Worksheet: Polar Form of Complex Numbers

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In this worksheet, we will practice representing a complex number in polar form, calculating the modulus and argument, and using this to change the form of a complex number.

Q1:

Find the trigonometric form of the complex number 𝑍 represented by the given Argand diagram.

  • A 4 − 𝜋 3 + 𝑖 − 𝜋 3 c o s s i n
  • B − 4 − 𝜋 3 + 𝑖 − 𝜋 3 c o s s i n
  • C 4 𝜋 3 + 𝑖 𝜋 3 c o s s i n
  • D − 4 𝜋 3 + 𝑖 𝜋 3 c o s s i n

Q2:

Express 1 2 5 𝜋 6 + 𝑖 5 𝜋 6 c o s s i n in algebraic form.

  • A − 6 − 6 √ 3 𝑖
  • B 6 √ 3 − 6 𝑖
  • C 6 − 6 √ 3 𝑖
  • D − 6 √ 3 + 6 𝑖

Q3:

Find the modulus of the complex number 1 + 𝑖 .

  • A4
  • B2
  • C1
  • D √ 2
  • E √ 3

Find the argument of the complex number 1 + 𝑖 .

  • A 𝜋
  • B 𝜋 2
  • C − 𝜋 2
  • D − 𝜋 4
  • E 𝜋 4

Hence, write the complex number 1 + 𝑖 in polar form.

  • A √ 2 𝜋 4 + 𝑖 𝜋 4 c o s s i n
  • B 2 𝜋 4 + 𝑖 𝜋 4 c o s s i n
  • C 2 𝜋 2 + 𝑖 𝜋 2 c o s s i n
  • D √ 2 ( 𝜋 + 𝑖 𝜋 ) c o s s i n
  • E √ 2 𝜋 2 + 𝑖 𝜋 2 c o s s i n

Q4:

Consider the diagram.

Which of the following correctly describes the relationship between 𝑎 , 𝑟 , and 𝜃 ?

  • A 𝑎 = 𝜃 𝑟 s i n
  • B 𝑎 = 𝑟 𝜃 c o s
  • C 𝑎 = 𝑟 𝜃 t a n
  • D 𝑎 = 𝑟 𝜃 s i n
  • E 𝑎 = 𝜃 𝑟 c o s

Which of the following correctly describes the relationship between 𝑏 , 𝑟 , and 𝜃 ?

  • A 𝑏 = 𝜃 𝑟 c o s
  • B 𝑏 = 𝑟 𝜃 c o s
  • C 𝑏 = 𝜃 𝑟 s i n
  • D 𝑏 = 𝑟 𝜃 t a n
  • E 𝑏 = 𝑟 𝜃 s i n

Hence, express 𝑧 in terms of 𝑟 and 𝜃 .

  • A 𝑧 = 𝜃 𝑟 + 𝑖 𝜃 𝑟 s i n c o s
  • B 𝑧 = 𝜃 𝑟 + 𝑖 𝜃 𝑟 c o s s i n
  • C 𝑧 = 𝑟 𝜃 + 𝑟 𝑖 𝜃 s i n c o s
  • D 𝑧 = 𝑟 𝜃 + 𝑟 𝑖 𝜃 c o s s i n
  • E 𝑧 = 𝑟 𝜃 + 𝑖 𝜃 𝑟 c o s s i n

Q5:

The Argand diagram shows the complex number 𝑧 .

Write 𝑧 in rectangular form.

  • A 5 − 3 𝑖
  • B − ( 3 + 5 𝑖 )
  • C 3 + 5 𝑖
  • D 5 + 3 𝑖
  • E 3 − 5 𝑖

Convert 𝑧 to polar form, rounding the argument to two decimal places.

  • A 3 4 ( 1 . 0 3 + 𝑖 1 . 0 3 ) c o s s i n
  • B √ 3 4 ( 1 . 0 3 − 𝑖 1 . 0 3 ) c o s s i n
  • C √ 3 4 ( 1 . 0 3 + 𝑖 1 . 0 3 ) c o s s i n
  • D √ 8 ( 1 . 0 3 + 𝑖 1 . 0 3 ) c o s s i n
  • E 8 ( 1 . 0 3 − 𝑖 1 . 0 3 ) c o s s i n

Q6:

Express the complex number 𝑍 = 4 𝑖 in trigonometric form.

