Worksheet: Polar Form of Complex Numbers

In this worksheet, we will practice representing a complex number in polar form, calculating the modulus and argument, and using this to change the form of a complex number.

Q1:

Find the trigonometric form of the complex number 𝑍 represented by the given Argand diagram.

  • Aβˆ’4ο€»ο€»πœ‹3+π‘–ο€»πœ‹3cossin
  • Bβˆ’4ο€»ο€»βˆ’πœ‹3+π‘–ο€»βˆ’πœ‹3cossin
  • C4ο€»ο€»πœ‹3+π‘–ο€»πœ‹3cossin
  • D4ο€»ο€»βˆ’πœ‹3+π‘–ο€»βˆ’πœ‹3cossin

Q2:

Express 125πœ‹6+𝑖5πœ‹6cossin in algebraic form.

  • A6√3βˆ’6𝑖
  • Bβˆ’6√3+6𝑖
  • C6βˆ’6√3𝑖
  • Dβˆ’6βˆ’6√3𝑖

Q3:

Find the modulus of the complex number 1+𝑖.

  • A1
  • B√3
  • C2
  • D√2
  • E4

Find the argument of the complex number 1+𝑖.

  • Aβˆ’πœ‹4
  • Bπœ‹
  • Cπœ‹4
  • Dπœ‹2
  • Eβˆ’πœ‹2

Hence, write the complex number 1+𝑖 in polar form.

  • A√2(πœ‹+π‘–πœ‹)cossin
  • B√2ο€»πœ‹2+π‘–πœ‹2cossin
  • C2ο€»πœ‹2+π‘–πœ‹2cossin
  • D√2ο€»πœ‹4+π‘–πœ‹4cossin
  • E2ο€»πœ‹4+π‘–πœ‹4cossin

Q4:

Consider the diagram.

Which of the following correctly describes the relationship between π‘Ž,π‘Ÿ, and πœƒ?

  • Aπ‘Ž=π‘Ÿπœƒsin
  • Bπ‘Ž=π‘Ÿπœƒtan
  • Cπ‘Ž=π‘Ÿπœƒcos
  • Dπ‘Ž=πœƒπ‘Ÿcos
  • Eπ‘Ž=πœƒπ‘Ÿsin

Which of the following correctly describes the relationship between 𝑏,π‘Ÿ, and πœƒ?

  • A𝑏=π‘Ÿπœƒtan
  • B𝑏=πœƒπ‘Ÿcos
  • C𝑏=πœƒπ‘Ÿsin
  • D𝑏=π‘Ÿπœƒsin
  • E𝑏=π‘Ÿπœƒcos

Hence, express 𝑧 in terms of π‘Ÿ and πœƒ.

  • A𝑧=π‘Ÿπœƒ+π‘Ÿπ‘–πœƒcossin
  • B𝑧=πœƒπ‘Ÿ+π‘–πœƒπ‘Ÿsincos
  • C𝑧=π‘Ÿπœƒ+π‘–πœƒπ‘Ÿcossin
  • D𝑧=π‘Ÿπœƒ+π‘Ÿπ‘–πœƒsincos
  • E𝑧=πœƒπ‘Ÿ+π‘–πœƒπ‘Ÿcossin

Q5:

The Argand diagram shows the complex number 𝑧.

Write 𝑧 in rectangular form.

  • A5+3𝑖
  • Bβˆ’(3+5𝑖)
  • C3βˆ’5𝑖
  • D3+5𝑖
  • E5βˆ’3𝑖

Convert 𝑧 to polar form, rounding the argument to two decimal places.

  • A√34(1.03βˆ’π‘–1.03)cossin
  • B34(1.03+𝑖1.03)cossin
  • C√8(1.03+𝑖1.03)cossin
  • D√34(1.03+𝑖1.03)cossin
  • E8(1.03βˆ’π‘–1.03)cossin

Q6:

Express the complex number 𝑍=4𝑖 in trigonometric form.

  • A𝑍=4ο€»ο€»πœ‹2+π‘–ο€»πœ‹2cossin
  • B𝑍=4ο€»ο€»πœ‹2ο‡βˆ’π‘–ο€»πœ‹2cossin
  • C𝑍=4ο€»ο€»βˆ’πœ‹2ο‡βˆ’π‘–ο€»βˆ’πœ‹2cossin
  • D𝑍=4ο€»ο€»βˆ’πœ‹2+π‘–ο€»βˆ’πœ‹2cossin

Q7:

Given that 𝑍=√3+𝑖, determine the trigonometric form of 𝑍.

