Worksheet: Operations on Complex Numbers in Polar Form

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

Q1:

Given that 𝑧=20ο€»πœ‹2+π‘–πœ‹2ο‡οŠ§cossin and 𝑧=4ο€»πœ‹6+π‘–πœ‹6ο‡οŠ¨cossin, find π‘§π‘§οŠ§οŠ¨ in polar form.

  • A5ο€Ό2πœ‹3+𝑖2πœ‹3cossin
  • B16ο€»πœ‹3+π‘–πœ‹3cossin
  • C5ο€»πœ‹2+π‘–πœ‹2cossin
  • D5ο€»πœ‹3+π‘–πœ‹3cossin
  • E80ο€»πœ‹3+π‘–πœ‹3cossin

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=2ο€»πœ‹6+π‘–πœ‹6ο‡οŠ§cossin and 𝑧=1√3ο€»πœ‹3+π‘–πœ‹3ο‡οŠ¨cossin, find π‘§π‘§οŠ§οŠ¨.

  • A2√33ο€»πœ‹2+π‘–πœ‹2cossin
  • B2√33ο€Ό11πœ‹6+𝑖11πœ‹6cossin
  • Cο€Ώ2+1√3ο‹ο€»πœ‹2+π‘–πœ‹2cossin
  • Dο€Ώ2+1√311πœ‹6+𝑖11πœ‹6cossin
  • E2√33ο€»πœ‹2+π‘–πœ‹2cossin

Q4:

What is the argument of the product of 𝑧=π‘Ÿ(πœƒ+π‘–πœƒ)cossin and 𝑧=𝑠(πœ‘+π‘–πœ‘)cossin?

  • Aπ‘Ÿ+𝑠
  • Bπœƒ+πœ‘
  • Cπ‘Ÿπ‘ 
  • Dπ‘Ÿπœƒ+π‘ πœ‘
  • EπœƒΓ—πœ‘

Q5:

Given that 𝑧=ο€Ό7πœ‹6+𝑖7πœ‹6cossin, find 1𝑧.

  • Acossinο€»πœ‹6+π‘–ο€»πœ‹6
  • Bcossinο€Ό5πœ‹6+𝑖5πœ‹6
  • Ccossinο€Ό7πœ‹6+𝑖7πœ‹6
  • Dsincosο€Ό5πœ‹6+𝑖5πœ‹6

Q6:

Given that 𝑧=16(45+𝑖45)∘∘cossin and 𝑧=2(βˆ’285βˆ’π‘–285)∘∘sincos, find π‘§π‘§οŠ§οŠ¨.

  • A32(60+𝑖60)cossin∘∘
  • B8(0+𝑖0)cossin∘∘
  • C8(60+𝑖60)cossin∘∘
  • D32(120+𝑖120)cossin∘∘
  • E8(120+𝑖120)cossin∘∘

Q7:

Given that 𝑍=5(5πœƒ+𝑖5πœƒ)cossin, 𝑍=4πœƒ+𝑖4πœƒοŠ¨cossin, tanπœƒ=43, and πœƒβˆˆο€»0,πœ‹2, find π‘π‘οŠ§οŠ¨.

  • A3+4𝑖
  • B4+3𝑖
  • C35+45𝑖
  • D45+35𝑖

Q8:

Given that 𝑍=9(3πœƒ+𝑖3πœƒ)cossin, 𝑍=4(5πœƒ+𝑖5πœƒ)cossin, and sinπœƒ=12, where πœƒβˆˆο€»πœ‹2,πœ‹οŸ, find π‘π‘οŠ§οŠ¨.

  • A36ο€»πœ‹3+π‘–πœ‹3cossin
  • B94ο€Ό2πœ‹3+𝑖2πœ‹3cossin
  • C94ο€»πœ‹3+π‘–πœ‹3sincos
  • D36ο€Ό2πœ‹3+𝑖2πœ‹3cossin
  • E94ο€»πœ‹3+π‘–πœ‹3cossin

Q9:

Given that 𝑧=4(45+𝑖45)∘∘cossin and 𝑧=6(90+𝑖90)∘∘cossin, find the exponential form of π‘§π‘§οŠ¨οŠ§.