  • A 𝑍 = 4 𝜋 2 + 𝑖 𝜋 2 c o s s i n
  • B 𝑍 = 4 − 𝜋 2 + 𝑖 − 𝜋 2 c o s s i n
  • C 𝑍 = 4 − 𝜋 2 − 𝑖 − 𝜋 2 c o s s i n
  • D 𝑍 = 4 𝜋 2 − 𝑖 𝜋 2 c o s s i n

Q7:

Given that 𝑍 = √ 3 + 𝑖 , determine the trigonometric form of 𝑍 .

  • A 2 1 1 𝜋 6 + 𝑖 1 1 𝜋 6 c o s s i n
  • B 1 3 1 1 𝜋 6 + 𝑖 1 1 𝜋 6 c o s s i n
  • C 2 1 1 𝜋 6 − 𝑖 1 1 𝜋 6 c o s s i n
  • D 2 7 𝜋 3 + 𝑖 7 𝜋 3 c o s s i n
  • E 2 1 7 𝜋 6 + 𝑖 1 7 𝜋 6 c o s s i n

Q8:

Simplify 6 − 6 𝑖 − 2 𝑖 , giving your answer in both algebraic and trigonometric form.

  • A − 3 + 3 𝑖 , 3 √ 2 3 𝜋 4 + 𝑖 3 𝜋 4 c o s s i n
  • B − 3 − 3 𝑖 , 3 √ 2 − 3 𝜋 4 + 𝑖 − 3 𝜋 4 c o s s i n
  • C 3 − 3 𝑖 , 3 √ 2 − 𝜋 4 + 𝑖 − 𝜋 4 c o s s i n
  • D 3 + 3 𝑖 , 3 √ 2 𝜋 4 + 𝑖 𝜋 4 c o s s i n

Q9:

Simplify − 5 + 5 √ 3 𝑖 − √ 3 − 𝑖 , giving your answer in both algebraic and trigonometric form.

  • A − 5 𝑖 , 5 − 𝜋 2 + 𝑖 − 𝜋 2 c o s s i n
  • B 5 𝑖 , 5 𝜋 2 + 𝑖 𝜋 2 c o s s i n
  • C 5 𝑖 , 5 ( 0 + 𝑖 0 ) c o s s i n
  • D − 5 𝑖 , 5 ( 𝜋 + 𝑖 𝜋 ) c o s s i n

Q10:

Given that 𝑍 − 2 = ( 𝑍 + 2 ) 𝑖 , determine the trigonometric form of the complex number 𝑍 .

  • A 2 ( 0 + 𝑖 0 ) c o s s i n
  • B 2 − 𝜋 2 + 𝑖 − 𝜋 2 c o s s i n
  • C 2 ( 𝜋 + 𝑖 𝜋 ) c o s s i n
  • D 2 𝜋 2 + 𝑖 𝜋 2 c o s s i n

Q11:

Given that 𝑍 = ( 6 𝑖 − 6 ) ( 4 + 3 𝑖 ) ( 1 + 2 𝑖 ) , express the complex number 𝑍 in the form of 𝑥 + 𝑦 𝑖 , and then determine its trigonometric form.

  • A 6 + 6 𝑖 , 6 √ 2 𝜋 4 + 𝑖 𝜋 4 c o s s i n
  • B 6 + 6 𝑖 , 𝜋 4 + 𝑖 𝜋 4 c o s s i n
  • C 6 − 6 𝑖 , − 𝜋 4 + 𝑖 − 𝜋 4 c o s s i n
  • D 6 − 6 𝑖 , 6 √ 2 − 𝜋 4 + 𝑖 − 𝜋 4 c o s s i n

Q12:

Simplify − 7 + 4 √ 3 + − 7 √ 3 − 4 𝑖 7 + 4 𝑖 , giving your answer in both algebraic and trigonometric form.