  • A1311πœ‹6+𝑖11πœ‹6cossin
  • B217πœ‹6+𝑖17πœ‹6cossin
  • C211πœ‹6+𝑖11πœ‹6cossin
  • D27πœ‹3+𝑖7πœ‹3cossin
  • E211πœ‹6βˆ’π‘–11πœ‹6cossin

Q8:

Simplify 6βˆ’6π‘–βˆ’2𝑖, giving your answer in both algebraic and trigonometric form.

  • Aβˆ’3βˆ’3𝑖, 3√2ο€Όο€Όβˆ’3πœ‹4+π‘–ο€Όβˆ’3πœ‹4cossin
  • B3+3𝑖, 3√2ο€»ο€»πœ‹4+π‘–ο€»πœ‹4cossin
  • C3βˆ’3𝑖, 3√2ο€»ο€»βˆ’πœ‹4+π‘–ο€»βˆ’πœ‹4cossin
  • Dβˆ’3+3𝑖, 3√2ο€Όο€Ό3πœ‹4+𝑖3πœ‹4cossin

Q9:

Simplify βˆ’5+5√3π‘–βˆ’βˆš3βˆ’π‘–, giving your answer in both algebraic and trigonometric form.

  • Aβˆ’5𝑖, 5ο€»ο€»βˆ’πœ‹2+π‘–ο€»βˆ’πœ‹2cossin
  • B5𝑖, 5(0+𝑖0)cossin
  • Cβˆ’5𝑖, 5(πœ‹+π‘–πœ‹)cossin
  • D5𝑖, 5ο€»ο€»πœ‹2+π‘–ο€»πœ‹2cossin

Q10:

Given that π‘βˆ’2=(𝑍+2)𝑖, determine the trigonometric form of the complex number 𝑍.

  • A2ο€»ο€»βˆ’πœ‹2+π‘–ο€»βˆ’πœ‹2cossin
  • B2(πœ‹+π‘–πœ‹)cossin
  • C2ο€»ο€»πœ‹2+π‘–ο€»πœ‹2cossin
  • D2(0+𝑖0)cossin

Q11:

Given that 𝑍=(6π‘–βˆ’6)(4+3𝑖)(1+2𝑖), express the complex number 𝑍 in the form of π‘₯+𝑦𝑖, and then determine its trigonometric form.

  • A6βˆ’6𝑖, ο€»ο€»βˆ’πœ‹4+π‘–ο€»βˆ’πœ‹4cossin
  • B6βˆ’6𝑖, 6√2ο€»ο€»βˆ’πœ‹4+π‘–ο€»βˆ’πœ‹4cossin
  • C6+6𝑖, 6√2ο€»ο€»πœ‹4+π‘–ο€»πœ‹4cossin
  • D6+6𝑖, ο€»ο€»πœ‹4+π‘–ο€»πœ‹4cossin

Q12:

Simplify βˆ’7+4√3+ο€»βˆ’7√3βˆ’4𝑖7+4𝑖, giving your answer in both algebraic and trigonometric form.

  • A1βˆ’βˆš3𝑖, 2ο€»ο€»βˆ’πœ‹3+π‘–ο€»βˆ’πœ‹3cossin
  • B1+√3𝑖, 2ο€»ο€»πœ‹3+π‘–ο€»πœ‹3cossin
  • Cβˆ’1+√3𝑖, 2ο€Όο€Ό2πœ‹3+𝑖2πœ‹3cossin
  • Dβˆ’1βˆ’βˆš3𝑖, 2ο€Όο€Όβˆ’2πœ‹3+π‘–ο€Όβˆ’2πœ‹3cossin

Q13:

Given that |𝑍|=9 and the argument of 𝑍 is πœƒ=πœ‹6, find 𝑍, giving your answer in trigonometric form.

  • A𝑍=9ο“ο€»πœ‹6+π‘–ο€»πœ‹6ο‡οŸsincos
  • B𝑍=9ο€»πœ‹6+π‘–ο€»πœ‹6sincos
  • C𝑍=9ο€»πœ‹6+π‘–ο€»πœ‹6cossin
  • D𝑍=9ο“ο€»πœ‹6ο‡βˆ’π‘–ο€»πœ‹6ο‡οŸcossin
  • E𝑍=9ο“ο€»πœ‹6+π‘–ο€»πœ‹6ο‡οŸcossin

Q14:

Given that |𝑍|=8 and the argument of 𝑍 is πœƒ=360∘, find 𝑍, giving your answer in trigonometric form.

  • A𝑍=8πœ‹+π‘–πœ‹cossin
  • B𝑍=8[2πœ‹+𝑖2πœ‹]sincos
  • C𝑍=8[πœ‹+π‘–πœ‹]cossin
  • D𝑍=82πœ‹+𝑖2πœ‹cossin
  • E𝑍=8[2πœ‹+𝑖2πœ‹]cossin

Q15:

Given that |𝑍|=5 and the argument of 𝑍 is πœƒ=2πœ‹+2π‘›πœ‹, where π‘›βˆˆβ„€, find 𝑍, giving your answer in trigonometric form.