  • A23π‘’οŽ¦ο‘½οŽ£οƒ
  • B23π‘’ο‘½οŽ£οƒ
  • C32π‘’ο‘½οŽ£οƒ
  • D6π‘’οŽ¦ο‘½οŽ£οƒ
  • E4π‘’οŽ¦ο‘½οŽ£οƒ

Q10:

If 𝑍=7(πœƒ+π‘–πœƒ)cossin, 𝑍=16(πœƒ+π‘–πœƒ)cossin, and πœƒ+πœƒ=πœ‹οŠ§οŠ¨, then what is π‘π‘οŠ§οŠ¨?

  • Aβˆ’112𝑖
  • Bβˆ’112
  • C112𝑖
  • D112

Q11:

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

  • A80(215+𝑖215)sincos∘∘
  • B13(215+𝑖215)cossin∘∘
  • C80(215+𝑖215)cossin∘∘
  • D80(125+𝑖125)cossin∘∘

Q12:

Given that 𝑧=5(2π‘Ž+𝑖2π‘Ž)cossin and 𝑧=14(4π‘Ž+𝑖4π‘Ž)cossin, find π‘§π‘§οŠ§οŠ¨.

  • A20((βˆ’2π‘Ž)+𝑖(βˆ’2π‘Ž))cossin
  • B214(6π‘Ž+𝑖6π‘Ž)cossin
  • C54ο€Ή8π‘Ž+𝑖8π‘Žο…cossin
  • D54(6π‘Ž+𝑖6π‘Ž)cossin
  • E214ο€Ή8π‘Ž+𝑖8π‘Žο…cossin

Q13:

Given that 𝑧=6(4πœƒ+𝑖4πœƒ)cossin and 𝑧=13(2πœƒ+𝑖2πœƒ)sincos, where 0<πœƒ<90∘, determine the trigonometric form of π‘§π‘§οŠ§οŠ¨.

  • A2((90βˆ’2πœƒ)+𝑖(90βˆ’2πœƒ))cossin∘∘
  • B2(2πœƒ+𝑖2πœƒ)cossin
  • C193(2πœƒ+𝑖2πœƒ)cossin
  • D2((90+2πœƒ)+𝑖(90+2πœƒ))cossin∘∘
  • E193((90+2πœƒ)+𝑖(90+2πœƒ))cossin∘∘

Q14:

Given that 𝑧=2(5π‘Ž+𝑖5π‘Ž)cossin and 𝑧=4(3π‘Žβˆ’π‘–3π‘Ž)sincos, determine π‘§π‘§οŠ§οŠ¨.

  • A8((270βˆ’8π‘Ž)+𝑖(270βˆ’8π‘Ž))cossin∘∘
  • B8(8π‘Ž+𝑖8π‘Ž)cossin
  • C8((270+8π‘Ž)+𝑖(270+8π‘Ž))cossin∘∘
  • D6((90βˆ’8π‘Ž)+𝑖(90βˆ’8π‘Ž))cossin∘∘
  • E6((90+8π‘Ž)+𝑖(90+8π‘Ž))cossin∘∘

Q15:

Given that 𝑧=5ο€Ό5πœ‹6+𝑖5πœ‹6cossin and 𝑧=4(180+𝑖180)∘∘cossin, determine π‘§π‘§οŠ§οŠ¨.

  • A9(330+𝑖330)cossin∘∘
  • B20(330+𝑖330)cossin∘∘
  • C20ο€Ί330+𝑖330cossin∘∘
  • D9(330βˆ’π‘–330)cossin∘∘
  • E20(30+𝑖30)cossin∘∘

Q16:

Given that 𝑧=2((5π‘Žβˆ’2𝑏)+𝑖(5π‘Žβˆ’2𝑏))cossin and 𝑧=4((4π‘Žβˆ’3𝑏)+𝑖(4π‘Žβˆ’3𝑏))cossin, find π‘§π‘§οŠ§οŠ¨.

  • A6((20π‘Ž+6𝑏)+𝑖(20π‘Ž+6𝑏))cossin
  • B12((π‘Ž+𝑏)+𝑖(π‘Ž+𝑏))cossin
  • C8((20π‘Ž+6𝑏)+𝑖(20π‘Ž+6𝑏))cossin
  • D8((9π‘Žβˆ’5𝑏)+𝑖(9π‘Žβˆ’5𝑏))cossin
  • E6((9π‘Žβˆ’5𝑏)+𝑖(9π‘Žβˆ’5𝑏))cossin

Q17:

Given that 𝑧=βˆ’150βˆ’π‘–150∘∘sincos and that 𝑧=2(120βˆ’π‘–120)∘∘sincos, find π‘§π‘§οŠ§οŠ¨.

