Worksheet: The Argument of a Complex Number

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

Q1:

Find the argument of the complex number 2−7𝑖 in radians. Give your answer correct to two decimal places.

  • A − 0 . 7 7
  • B − 1 . 2 9
  • C1.3
  • D − 0 . 2 8
  • E0.27

Q2:

Consider the complex number 𝑧=7+7𝑖.

Find the argument of 𝑧.

  • A 𝜋 4
  • B 𝜋 2
  • C 7 √ 2
  • D 3 𝜋 4
  • E7

Hence, find the argument of 𝑧.

  • A 2 𝜋
  • B 𝜋 4
  • C 𝜋 4
  • D 𝜋
  • E 𝜋 1 6

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.

  • A0.64
  • B0.67
  • C0.89
  • D0.93
  • E0.54

Q5:

What is the argument of the complex number 4𝑖?

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

Q6:

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

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

Q7:

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

  • A t a n 𝑎 𝑏
  • B t a n 𝑏 √ 𝑎 + 𝑏
  • C t a n 𝑎 √ 𝑎 + 𝑏
  • D t a n √ 𝑎 + 𝑏 𝑎
  • E t a n 𝑏 𝑎

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.

  • A 2 9 𝜋 3
  • B 1 7 𝜋 1 2
  • C 5 5 𝜋 6
  • D 1 1 𝜋 1 2

Q10:

What is the argument of the complex number −6?

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

Q11:

Given that 𝑍=−9−9√3𝑖 and 𝑍=4+4√3𝑖, determine the principal argument of (𝑍−𝑍).

  • A 3 0 0 ∘
  • B 2 4 0 ∘
  • C 6 0 ∘
  • D 1 8 0 ∘

Q12:

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

  • A − 𝜃
  • B 𝜃
  • C 𝜋 − 𝜃
  • D − 𝜋 + 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=−3−7𝑖, find the principal argument of 𝑍 rounded to the nearest two decimal places.

Q16:

Given that 𝑍=6−4𝑖, 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
  • B 𝜋
  • C − 𝜋 2
  • D 𝜋 2

Q18:

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

  • A 2 𝜋 3
  • B − 𝜋 3
  • C − 5 𝜋 6
  • D 𝜋 3

Q19:

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

  • A − 𝜋 2 , 𝜋 2
  • B ( − 𝜋 , 𝜋 ]
  • C 0 , 𝜋 2
  • 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?

  • A t a n 𝑏 𝑎
  • B 𝜋 − 𝑏 𝑎 t a n
  • C 𝜋 + 𝑎 𝑏 t a n
  • D t a n 𝑏 𝑎 − 𝜋
  • E 𝜋 + 𝑏 𝑎 t a n

Q21:

Consider the complex number 𝑧=−4−5𝑖.

Calculate arg(𝑧), giving your answer correct to three significant figures.

Calculate arg(𝑧)∗, giving your answer correct to three significant figures.

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