Lesson Worksheet: Polar Form of Complex Numbers Mathematics

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.

  • A−4𝜋3+𝑖𝜋3cossin
  • B−4−𝜋3+𝑖−𝜋3cossin
  • C4𝜋3+𝑖𝜋3cossin
  • D4−𝜋3+𝑖−𝜋3cossin

Q2:

Express 125𝜋6+𝑖5𝜋6cossin in algebraic form.

  • A6√3−6𝑖
  • B−6√3+6𝑖
  • C6−6√3𝑖
  • D−6−6√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.

  • A−𝜋4
  • B𝜋
  • C𝜋4
  • D𝜋2
  • E−𝜋2

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

  • A√2(𝜋+𝑖𝜋)cossin
  • B√2𝜋2+𝑖𝜋2cossin
  • C2𝜋2+𝑖𝜋2cossin
  • D√2𝜋4+𝑖𝜋4cossin
  • E2𝜋4+𝑖𝜋4cossin

Q4:

Consider the diagram.

Which of the following correctly describes the relationship between 𝑎,𝑟, and 𝜃?

  • A𝑎=𝑟𝜃sin
  • B𝑎=𝑟𝜃tan
  • C𝑎=𝑟𝜃cos
  • D𝑎=𝜃𝑟cos
  • E𝑎=𝜃𝑟sin

Which of the following correctly describes the relationship between 𝑏,𝑟, and 𝜃?

  • A𝑏=𝑟𝜃tan
  • B𝑏=𝜃𝑟cos
  • C𝑏=𝜃𝑟sin
  • D𝑏=𝑟𝜃sin
  • E𝑏=𝑟𝜃cos

Hence, express 𝑧 in terms of 𝑟 and 𝜃.

  • A𝑧=𝑟𝜃+𝑟𝑖𝜃cossin
  • B𝑧=𝜃𝑟+𝑖𝜃𝑟sincos
  • C𝑧=𝑟𝜃+𝑖𝜃𝑟cossin
  • D𝑧=𝑟𝜃+𝑟𝑖𝜃sincos
  • E𝑧=𝜃𝑟+𝑖𝜃𝑟cossin

Q5:

The Argand diagram shows the complex number 𝑧.

Write 𝑧 in rectangular form.

  • A5+3𝑖
  • B−(3+5𝑖)
  • C3−5𝑖
  • D3+5𝑖
  • E5−3𝑖

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

  • A√34(1.03−𝑖1.03)cossin
  • B34(1.03+𝑖1.03)cossin
  • C√8(1.03+𝑖1.03)cossin
  • D√34(1.03+𝑖1.03)cossin
  • E8(1.03−𝑖1.03)cossin

Q6:

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

  • A𝑍=4𝜋2+𝑖𝜋2cossin
  • B𝑍=4𝜋2−𝑖𝜋2cossin
  • C𝑍=4−𝜋2−𝑖−𝜋2cossin
  • D𝑍=4−𝜋2+𝑖−𝜋2cossin

Q7:

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

  • A1311𝜋6+𝑖11𝜋6cossin
  • B217𝜋6+𝑖17𝜋6cossin
  • C211𝜋6+𝑖11𝜋6cossin
  • D27𝜋3+𝑖7𝜋3cossin
  • E211𝜋6−𝑖11𝜋6cossin

Q8:

Simplify 6−6𝑖−2𝑖, giving your answer in both algebraic and trigonometric form.

  • A−3−3𝑖, 3√2−3𝜋4+𝑖−3𝜋4cossin
  • B3+3𝑖, 3√2𝜋4+𝑖𝜋4cossin
  • C3−3𝑖, 3√2−𝜋4+𝑖−𝜋4cossin
  • D−3+3𝑖, 3√23𝜋4+𝑖3𝜋4cossin

Q9:

Simplify −5+5√3𝑖−√3−𝑖, giving your answer in both algebraic and trigonometric form.

  • A−5𝑖, 5−𝜋2+𝑖−𝜋2cossin
  • B5𝑖, 5(0+𝑖0)cossin
  • C−5𝑖, 5(𝜋+𝑖𝜋)cossin
  • D5𝑖, 5𝜋2+𝑖𝜋2cossin

Q10:

Given that 𝑍−2=(𝑍+2)𝑖, determine the trigonometric form of the complex number 𝑍.

