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Worksheet: Polar Form of Complex Numbers

Q1:

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

  • A βˆ’ 4 ο€» ο€» πœ‹ 3  + 𝑖 ο€» πœ‹ 3   c o s s i n
  • B 4 ο€» ο€» πœ‹ 3  + 𝑖 ο€» πœ‹ 3   c o s s i n
  • C βˆ’ 4 ο€» ο€» βˆ’ πœ‹ 3  + 𝑖 ο€» βˆ’ πœ‹ 3   c o s s i n
  • D 4 ο€» ο€» βˆ’ πœ‹ 3  + 𝑖 ο€» βˆ’ πœ‹ 3   c o s s i n

Q2:

Given that 𝑧 = 1 1 and 𝑧 = ( 3 πœƒ + 𝑖 3 πœƒ ) 2 2 c o s s i n , find the trigonometric form of 𝑧 𝑧 1 2 .

  • A c o s s i n ( 2 πœ‹ βˆ’ 3 πœƒ ) + 𝑖 ( 2 πœ‹ βˆ’ 3 πœƒ )
  • B c o s s i n ( 2 πœ‹ + 6 πœƒ ) + 𝑖 ( 2 πœ‹ + 6 πœƒ )
  • C c o s s i n ( πœ‹ βˆ’ 6 πœƒ ) + 𝑖 ( πœ‹ βˆ’ 6 πœƒ )
  • D c o s s i n ( 2 πœ‹ βˆ’ 6 πœƒ ) + 𝑖 ( 2 πœ‹ βˆ’ 6 πœƒ )

Q3:

Express 1 2  ο€Ό 5 πœ‹ 6  + 𝑖 ο€Ό 5 πœ‹ 6   c o s s i n in algebraic form.

  • A βˆ’ 6 βˆ’ 6 √ 3 𝑖
  • B 6 βˆ’ 6 √ 3 𝑖
  • C 6 √ 3 βˆ’ 6 𝑖
  • D βˆ’ 6 √ 3 + 6 𝑖

Q4:

Find the modulus of the complex number 1 + 𝑖 .

  • A1
  • B2
  • C4
  • D √ 2
  • E √ 3

Find the argument of the complex number 1 + 𝑖 .

  • A πœ‹ 4
  • B βˆ’ πœ‹ 4
  • C πœ‹
  • D πœ‹ 2
  • E βˆ’ πœ‹ 2

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

  • A √ 2 ( πœ‹ + 𝑖 πœ‹ ) c o s s i n
  • B 2 ο€» πœ‹ 4 + 𝑖 πœ‹ 4  c o s s i n
  • C √ 2 ο€» πœ‹ 4 + 𝑖 πœ‹ 4  c o s s i n
  • D √ 2 ο€» πœ‹ 2 + 𝑖 πœ‹ 2  c o s s i n
  • E 2 ο€» πœ‹ 2 + 𝑖 πœ‹ 2  c o s s i n

Q5:

Consider the diagram.

Which of the following correctly describes the relationship between π‘Ž , π‘Ÿ , and πœƒ ?

  • A π‘Ž = π‘Ÿ πœƒ t a n
  • B π‘Ž = π‘Ÿ πœƒ s i n
  • C π‘Ž = πœƒ π‘Ÿ c o s
  • D π‘Ž = π‘Ÿ πœƒ c o s
  • E π‘Ž = πœƒ π‘Ÿ s i n

Which of the following correctly describes the relationship between 𝑏 , π‘Ÿ , and πœƒ ?

  • A 𝑏 = π‘Ÿ πœƒ s i n
  • B 𝑏 = πœƒ π‘Ÿ c o s
  • C 𝑏 = π‘Ÿ πœƒ c o s
  • D 𝑏 = πœƒ π‘Ÿ s i n
  • E 𝑏 = π‘Ÿ πœƒ t a n

Hence, express 𝑧 in terms of π‘Ÿ and πœƒ .

  • A 𝑧 = π‘Ÿ πœƒ + 𝑖 πœƒ π‘Ÿ c o s s i n
  • B 𝑧 = π‘Ÿ πœƒ + π‘Ÿ 𝑖 πœƒ s i n c o s
  • C 𝑧 = π‘Ÿ πœƒ + π‘Ÿ 𝑖 πœƒ c o s s i n
  • D 𝑧 = πœƒ π‘Ÿ + 𝑖 πœƒ π‘Ÿ c o s s i n
  • E 𝑧 = πœƒ π‘Ÿ + 𝑖 πœƒ π‘Ÿ s i n c o s

Q6:

The Argand diagram shows the complex number 𝑧 .

Write 𝑧 in rectangular form.

