Worksheet: Operations with Complex Numbers in Polar Form

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

Q1:

Given that 𝑧 = 2 0 𝜋 2 + 𝑖 𝜋 2 1 c o s s i n and 𝑧 = 4 𝜋 6 + 𝑖 𝜋 6 2 c o s s i n , find 𝑧 𝑧 1 2 in polar form.

  • A 5 2 𝜋 3 + 𝑖 2 𝜋 3 c o s s i n
  • B 8 0 𝜋 3 + 𝑖 𝜋 3 c o s s i n
  • C 5 𝜋 2 + 𝑖 𝜋 2 c o s s i n
  • D 5 𝜋 3 + 𝑖 𝜋 3 c o s s i n
  • E 1 6 𝜋 3 + 𝑖 𝜋 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 𝜋 6 + 𝑖 𝜋 6 c o s s i n and 𝑧 = 1 √ 3 𝜋 3 + 𝑖 𝜋 3 c o s s i n , find 𝑧 𝑧 .

  • A 2 √ 3 3 1 1 𝜋 6 + 𝑖 1 1 𝜋 6 c o s s i n
  • B 2 + 1 √ 3 𝜋 2 + 𝑖 𝜋 2 c o s s i n
  • C 2 + 1 √ 3 1 1 𝜋 6 + 𝑖 1 1 𝜋 6 c o s s i n
  • D 2 √ 3 3 𝜋 2 + 𝑖 𝜋 2 c o s s i n
  • E 2 √ 3 3 𝜋 2 + 𝑖 𝜋 2 c o s s i n

Q4:

What is the argument of the product of 𝑧 = 𝑟 ( 𝜃 + 𝑖 𝜃 ) 1 c o s s i n and 𝑧 = 𝑠 ( 𝜑 + 𝑖 𝜑 ) 2 c o s s i n ?

  • A 𝑟 𝑠
  • B 𝜃 × 𝜑
  • C 𝑟 + 𝑠
  • D 𝜃 + 𝜑
  • E 𝑟 𝜃 + 𝑠 𝜑

Q5:

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

Find the modulus of 𝑧 .

Find the argument of 𝑧 .

  • A 𝜋 3
  • B 2 𝜋 3
  • C √ 1 0
  • D 𝜋 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
  • Bmodulus = √ 1 0 , argument = 𝜋 2
  • Cmodulus = 8, argument = 𝜋
  • Dmodulus = 8, argument = 𝜋 2
  • Emodulus = √ 1 0 , argument = 𝜋

Hence, find the value of 𝑧 3 .

Q6:

Given that 𝑧 = 3 1 1 𝜋 6 + 𝑖 1 1 𝜋 6 c o s s i n , find 1 𝑧 in exponential form.

  • A 1 𝑧 = 𝑒
  • B 1 𝑧 = 1 3 𝑒
  • C 1 𝑧 = 3 𝑒
  • D 1 𝑧 = 1 3 𝑒

Q7:

Given that 𝑧 = c o s 7 𝜋 6 + 𝑖 s i n 7 𝜋 6 , find 1 𝑧 .

  • A s i n 5 𝜋 6 + 𝑖 c o s 5 𝜋 6
  • B c o s 7 𝜋 6 + 𝑖 s i n 7 𝜋 6
  • C c o s 𝜋 6 + 𝑖 s i n 𝜋 6
  • D c o s 5 𝜋 6 + 𝑖 s i n 5 𝜋 6

Q8:

Given that 𝑧 = 1 6 ( 4 5 + 𝑖 4 5 ) ∘ ∘ c o s s i n and 𝑧 = 2 ( − 2 8 5 − 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
  • B 2 3 𝑒
  • C 4 𝑒
  • D 3 2 𝑒
  • E 6 𝑒

Q12:

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

  • A 𝑧 𝑧 = 4 𝑒
  • B 𝑧 𝑧 = 1 4 𝑒
  • C 𝑧 𝑧 = 𝑒
  • D 𝑧 𝑧 = 1 4 𝑒
  • E 𝑧 𝑧 = 1 4 𝑒

Q13:

If 𝑍 = 7 ( 𝜃 + 𝑖 𝜃 ) c o s s i n , 𝑍 = 1 6 ( 𝜃 + 𝑖 𝜃 ) c o s s i n , and 𝜃 + 𝜃 = 𝜋 , then what is 𝑍 𝑍 ?