  • A𝑍=5(4πœ‹+𝑖4πœ‹)cossin
  • B𝑍=5(2πœ‹+𝑖2πœ‹)sincos
  • C𝑍=5(2πœ‹+𝑖2πœ‹)cossin
  • D𝑍=10(2πœ‹+𝑖2πœ‹)sincos
  • E𝑍=10(2πœ‹+𝑖2πœ‹)cossin

Q16:

Given that |𝑍|=3 and the argument of 𝑍 is πœƒ=πœ‹3, find 𝑍, giving your answer in algebraic form.

  • A𝑍=βˆ’32βˆ’3√32𝑖
  • B𝑍=32+3√32𝑖
  • C𝑍=βˆ’3√32+32𝑖
  • D𝑍=32βˆ’3√32𝑖
  • E𝑍=3√32+32𝑖

Q17:

Given that |𝑍|=12 and the argument of 𝑍 is πœƒ=120∘, find 𝑍, giving your answer in algebraic form.

  • A𝑍=6βˆ’6√3𝑖
  • B𝑍=βˆ’6√3βˆ’6𝑖
  • C𝑍=βˆ’6βˆ’6√3𝑖
  • D𝑍=βˆ’6+6√3𝑖
  • E𝑍=6√3βˆ’6𝑖

Q18:

Given that |𝑧|=5 and the argument of 𝑧 is πœƒ=270∘, find 𝑧, giving your answer in algebraic form.

  • A𝑧=βˆ’5𝑖
  • B𝑧=βˆ’5
  • C𝑧=5+5𝑖
  • D𝑧=5𝑖
  • E𝑧=5

Q19:

Given that 𝑍=7[(βˆ’58)+𝑖(βˆ’58)]cossin∘∘, determine the algebraic form of 𝑍, approximating the real and imaginary parts to the nearest two decimal places.

  • A𝑍=βˆ’3.71+5.94𝑖
  • B𝑍=βˆ’5.94+5.94𝑖
  • C𝑍=3.71βˆ’5.94𝑖
  • D𝑍=5.94+3.71𝑖

Q20:

Given that 𝑍=πœƒβˆ’π‘–πœƒsincos, find the principal argument of 𝑍, where πœƒβˆˆο“0,πœ‹2.

  • Aπœƒβˆ’πœ‹2
  • B2πœ‹βˆ’πœƒ
  • Cπœ‹+πœƒ
  • Dπœ‹βˆ’πœƒ
  • Eπœƒ

Q21:

Find cosπœ‹6.

  • A√33
  • B3√33
  • C√32
  • D12
  • E2√32

Find sinπœ‹6.

  • A2√32
  • B√33
  • C12
  • D√32
  • E3√33

Hence, express the complex number 10ο€»πœ‹6+π‘–πœ‹6cossin in rectangular form.

  • A10√33+5𝑖
  • B5+5𝑖
  • C5+10√33𝑖
  • D5+5√3𝑖
  • E5√3+5𝑖

Q22:

Given that 𝑍=6ο€Όο€Ό3πœ‹4+𝑖3πœ‹4cossin, find |𝑍|.

Q23:

Find the modulus and the principal amplitude of the number 𝑍=βˆ’41(30+𝑖30)cossin∘∘.

  • A|𝑍|=√41, principal amplitude 150∘
  • B|𝑍|=41, principal amplitude βˆ’150∘
  • C|𝑍|=√41, principal amplitude βˆ’150∘
  • D|𝑍|=41, principal amplitude 150∘

Q24:

Find the modulus and the principal amplitude of the number 𝑍=βˆ’37ο€Όο€Ό5πœ‹3οˆβˆ’π‘–ο€Ό5πœ‹3sincos.

  • A|𝑍|=37, principal amplitude πœƒ=βˆ’πœ‹6
  • B|𝑍|=√37, principal amplitude πœƒ=βˆ’πœ‹6
  • C|𝑍|=37, principal amplitude πœƒ=πœ‹6
  • D|𝑍|=√37, principal amplitude πœƒ=πœ‹6

Q25:

Find the modulus and the principal amplitude of the number 𝑍=16+16𝑖305tan∘.

  • A|𝑍|=16305csc∘, principal amplitude πœƒ=235∘
  • B|𝑍|=16305cos∘, principal amplitude πœƒ=βˆ’55∘
  • C|𝑍|=16305sec∘, principal amplitude πœƒ=235∘
  • D|𝑍|=16305sec∘, principal amplitude πœƒ=βˆ’55∘

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