  • A 3 βˆ’ 5 𝑖
  • B 5 + 3 𝑖
  • C 5 βˆ’ 3 𝑖
  • D 3 + 5 𝑖
  • E βˆ’ ( 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 8 ( 1 . 0 3 βˆ’ 𝑖 1 . 0 3 ) c o s s i n
  • C √ 3 4 ( 1 . 0 3 βˆ’ 𝑖 1 . 0 3 ) c o s s i n
  • D 3 4 ( 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

Q7:

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

  • A 𝑍 = 4 ο€» ο€» πœ‹ 2  βˆ’ 𝑖 ο€» πœ‹ 2   c o s s i n
  • B 𝑍 = 4 ο€» ο€» βˆ’ πœ‹ 2  + 𝑖 ο€» βˆ’ πœ‹ 2   c o s s i n
  • C 𝑍 = 4 ο€» ο€» βˆ’ πœ‹ 2  βˆ’ 𝑖 ο€» βˆ’ πœ‹ 2   c o s s i n
  • D 𝑍 = 4 ο€» ο€» πœ‹ 2  + 𝑖 ο€» πœ‹ 2   c o s s i n

Q8:

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

  • A 2  1 7 πœ‹ 6 + 𝑖 1 7 πœ‹ 6  c o s s i n
  • B 2  7 πœ‹ 3 + 𝑖 7 πœ‹ 3  c o s s i n
  • C 2  1 1 πœ‹ 6 βˆ’ 𝑖 1 1 πœ‹ 6  c o s s i n
  • D 2  1 1 πœ‹ 6 + 𝑖 1 1 πœ‹ 6  c o s s i n
  • E 1 3  1 1 πœ‹ 6 + 𝑖 1 1 πœ‹ 6  c o s s i n

Q9:

If 𝑍 = π‘Ÿ ( πœƒ + 𝑖 πœƒ ) c o s s i n , what is 1 𝑍 ?

  • A 1 π‘Ÿ ( πœƒ + 𝑖 πœƒ ) c o s s i n
  • B π‘Ÿ ( ( βˆ’ πœƒ ) + 𝑖 ( βˆ’ πœƒ ) ) c o s s i n
  • C π‘Ÿ ( ( βˆ’ πœƒ ) βˆ’ 𝑖 ( βˆ’ πœƒ ) ) c o s s i n
  • D 1 π‘Ÿ ( ( βˆ’ πœƒ ) + 𝑖 ( βˆ’ πœƒ ) ) c o s s i n

Q10:

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

  • A βˆ’ 3 βˆ’ 3 𝑖 , 3 √ 2 ο€Ό ο€Ό βˆ’ 3 πœ‹ 4  + 𝑖 ο€Ό βˆ’ 3 πœ‹ 4   c o s s i n
  • B βˆ’ 3 + 3 𝑖 , 3 √ 2 ο€Ό ο€Ό 3 πœ‹ 4  + 𝑖 ο€Ό 3 πœ‹ 4   c o s s i n
  • C 3 βˆ’ 3 𝑖 , 3 √ 2 ο€» ο€» βˆ’ πœ‹ 4  + 𝑖 ο€» βˆ’ πœ‹ 4   c o s s i n
  • D 3 + 3 𝑖 , 3 √ 2 ο€» ο€» πœ‹ 4  + 𝑖 ο€» πœ‹ 4   c o s s i n

Q11:

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

  • A βˆ’ 5 𝑖 , 5 ( πœ‹ + 𝑖 πœ‹ ) c o s s i n
  • B 5 𝑖 , 5 ο€» ο€» πœ‹ 2  + 𝑖 ο€» πœ‹ 2   c o s s i n
  • C 5 𝑖 , 5 ( 0 + 𝑖 0 ) c o s s i n
  • D βˆ’ 5 𝑖 , 5 ο€» ο€» βˆ’ πœ‹ 2  + 𝑖 ο€» βˆ’ πœ‹ 2   c o s s i n

Q12:

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

  • A 2 ( πœ‹ + 𝑖 πœ‹ ) c o s s i n
  • B 2 ο€» ο€» βˆ’ πœ‹ 2  + 𝑖 ο€» βˆ’ πœ‹ 2   c o s s i n
  • C 2 ( 0 + 𝑖 0 ) c o s s i n
  • D 2 ο€» ο€» πœ‹ 2  + 𝑖 ο€» πœ‹ 2   c o s s i n

Q13:

Given that 𝑍 = ( 6 𝑖 βˆ’ 6 ) ( 4 + 3 𝑖 ) ( 1 + 2 𝑖 ) 2 , express the complex number 𝑍 in the form of π‘₯ + 𝑦 𝑖 , and then determine its trigonometric form.