  • A2−𝜋2+𝑖−𝜋2cossin
  • B2(𝜋+𝑖𝜋)cossin
  • C2𝜋2+𝑖𝜋2cossin
  • D2(0+𝑖0)cossin

Q11:

Given that 𝑍=(6𝑖−6)(4+3𝑖)(1+2𝑖), express the complex number 𝑍 in the form of 𝑥+𝑦𝑖, and then determine its trigonometric form.

  • A6−6𝑖, −𝜋4+𝑖−𝜋4cossin
  • B6−6𝑖, 6√2−𝜋4+𝑖−𝜋4cossin
  • C6+6𝑖, 6√2𝜋4+𝑖𝜋4cossin
  • D6+6𝑖, 𝜋4+𝑖𝜋4cossin

Q12:

Simplify −7+4√3+−7√3−4𝑖7+4𝑖, giving your answer in both algebraic and trigonometric form.

  • A1−√3𝑖, 2−𝜋3+𝑖−𝜋3cossin
  • B1+√3𝑖, 2𝜋3+𝑖𝜋3cossin
  • C−1+√3𝑖, 22𝜋3+𝑖2𝜋3cossin
  • D−1−√3𝑖, 2−2𝜋3+𝑖−2𝜋3cossin

Q13:

Given that |𝑍|=9 and the argument of 𝑍 is 𝜃=𝜋6, find 𝑍, giving your answer in trigonometric form.

  • A𝑍=9𝜋6+𝑖𝜋6sincos
  • B𝑍=9𝜋6+𝑖𝜋6sincos
  • C𝑍=9𝜋6+𝑖𝜋6cossin
  • D𝑍=9𝜋6−𝑖𝜋6cossin
  • E𝑍=9𝜋6+𝑖𝜋6cossin

Q14:

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

  • A𝑍=8𝜋+𝑖𝜋cossin
  • B𝑍=8[2𝜋+𝑖2𝜋]sincos
  • C𝑍=8[𝜋+𝑖𝜋]cossin
  • D𝑍=82𝜋+𝑖2𝜋cossin
  • E𝑍=8[2𝜋+𝑖2𝜋]cossin

Q15:

Given that |𝑍|=5 and the argument of 𝑍 is 𝜃=2𝜋+2𝑛𝜋, where 𝑛∈ℤ, find 𝑍, giving your answer in trigonometric form.

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

Q16:

Given that |𝑍|=3 and the argument of 𝑍 is 𝜃=𝜋3, find 𝑍, giving your answer in algebraic form.

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

Q17:

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

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

Q18:

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

  • A𝑧=−5𝑖
  • B𝑧=−5
  • C𝑧=5+5𝑖
  • D𝑧=5𝑖
  • E𝑧=5

Q19:

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

  • A𝑍=−3.71+5.94𝑖
  • B𝑍=−5.94+5.94𝑖
  • C𝑍=3.71−5.94𝑖
  • D𝑍=5.94+3.71𝑖

Q20:

Given that 𝑍=𝜃−𝑖𝜃sincos, find the principal argument of 𝑍, where 𝜃∈0,𝜋2.

  • A𝜃−𝜋2
  • B2𝜋−𝜃
  • C𝜋+𝜃
  • D𝜋−𝜃
  • E𝜃

Q21:

Find cos𝜋6.

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

Find sin𝜋6.

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

Hence, express the complex number 10𝜋6+𝑖𝜋6cossin in rectangular form.

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

Q22:

Given that 𝑍=63𝜋4+𝑖3𝜋4cossin, find |𝑍|.

Q23:

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

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

Q24:

Find the modulus and the principal amplitude of the number 𝑍=−375𝜋3−𝑖5𝜋3sincos.

  • A|𝑍|=37, principal amplitude 𝜃=−𝜋6
  • B|𝑍|=√37, principal amplitude 𝜃=−𝜋6
  • C|𝑍|=37, principal amplitude 𝜃=𝜋6
  • D|𝑍|=√37, principal amplitude 𝜃=𝜋6

Q25:

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

  • A|𝑍|=16305csc∘, principal amplitude 𝜃=235∘
  • B|𝑍|=16305cos∘, principal amplitude 𝜃=−55∘
  • C|𝑍|=16305sec∘, principal amplitude 𝜃=235∘
  • D|𝑍|=16305sec∘, principal amplitude 𝜃=−55∘

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