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.

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

  • 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=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
  • 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=45𝑖.

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

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

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.