Worksheet: Operations on 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=20𝜋2+𝑖𝜋2cossin and 𝑧=4𝜋6+𝑖𝜋6cossin, find 𝑧𝑧 in polar form.

  • A 5 2 𝜋 3 + 𝑖 2 𝜋 3 c o s s i n
  • B 1 6 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 8 0 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?

  • Amultiply their moduli together and add their arguments
  • Badd their moduli together and multiply their arguments
  • Cmultiply their moduli together and multiply their arguments
  • Dadd their moduli together and add their arguments
  • Emultiply their moduli together and subtract their arguments

Q3:

Given that 𝑧=2𝜋6+𝑖𝜋6cossin and 𝑧=1√3𝜋3+𝑖𝜋3cossin, find 𝑧𝑧.

  • A 2 √ 3 3 𝜋 2 + 𝑖 𝜋 2 c o s s i n
  • B 2 √ 3 3 1 1 𝜋 6 + 𝑖 1 1 𝜋 6 c o s s i n
  • C 2 + 1 √ 3 𝜋 2 + 𝑖 𝜋 2 c o s s i n
  • D 2 + 1 √ 3 1 1 𝜋 6 + 𝑖 1 1 𝜋 6 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+𝑖𝜃)cossin and 𝑧=𝑠(𝜑+𝑖𝜑)cossin?

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

Q5:

Given that 𝑧=7𝜋6+𝑖7𝜋6cossin, find 1𝑧.

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

Q6:

Given that 𝑧=16(45+𝑖45)∘∘cossin and 𝑧=2(−285−𝑖285)∘∘sincos, find 𝑧𝑧.

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

Q7:

Given that 𝑍=5(5𝜃+𝑖5𝜃)cossin, 𝑍=4𝜃+𝑖4𝜃cossin, tan𝜃=43, and 𝜃∈0,𝜋2, find 𝑍𝑍.

  • A 3 + 4 𝑖
  • B 4 + 3 𝑖
  • C 3 5 + 4 5 𝑖
  • D 4 5 + 3 5 𝑖

Q8:

Given that 𝑍=9(3𝜃+𝑖3𝜃)cossin, 𝑍=4(5𝜃+𝑖5𝜃)cossin, and sin𝜃=12, where 𝜃∈𝜋2,𝜋, find 𝑍𝑍.

  • A 3 6 𝜋 3 + 𝑖 𝜋 3 c o s s i n
  • B 9 4 2 𝜋 3 + 𝑖 2 𝜋 3 c o s s i n
  • C 9 4 𝜋 3 + 𝑖 𝜋 3 s i n c o s
  • D 3 6 2 𝜋 3 + 𝑖 2 𝜋 3 c o s s i n
  • E 9 4 𝜋 3 + 𝑖 𝜋 3 c o s s i n

Q9:

Given that 𝑧=4(45+𝑖45)∘∘cossin and 𝑧=6(90+𝑖90)∘∘cossin, find the exponential form of 𝑧𝑧.

  • A 2 3 𝑒
  • B 2 3 𝑒
  • C 3 2 𝑒
  • D 6 𝑒
  • E 4 𝑒

Q10:

If 𝑍=7(𝜃+𝑖𝜃)cossin, 𝑍=16(𝜃+𝑖𝜃)cossin, and 𝜃+𝜃=𝜋, then what is 𝑍𝑍?

  • A − 1 1 2 𝑖
  • B − 1 1 2
  • C 1 1 2 𝑖
  • D112

Q11:

Simplify 4(90+𝑖90)×5(80+𝑖80)×4(45+𝑖45)cossincossincossin∘∘∘∘∘∘, giving your answer in trigonometric form.

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

Q12:

Given that 𝑧=5(2𝑎+𝑖2𝑎)cossin and 𝑧=14(4𝑎+𝑖4𝑎)cossin, find 𝑧𝑧.

  • A 2 0 ( ( − 2 𝑎 ) + 𝑖 ( − 2 𝑎 ) ) c o s s i n
  • B 2 1 4 ( 6 𝑎 + 𝑖 6 𝑎 ) c o s s i n
  • C 5 4 8 𝑎 + 𝑖 8 𝑎 c o s s i n
  • D 5 4 ( 6 𝑎 + 𝑖 6 𝑎 ) c o s s i n
  • E 2 1 4 8 𝑎 + 𝑖 8 𝑎 c o s s i n

Q13:

Given that 𝑧=6(4𝜃+𝑖4𝜃)cossin and 𝑧=13(2𝜃+𝑖2𝜃)sincos, where 0<𝜃<90∘, determine the trigonometric form of 𝑧𝑧.

