Worksheet: The Argument of a Complex Number

In this worksheet, we will practice identifying the argument of a complex number calculating it.

Q1:

Find the argument of the complex number 27𝑖 in radians. Give your answer correct to two decimal places.

Q2:

Consider the complex number 𝑧=7+7𝑖.

Find the argument of 𝑧.

  • A𝜋4
  • B𝜋2
  • C72
  • D3𝜋4
  • E7

Hence, find the argument of 𝑧.

  • A2𝜋
  • B𝜋4
  • C𝜋4
  • D𝜋
  • E𝜋16

Q3:

A complex number is multiplied by another complex number 𝑧, and then by the complex conjugate 𝑧. How is the argument of the resulting complex number related to the argument of the original complex number?

  • AIt is unchanged.
  • BIt is increased by the argument of 𝑧.
  • CIt is increased by twice the argument of 𝑧.
  • DIt is increased by twice the argument of 𝑧.
  • EIt is increased by 𝜋.

Q4:

Find the argument of the complex number 4+3𝑖 in radians. Give your answer correct to two decimal places.

Q5:

What is the argument of the complex number 4𝑖?

  • A𝜋2
  • B𝜋
  • C𝜋
  • D𝜋2
  • E𝜋3

Q6:

What is the argument of the complex number 𝑏𝑖, where 𝑏<0?

  • A𝜋2
  • B𝜋
  • C𝜋2
  • D𝜋3
  • E𝜋

Q7:

What is the argument of the complex number 𝑎+𝑏𝑖, where 𝑎>0 and 𝑏>0?

  • Atan𝑎𝑏
  • Btan𝑏𝑎+𝑏
  • Ctan𝑎𝑎+𝑏
  • Dtan𝑎+𝑏𝑎
  • Etan𝑏𝑎

Q8:

What does the argument of a complex number represent?

  • Aits imaginary coordinate in the complex plane
  • Bits real coordinate in the complex plane
  • Cthe angle it makes with the positive real axis
  • Dthe angle it makes with the positive imaginary axis
  • Eits distance from the origin in the complex plane

Q9:

Given a complex number 𝑍, where the principal argument of 𝑍 is 𝜃=11𝜋12, determine the principal argument of 10𝑍.

  • A29𝜋3
  • B17𝜋12
  • C55𝜋6
  • D11𝜋12

Q10:

What is the argument of the complex number 6?

  • A𝜋3
  • B𝜋
  • C𝜋2
  • D2𝜋
  • E𝜋2

Q11:

Given that 𝑍=993𝑖 and 𝑍=4+43𝑖, determine the principal argument of (𝑍𝑍).

  • A300
  • B240
  • C60
  • D180

Q12:

If 𝜃 is the principal argument of the complex number 𝑍 then what is the principal argument of 𝑍?

  • A𝜃
  • B𝜃
  • C𝜋𝜃
  • D𝜋+𝜃

Q13:

Given that 𝑍=9+3𝑖, find the principal argument of 𝑍 rounded to the nearest two decimal places.

Q14:

Given that 𝑍=5+9𝑖, find the principal argument of 𝑍 rounded to the nearest two decimal places.

Q15:

Given that 𝑍=37𝑖, find the principal argument of 𝑍 rounded to the nearest two decimal places.

Q16:

Given that 𝑍=64𝑖, find the principal argument of 𝑍 rounded to the nearest two decimal places.

Q17:

Given that 𝑍=7𝑖, find the principal argument of 𝑍.

  • A0
  • B𝜋
  • C𝜋2
  • D𝜋2

Q18:

Given that 𝑍=12+32𝑖, find the principal argument of 𝑍.

  • A2𝜋3
  • B𝜋3
  • C5𝜋6
  • D𝜋3

Q19:

In what interval does the principal argument of a complex number lie?

  • A𝜋2,𝜋2
  • B(𝜋,𝜋]
  • C0,𝜋2
  • D[0,𝜋]
  • E(𝜋,0]

Q20:

What is the principal argument of the complex number 𝑧=𝑎+𝑏𝑖, where 𝑎 and 𝑏 are real, which lies in the second quadrant of the Argand diagram?

  • Atan𝑏𝑎
  • B𝜋𝑏𝑎tan
  • C𝜋+𝑎𝑏tan
  • Dtan𝑏𝑎𝜋
  • E𝜋+𝑏𝑎tan

Q21:

Consider the complex number 𝑧=45𝑖.

Calculate arg(𝑧), giving your answer correct to two decimal places in an interval from 𝜋 to 𝜋.

Calculate arg𝑧, giving your answer correct to two decimal places in an interval from 𝜋 to 𝜋.

Q22:

Consider the complex numbers 𝑧=1+3𝑖 and 𝑤=22𝑖.

Find arg(𝑧) and arg(𝑤).

  • Aarg(𝑧)=𝜋3, arg(𝑤)=𝜋4
  • Barg(𝑧)=𝜋6, arg(𝑤)=𝜋2
  • Carg(𝑧)=𝜋3, arg(𝑤)=𝜋2
  • Darg(𝑧)=𝜋6, arg(𝑤)=𝜋4
  • Earg(𝑧)=𝜋3, arg(𝑤)=𝜋4

Calculate arg(𝑧𝑤). How does this compare to arg(𝑧) and arg(𝑤)?

  • Aarg(𝑧𝑤)=𝜋3, argmaxargarg(𝑧𝑤)=((𝑧),(𝑤))
  • Barg(𝑧𝑤)=𝜋12, argargarg(𝑧𝑤)=(𝑧)+(𝑤)
  • Carg(𝑧𝑤)=𝜋12, argargarg(𝑧𝑤)=(𝑧)(𝑤)
  • Darg(𝑧𝑤)=𝜋12, argargarg(𝑧𝑤)=(𝑧)+(𝑤)
  • Earg(𝑧𝑤)=7𝜋12, argargarg(𝑧𝑤)=|(𝑧)(𝑤)|

Calculate arg𝑧𝑤. How does this compare to arg(𝑧) and arg(𝑤)?

  • Aarg𝑧𝑤=7𝜋12, argargarg𝑧𝑤=(𝑧)(𝑤)
  • Barg𝑧𝑤=𝜋4, argminargarg𝑧𝑤=((𝑧),(𝑤))
  • Carg𝑧𝑤=𝜋12, argargarg𝑧𝑤=(𝑧)+(𝑤)
  • Darg𝑧𝑤=43, argargarg𝑧𝑤=(𝑧)(𝑤)
  • Earg𝑧𝑤=7𝜋12, argargarg𝑧𝑤=(𝑤)(𝑧)

Q23:

If 𝜃 is the principal argument of a complex number 𝑍, determine the argument of 1𝑍.

  • A𝜋𝜃
  • B𝜃
  • C𝜋+𝜃
  • D𝜃

Q24:

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

  • A𝜋3
  • B𝜋3
  • C𝜋6
  • D2𝜋3
  • E𝜋6

Q25:

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

  • A11𝜋12
  • B𝜋12
  • C𝜋6
  • D7𝜋12

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