  • A 1 + √ 3 𝑖 , 2 𝜋 3 + 𝑖 𝜋 3 c o s s i n
  • B − 1 − √ 3 𝑖 , 2 − 2 𝜋 3 + 𝑖 − 2 𝜋 3 c o s s i n
  • C − 1 + √ 3 𝑖 , 2 2 𝜋 3 + 𝑖 2 𝜋 3 c o s s i n
  • D 1 − √ 3 𝑖 , 2 − 𝜋 3 + 𝑖 − 𝜋 3 c o s s i n

Q13:

Given that | 𝑍 | = 9 and the argument of 𝑍 is 𝜃 = 𝜋 6 , find 𝑍 , giving your answer in trigonometric form.

  • A 𝑍 = 9 𝜋 6 + 𝑖 𝜋 6 c o s s i n
  • B 𝑍 = 9 𝜋 6 − 𝑖 𝜋 6 c o s s i n
  • C 𝑍 = 9 𝜋 6 + 𝑖 𝜋 6 c o s s i n
  • D 𝑍 = 9 𝜋 6 + 𝑖 𝜋 6 s i n c o s
  • E 𝑍 = 9 𝜋 6 + 𝑖 𝜋 6 s i n c o s

Q14:

Given that | 𝑍 | = 8 and the argument of 𝑍 is 𝜃 = 3 6 0 ∘ , find 𝑍 , giving your answer in trigonometric form.

  • A 𝑍 = 8 𝜋 + 𝑖 𝜋 c o s s i n
  • B 𝑍 = 8 [ 2 𝜋 + 𝑖 2 𝜋 ] c o s s i n
  • C 𝑍 = 8 [ 𝜋 + 𝑖 𝜋 ] c o s s i n
  • D 𝑍 = 8 2 𝜋 + 𝑖 2 𝜋 c o s s i n
  • E 𝑍 = 8 [ 2 𝜋 + 𝑖 2 𝜋 ] s i n c o s

Q15:

Given that | 𝑍 | = 5 and the argument of 𝑍 is 𝜃 = 2 𝜋 + 2 𝑛 𝜋 , where 𝑛 ∈ ℤ , find 𝑍 , giving your answer in trigonometric form.

  • A 𝑍 = 5 ( 2 𝜋 + 𝑖 2 𝜋 ) c o s s i n
  • B 𝑍 = 5 ( 4 𝜋 + 𝑖 4 𝜋 ) c o s s i n
  • C 𝑍 = 1 0 ( 2 𝜋 + 𝑖 2 𝜋 ) s i n c o s
  • D 𝑍 = 5 ( 2 𝜋 + 𝑖 2 𝜋 ) s i n c o s
  • E 𝑍 = 1 0 ( 2 𝜋 + 𝑖 2 𝜋 ) c o s s i n

Q16:

Given that | 𝑍 | = 3 and the argument of 𝑍 is 𝜃 = 𝜋 3 , find 𝑍 , giving your answer in algebraic form.

  • A 𝑍 = − 3 2 − 3 √ 3 2 𝑖
  • B 𝑍 = 3 2 + 3 √ 3 2 𝑖
  • C 𝑍 = 3 √ 3 2 + 3 2 𝑖
  • D 𝑍 = 3 2 − 3 √ 3 2 𝑖
  • E 𝑍 = − 3 √ 3 2 + 3 2 𝑖

Q17:

Given that | 𝑍 | = 1 2 and the argument of 𝑍 is 𝜃 = 1 2 0 ∘ , find 𝑍 , giving your answer in algebraic form.

  • A 𝑍 = 6 − 6 √ 3 𝑖
  • B 𝑍 = − 6 − 6 √ 3 𝑖
  • C 𝑍 = − 6 + 6 √ 3 𝑖
  • D 𝑍 = − 6 √ 3 − 6 𝑖
  • E 𝑍 = 6 √ 3 − 6 𝑖

Q18:

Given that | 𝑧 | = 5 and the argument of 𝑧 is 𝜃 = 2 7 0 ∘ , find 𝑧 , giving your answer in algebraic form.

