Worksheet: Argument of a Complex Number

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

Q1:

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

  • A − 0 . 7 7
  • B0.27
  • C − 0 . 2 8
  • D − 1 . 2 9
  • E1.3

Q2:

Consider the complex number 𝑧 = 7 + 7 𝑖 .

Find the argument of 𝑧 .

  • A 3 𝜋 4
  • B 7 √ 2
  • C7
  • D 𝜋 4
  • E 𝜋 2

Hence, find the argument of 𝑧 4 .

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

Q3:

A complex number is multiplied by another complex number 𝑧 , and then by the complex conjugate 𝑧 ∗ . How is the argument of the original complex number affected?

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

Q4:

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

  • A0.67
  • B 0.54
  • C0.93
  • D 0.64
  • E0.89

Q5:

What is the argument of the complex number 4 𝑖 ?

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

Q6:

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

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

Q7:

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

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

Q8:

What does the argument of a complex number represent?

  • A the angle it makes with the positive imaginary axis
  • B its distance from the origin in the complex plane
  • C its real coordinate in the complex plane
  • D the angle it makes with the positive real axis
  • E its imaginary coordinate in the complex plane

Q9:

Given a complex number 𝑍 , where the principal argument of 𝑍 is 𝜃 = 1 1 𝜋 1 2 , determine the principal argument of 1 0 𝑍 .

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

Q10:

What is the argument of the complex number − 6 ?

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

Q11:

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

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

Q12:

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

  • A 𝜋 − 𝜃
  • B 𝜃
  • C − 𝜋 + 𝜃
  • D − 𝜃

Q13:

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

Q14:

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

Q15:

Given that 𝑍 = − 3 − 7 𝑖 , find the principal argument of 𝑍 rounded to the nearest two decimal places.

Q16:

Given that 𝑍 = 6 − 4 𝑖 , find the principal argument of 𝑍 rounded to the nearest two decimal places.

Q17:

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

  • A 𝜋
  • B 𝜋 2
  • C0
  • D − 𝜋 2

Q18:

Given that 𝑍 = − 1 2 + √ 3 2 𝑖 , find the principal argument of 𝑍 .

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

Q19:

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

  • A − 𝜋 2 , 𝜋 2
  • B [ 0 , 𝜋 ]
  • C ( − 𝜋 , 0 ]
  • D ( − 𝜋 , 𝜋 ]
  • E 0 , 𝜋 2

Q20:

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

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

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