 𝑖 5 π‘Ž )  c o s s i n and 𝑧 = 4 ( 3 π‘Ž βˆ’ 𝑖 3 π‘Ž )  s i n c o s , determine 𝑧 𝑧   .

  • A 8 ( ( 2 7 0 βˆ’ 8 π‘Ž ) + 𝑖 ( 2 7 0 βˆ’ 8 π‘Ž ) ) c o s s i n ∘ ∘
  • B 6 ( ( 9 0 + 8 π‘Ž ) + 𝑖 ( 9 0 + 8 π‘Ž ) ) c o s s i n ∘ ∘
  • C 6 ( ( 9 0 βˆ’ 8 π‘Ž ) + 𝑖 ( 9 0 βˆ’ 8 π‘Ž ) ) c o s s i n ∘ ∘
  • D 8 ( ( 2 7 0 + 8 π‘Ž ) + 𝑖 ( 2 7 0 + 8 π‘Ž ) ) c o s s i n ∘ ∘
  • E 8 ( 8 π‘Ž + 𝑖 8 π‘Ž ) c o s s i n

Q18:

If and , is it true that ?

  • Ayes
  • Bno

Q19:

Given that 𝑧 = 5 ο€Ό 5 πœ‹ 6 + 𝑖 5 πœ‹ 6   c o s s i n and 𝑧 = 4 ( 1 8 0 + 𝑖 1 8 0 )  ∘ ∘ c o s s i n , determine 𝑧 𝑧   .

  • A 2 0 ( 3 0 + 𝑖 3 0 ) c o s s i n ∘ ∘
  • B 9 ( 3 3 0 + 𝑖 3 3 0 ) c o s s i n ∘ ∘
  • C 9 ( 3 3 0 βˆ’ 𝑖 3 3 0 ) c o s s i n ∘ ∘
  • D 2 0 ( 3 3 0 + 𝑖 3 3 0 ) c o s s i n ∘ ∘
  • E 2 0 ο€Ί 3 3 0 + 𝑖 3 3 0  c o s s i n  ∘  ∘

Q20:

Given that 𝑧 = 2 ( ( 5 π‘Ž βˆ’ 2 𝑏 ) + 𝑖 ( 5 π‘Ž βˆ’ 2 𝑏 ) )  c o s s i n and 𝑧 = 4 ( ( 4 π‘Ž βˆ’ 3 𝑏 ) + 𝑖 ( 4 π‘Ž βˆ’ 3 𝑏 ) )  c o s s i n , find 𝑧 𝑧   .

  • A 8 ( ( 2 0 π‘Ž + 6 𝑏 ) + 𝑖 ( 2 0 π‘Ž + 6 𝑏 ) ) c o s s i n
  • B 6 ( ( 9 π‘Ž βˆ’ 5 𝑏 ) + 𝑖 ( 9 π‘Ž βˆ’ 5 𝑏 ) ) c o s s i n
  • C 6 ( ( 2 0 π‘Ž + 6 𝑏 ) + 𝑖 ( 2 0 π‘Ž + 6 𝑏 ) ) c o s s i n
  • D 8 ( ( 9 π‘Ž βˆ’ 5 𝑏 ) + 𝑖 ( 9 π‘Ž βˆ’ 5 𝑏 ) ) c o s s i n
  • E 1 2 ( ( π‘Ž + 𝑏 ) + 𝑖 ( π‘Ž + 𝑏 ) ) c o s s i n

Q21:

Given that 𝑧 = βˆ’ 1 5 0 βˆ’ 𝑖 1 5 0  ∘ ∘ s i n c o s and that 𝑧 = 2 ( 1 2 0 βˆ’ 𝑖 1 2 0 )  ∘ ∘ s i n c o s , find 𝑧 𝑧   .

  • A 2 ( 2 7 0 + 𝑖 2 7 0 ) c o s s i n ∘ ∘
  • B 3 ( 1 5 0 + 𝑖 1 5 0 ) c o s s i n ∘ ∘
  • C 3 ( 2 7 0 + 𝑖 2 7 0 ) c o s s i n ∘ ∘
  • D 2 ( 1 5 0 + 𝑖 1 5 0 ) c o s s i n ∘ ∘

Q22:

Given that 𝑧 = 7 ( s i n 3 1 5 ∘ + 𝑖 c o s 3 1 5 ∘ ) , find 𝑧  , giving your answer in exponential form.

  • A 4 9 𝑒  ο‘½  
  • B 4 9 𝑒  ο‘½  
  • C 7 𝑒  ο‘½  
  • D 4 9 𝑒  ο‘½  
  • E 1 4 𝑒  ο‘½  

Q23:

Given that 𝑧 = 3 √ 2 ( 2 2 5 βˆ’ 𝑖 2 2 5 ) c o s s i n ∘ ∘ , find 𝑧  , giving your answer in exponential form.

  • A 3 √ 2 𝑒  ο‘½  
  • B 1 8 𝑒  ο‘½  
  • C 6 √ 2 𝑒  ο‘½  
  • D 1 8 𝑒  ο‘½  

Q24:

Given that 𝑧 = 2 ( 9 0 βˆ’ 𝑖 9 0 )  ∘ ∘ c o s s i n and 𝑧 = 4 ( 3 0 + 𝑖 3 0 )  ∘ ∘ s i n c o s , find 𝑧 𝑧   , giving your answer in exponential form.

  • A 𝑧 𝑧 = 8 𝑒     ο‘½ 
  • B 𝑧 𝑧 = 8 𝑒     ο‘½ 
  • C 𝑧 𝑧 = 8 𝑒     ο‘½ οŽ₯
  • D 𝑧 𝑧 = 8 𝑒      ο‘½ οŽ₯
  • E 𝑧 𝑧 = 6 𝑒      ο‘½ οŽ₯

Q25:

If 𝑧 = 6 ( 2 2 5 + 𝑖 2 2 5 )  ∘ ∘ c o s s i n , 𝑧 = 9 0 + 𝑖 9 0  ∘ ∘ c o s s i n , and 𝑧 = 2 7 0 + 𝑖 2 7 0  ∘ ∘ c o s s i n , what is the exponential form of ( 𝑧 𝑧 𝑧 )     ?

  • A 3 6 𝑒  ο‘½  
  • B 6 𝑒 ο‘½  
  • C 6 𝑒  ο‘½  
  • D 3 6 𝑒 ο‘½  
  • E 3 6 𝑒  ο‘½  

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