  • A 𝑧 = − 5 𝑖
  • B 𝑧 = − 5
  • C 𝑧 = 5 + 5 𝑖
  • D 𝑧 = 5
  • E 𝑧 = 5 𝑖

Q19:

Given that 𝑍 = 7 [ ( − 5 8 ) + 𝑖 ( − 5 8 ) ] c o s s i n ∘ ∘ , determine the algebraic form of 𝑍 , approximating the real and imaginary parts to the nearest two decimal places.

  • A 𝑍 = 5 . 9 4 + 3 . 7 1 𝑖
  • B 𝑍 = − 5 . 9 4 + 5 . 9 4 𝑖
  • C 𝑍 = − 3 . 7 1 + 5 . 9 4 𝑖
  • D 𝑍 = 3 . 7 1 − 5 . 9 4 𝑖

Q20:

Given that 𝑍 = 𝜃 − 𝑖 𝜃 s i n c o s , find the principal argument of 𝑍 , where 𝜃 ∈ 0 , 𝜋 2 .

  • A 𝜋 + 𝜃
  • B 𝜋 − 𝜃
  • C 𝜃 − 𝜋 2
  • D 2 𝜋 − 𝜃
  • E 𝜃

Q21:

Find c o s 𝜋 6 .

  • A √ 3 2
  • B √ 3 3
  • C 2 √ 3 2
  • D 3 √ 3 3
  • E 1 2

Find s i n 𝜋 6 .

  • A √ 3 3
  • B 1 2
  • C 3 √ 3 3
  • D √ 3 2
  • E 2 √ 3 2

Hence, express the complex number 1 0 𝜋 6 + 𝑖 𝜋 6 c o s s i n in rectangular form.

  • A 5 + 1 0 √ 3 3 𝑖
  • B 5 √ 3 + 5 𝑖
  • C 5 + 5 𝑖
  • D 5 + 5 √ 3 𝑖
  • E 1 0 √ 3 3 + 5 𝑖

Q22:

Given that 𝑍 = 6 3 𝜋 4 + 𝑖 3 𝜋 4 c o s s i n , find | 𝑍 | .

Q23:

Find the modulus and the principal amplitude of the number 𝑍 = − 4 1 ( 3 0 + 𝑖 3 0 ) c o s s i n ∘ ∘ .

  • A | 𝑍 | = √ 4 1 , principal amplitude − 1 5 0 ∘
  • B | 𝑍 | = 4 1 , principal amplitude − 1 5 0 ∘
  • C | 𝑍 | = 4 1 , principal amplitude 1 5 0 ∘
  • D | 𝑍 | = √ 4 1 , principal amplitude 1 5 0 ∘

Q24:

Find the modulus and the principal amplitude of the number 𝑍 = − 3 7 5 𝜋 3 − 𝑖 5 𝜋 3 s i n c o s .

  • A | 𝑍 | = 3 7 , principal amplitude 𝜃 = − 𝜋 6
  • B | 𝑍 | = 3 7 , principal amplitude 𝜃 = 𝜋 6
  • C | 𝑍 | = √ 3 7 , principal amplitude 𝜃 = − 𝜋 6
  • D | 𝑍 | = √ 3 7 , principal amplitude 𝜃 = 𝜋 6

Q25:

Find the modulus and the principal amplitude of the number 𝑍 = 1 6 + 1 6 𝑖 3 0 5 t a n ∘ .

  • A | 𝑍 | = 1 6 3 0 5 c o s ∘ , principal amplitude 𝜃 = − 5 5 ∘
  • B | 𝑍 | = 1 6 3 0 5 s e c ∘ , principal amplitude 𝜃 = − 5 5 ∘
  • C | 𝑍 | = 1 6 3 0 5 c s c ∘ , principal amplitude 𝜃 = 2 3 5 ∘
  • D | 𝑍 | = 1 6 3 0 5 s e c ∘ , principal amplitude 𝜃 = 2 3 5 ∘

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