  • A 1 1 2 𝑖
  • B112
  • C − 1 1 2 𝑖
  • D − 1 1 2

Q14:

Simplify 4 ( c o s 9 0 ∘ + 𝑖 s i n 9 0 ∘ ) × 5 ( c o s 8 0 ∘ + 𝑖 s i n 8 0 ∘ ) × 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 𝑎 + 𝑖 5 𝑎 ) c o s s i n and 𝑧 = 4 ( 3 𝑎 − 𝑖 3 𝑎 ) s i n c o s , determine 𝑧 𝑧 .

  • A 8 ( ( 2 7 0 − 8 𝑎 ) + 𝑖 ( 2 7 0 − 8 𝑎 ) ) c o s s i n ∘ ∘
  • B 6 ( ( 9 0 + 8 𝑎 ) + 𝑖 ( 9 0 + 8 𝑎 ) ) c o s s i n ∘ ∘
  • C 6 ( ( 9 0 − 8 𝑎 ) + 𝑖 ( 9 0 − 8 𝑎 ) ) c o s s i n ∘ ∘
  • D 8 ( ( 2 7 0 + 8 𝑎 ) + 𝑖 ( 2 7 0 + 8 𝑎 ) ) c o s s i n ∘ ∘
  • E 8 ( 8 𝑎 + 𝑖 8 𝑎 ) c o s s i n

Q18:

If and , is it true that ?

  • Ayes
  • Bno

Q19:

Given that 𝑧 = 5 5 𝜋 6 + 𝑖 5 𝜋 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 − 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 3 3 0 + 𝑖 3 3 0 c o s s i n ∘ ∘

Q20:

Given that 𝑧 = 2 ( ( 5 𝑎 − 2 𝑏 ) + 𝑖 ( 5 𝑎 − 2 𝑏 ) ) c o s s i n and 𝑧 = 4 ( ( 4 𝑎 − 3 𝑏 ) + 𝑖 ( 4 𝑎 − 3 𝑏 ) ) c o s s i n , find 𝑧 𝑧 .

  • A 8 ( ( 2 0 𝑎 + 6 𝑏 ) + 𝑖 ( 2 0 𝑎 + 6 𝑏 ) ) c o s s i n
  • B 6 ( ( 9 𝑎 − 5 𝑏 ) + 𝑖 ( 9 𝑎 − 5 𝑏 ) ) c o s s i n
  • C 6 ( ( 2 0 𝑎 + 6 𝑏 ) + 𝑖 ( 2 0 𝑎 + 6 𝑏 ) ) c o s s i n
  • D 8 ( ( 9 𝑎 − 5 𝑏 ) + 𝑖 ( 9 𝑎 − 5 𝑏 ) ) c o s s i n
  • E 1 2 ( ( 𝑎 + 𝑏 ) + 𝑖 ( 𝑎 + 𝑏 ) ) c o s s i n

Q21:

Given that 𝑧 = − 1 5 0 − 𝑖 1 5 0 ∘ ∘ s i n c o s and that 𝑧 = 2 ( 1 2 0 − 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
  • B 4 9 𝑒
  • C 7 𝑒
  • D 4 9 𝑒
  • E 1 4 𝑒

Q23:

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

  • A 3 √ 2 𝑒
  • B 1 8 𝑒
  • C 6 √ 2 𝑒
  • D 1 8 𝑒

Q24:

Given that 𝑧 = 2 ( 9 0 − 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
  • B 𝑧 𝑧 = 8 𝑒
  • C 𝑧 𝑧 = 8 𝑒
  • D 𝑧 𝑧 = 8 𝑒
  • E 𝑧 𝑧 = 6 𝑒

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
  • B 6 𝑒
  • C 6 𝑒
  • D 3 6 𝑒
  • E 3 6 𝑒

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