  • A2(270+𝑖270)cossin∘∘
  • B2(150+𝑖150)cossin∘∘
  • C3(150+𝑖150)cossin∘∘
  • D3(270+𝑖270)cossin∘∘

Q18:

What is the magnitude of the product of 𝑧=π‘Ÿ(πœƒ+π‘–πœƒ)cossin and 𝑧=𝑠(πœ‘+π‘–πœ‘)cossin?

  • Aπœƒ+πœ‘
  • BπœƒΓ—πœ‘
  • Cπ‘Ÿπ‘ 
  • Dπ‘Ÿπ‘ 
  • Eπ‘Ÿ+𝑠

Q19:

Given that principal argument of 𝑍=13πœ‹12 and principal argument of 𝑍=3πœ‹4, determine the principal argument of 𝑍𝑍οŠͺ.

  • Aβˆ’11πœ‹12
  • Bβˆ’πœ‹12
  • Cβˆ’πœ‹6
  • D7πœ‹12

Q20:

Given that the principal argument of 𝑍=13πœ‹12 and the principal argument of 𝑍=πœ‹2, determine the principal argument of 12π‘π‘οŠ§οŠ¨.

  • Aβˆ’5πœ‹12
  • Bπœ‹12
  • Cβˆ’πœ‹4
  • Dβˆ’πœ‹2
  • Eπœ‹

Q21:

Given that principal argument (𝑍)=5πœ‹6, determine principal argument ο€Ήπ‘ο…οŠ¨.

  • Aβˆ’πœ‹3
  • Bπœ‹3
  • Cπœ‹6
  • Dβˆ’2πœ‹3
  • Eβˆ’πœ‹6

Q22:

Given that 𝑧=1 and 𝑧=(3πœƒ+𝑖3πœƒ)cossin, find the trigonometric form of π‘§π‘§οŠ§οŠ¨.

  • Acossin(2πœ‹βˆ’6πœƒ)+𝑖(2πœ‹βˆ’6πœƒ)
  • Bcossin(2πœ‹βˆ’3πœƒ)+𝑖(2πœ‹βˆ’3πœƒ)
  • Ccossin(πœ‹βˆ’6πœƒ)+𝑖(πœ‹βˆ’6πœƒ)
  • Dcossin(2πœ‹+6πœƒ)+𝑖(2πœ‹+6πœƒ)

Q23:

If 𝑍=π‘Ÿ(πœƒ+π‘–πœƒ)cossin, what is 1𝑍?

  • Aπ‘Ÿ((βˆ’πœƒ)+𝑖(βˆ’πœƒ))cossin
  • B1π‘Ÿ(πœƒ+π‘–πœƒ)cossin
  • Cπ‘Ÿ((βˆ’πœƒ)βˆ’π‘–(βˆ’πœƒ))cossin
  • D1π‘Ÿ((βˆ’πœƒ)+𝑖(βˆ’πœƒ))cossin

Q24:

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

  • A3(30+𝑖30)cossin∘∘
  • B3(210+𝑖210)cossin∘∘
  • C13(330+𝑖330)cossin∘∘
  • D3(330+𝑖330)cossin∘∘
  • E13(210+𝑖210)cossin∘∘

Q25:

Given that 𝑧=ο€Ό5πœ‹6π‘Žοˆ+𝑖5πœ‹6π‘Žοˆcossin, find 1𝑧.

  • Acossinο€Ό5πœ‹6π‘Žοˆ+𝑖5πœ‹6π‘Žοˆ
  • Bcossinο€Ό7πœ‹6π‘Žοˆ+𝑖7πœ‹6
  • Ccossinο€Ό7πœ‹6π‘Žοˆ+𝑖7πœ‹6π‘Žοˆ
  • Dcossinο€Ό11πœ‹6π‘Žοˆ+𝑖11πœ‹6π‘Žοˆ
  • Esincosο€Ό7πœ‹6π‘Žοˆ+𝑖7πœ‹6π‘Žοˆ

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