  • A 2 ( ( 9 0 − 2 𝜃 ) + 𝑖 ( 9 0 − 2 𝜃 ) ) c o s s i n ∘ ∘
  • B 2 ( 2 𝜃 + 𝑖 2 𝜃 ) c o s s i n
  • C 1 9 3 ( 2 𝜃 + 𝑖 2 𝜃 ) c o s s i n
  • D 2 ( ( 9 0 + 2 𝜃 ) + 𝑖 ( 9 0 + 2 𝜃 ) ) c o s s i n ∘ ∘
  • E 1 9 3 ( ( 9 0 + 2 𝜃 ) + 𝑖 ( 9 0 + 2 𝜃 ) ) c o s s i n ∘ ∘

Q14:

Given that 𝑧=2(5𝑎+𝑖5𝑎)cossin and 𝑧=4(3𝑎−𝑖3𝑎)sincos, determine 𝑧𝑧.

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

Q15:

Given that 𝑧=55𝜋6+𝑖5𝜋6cossin and 𝑧=4(180+𝑖180)∘∘cossin, determine 𝑧𝑧.

  • A 2 0 ( 3 0 + 𝑖 3 0 ) c o s s i n ∘ ∘
  • B 2 0 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 9 ( 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 ∘ ∘

Q16:

Given that 𝑧=2((5𝑎−2𝑏)+𝑖(5𝑎−2𝑏))cossin and 𝑧=4((4𝑎−3𝑏)+𝑖(4𝑎−3𝑏))cossin, find 𝑧𝑧.

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

Q17:

Given that 𝑧=−150−𝑖150∘∘sincos and that 𝑧=2(120−𝑖120)∘∘sincos, find 𝑧𝑧.

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

Q18:

What is the magnitude of the product of 𝑧=𝑟(𝜃+𝑖𝜃)cossin and 𝑧=𝑠(𝜑+𝑖𝜑)cossin?

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

Q19:

Given that principal argument of 𝑍=13𝜋12 and principal argument of 𝑍=3𝜋4, determine the principal argument of 𝑍𝑍.

  • A − 1 1 𝜋 1 2
  • B − 𝜋 1 2
  • C − 𝜋 6
  • D 7 𝜋 1 2

Q20:

Given that the principal argument of 𝑍=13𝜋12 and the principal argument of 𝑍=𝜋2, determine the principal argument of 12𝑍𝑍.

  • A − 5 𝜋 1 2
  • B 𝜋 1 2
  • C − 𝜋 4
  • D − 𝜋 2
  • E 𝜋

Q21:

Given that principal argument (𝑍)=5𝜋6, determine principal argument 𝑍.

  • A − 𝜋 3
  • B 𝜋 3
  • C 𝜋 6
  • D − 2 𝜋 3
  • E − 𝜋 6

Q22:

Given that 𝑧=1 and 𝑧=(3𝜃+𝑖3𝜃)cossin, find the trigonometric form of 𝑧𝑧.

  • A c o s s i n ( 2 𝜋 − 6 𝜃 ) + 𝑖 ( 2 𝜋 − 6 𝜃 )
  • B c o s s i n ( 2 𝜋 − 3 𝜃 ) + 𝑖 ( 2 𝜋 − 3 𝜃 )
  • C c o s s i n ( 𝜋 − 6 𝜃 ) + 𝑖 ( 𝜋 − 6 𝜃 )
  • D c o s s i n ( 2 𝜋 + 6 𝜃 ) + 𝑖 ( 2 𝜋 + 6 𝜃 )

Q23:

If 𝑍=𝑟(𝜃+𝑖𝜃)cossin, what is 1𝑍?

  • A 𝑟 ( ( − 𝜃 ) + 𝑖 ( − 𝜃 ) ) c o s s i n
  • B 1 𝑟 ( 𝜃 + 𝑖 𝜃 ) c o s s i n
  • C 𝑟 ( ( − 𝜃 ) − 𝑖 ( − 𝜃 ) ) c o s s i n
  • D 1 𝑟 ( ( − 𝜃 ) + 𝑖 ( − 𝜃 ) ) c o s s i n

Q24:

Given that 𝑧=13(30+𝑖30)cossin∘∘, find 1𝑧.

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

Q25:

Given that 𝑧=5𝜋6𝑎+𝑖5𝜋6𝑎cossin, find 1𝑧.

  • A c o s s i n 5 𝜋 6 𝑎 + 𝑖 5 𝜋 6 𝑎
  • B c o s s i n 7 𝜋 6 𝑎 + 𝑖 7 𝜋 6
  • C c o s s i n 7 𝜋 6 𝑎 + 𝑖 7 𝜋 6 𝑎
  • D c o s s i n 1 1 𝜋 6 𝑎 + 𝑖 1 1 𝜋 6 𝑎
  • E s i n c o s 7 𝜋 6 𝑎 + 𝑖 7 𝜋 6 𝑎

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