Worksheet: Argument of a Complex Number

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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