  • A 6 + 6 𝑖 , ο€» ο€» πœ‹ 4  + 𝑖 ο€» πœ‹ 4   c o s s i n
  • B 6 βˆ’ 6 𝑖 , 6 √ 2 ο€» ο€» βˆ’ πœ‹ 4  + 𝑖 ο€» βˆ’ πœ‹ 4   c o s s i n
  • C 6 βˆ’ 6 𝑖 , ο€» ο€» βˆ’ πœ‹ 4  + 𝑖 ο€» βˆ’ πœ‹ 4   c o s s i n
  • D 6 + 6 𝑖 , 6 √ 2 ο€» ο€» πœ‹ 4  + 𝑖 ο€» πœ‹ 4   c o s s i n

Q14:

Determine, in trigonometric form, the square roots of ο€½ βˆ’ 5 βˆ’ 5 𝑖 βˆ’ 5 + 5 𝑖  9 .

  • A ο€» ο€» πœ‹ 4  + 𝑖 ο€» πœ‹ 4   c o s s i n , ο€Ό ο€Ό 3 πœ‹ 4  + 𝑖 ο€Ό 3 πœ‹ 4   c o s s i n
  • B ο€Ό ο€Ό 3 πœ‹ 4  + 𝑖 ο€Ό 3 πœ‹ 4   c o s s i n , ο€» ο€» βˆ’ πœ‹ 4  + 𝑖 ο€» βˆ’ πœ‹ 4   c o s s i n
  • C ο€» ο€» βˆ’ πœ‹ 4  + 𝑖 ο€» βˆ’ πœ‹ 4   c o s s i n , ο€» ο€» πœ‹ 4  + 𝑖 ο€» πœ‹ 4   c o s s i n
  • D ο€» ο€» πœ‹ 4  + 𝑖 ο€» πœ‹ 4   c o s s i n , ο€Ό ο€Ό βˆ’ 3 πœ‹ 4  + 𝑖 ο€Ό βˆ’ 3 πœ‹ 4   c o s s i n

Q15:

Simplify βˆ’ 7 + 4 √ 3 + ο€» βˆ’ 7 √ 3 βˆ’ 4  𝑖 7 + 4 𝑖 , giving your answer in both algebraic and trigonometric form.

  • A βˆ’ 1 + √ 3 𝑖 , 2 ο€Ό ο€Ό 2 πœ‹ 3  + 𝑖 ο€Ό 2 πœ‹ 3   c o s s i n
  • B 1 βˆ’ √ 3 𝑖 , 2 ο€» ο€» βˆ’ πœ‹ 3  + 𝑖 ο€» βˆ’ πœ‹ 3   c o s s i n
  • C 1 + √ 3 𝑖 , 2 ο€» ο€» πœ‹ 3  + 𝑖 ο€» πœ‹ 3   c o s s i n
  • D βˆ’ 1 βˆ’ √ 3 𝑖 , 2 ο€Ό ο€Ό βˆ’ 2 πœ‹ 3  + 𝑖 ο€Ό βˆ’ 2 πœ‹ 3   c o s s i n

Q16:

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

  • A 𝑍 = 9 ο€» πœ‹ 6  + 𝑖 ο€» πœ‹ 6  c o s s i n
  • B 𝑍 = 9  ο€» πœ‹ 6  + 𝑖 ο€» πœ‹ 6   s i n c o s
  • C 𝑍 = 9 ο€» πœ‹ 6  + 𝑖 ο€» πœ‹ 6  s i n c o s
  • D 𝑍 = 9  ο€» πœ‹ 6  + 𝑖 ο€» πœ‹ 6   c o s s i n
  • E 𝑍 = 9  ο€» πœ‹ 6  βˆ’ 𝑖 ο€» πœ‹ 6   c o s s i n

Q17:

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

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

Q18:

Given that | 𝑍 | = 5 and the argument of 𝑍 is πœƒ = 2 πœ‹ + 2 𝑛 πœ‹ , where 𝑛 ∈ β„€ , find 𝑍 , giving your answer in trigonometric form.

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

Q19:

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

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

Q20:

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

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

Q21:

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

  • A 𝑧 = 5 𝑖
  • B 𝑧 = 5
  • C 𝑧 = βˆ’ 5
  • D 𝑧 = βˆ’ 5 𝑖
  • E 𝑧 = 5 + 5 𝑖

Q22:

Given that 𝑍 = 7 [ ( βˆ’ 5 8 ) + 𝑖 ( βˆ’ 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 = βˆ’ 3 . 7 1 + 5 . 9 4 𝑖
  • B 𝑍 = 5 . 9 4 + 3 . 7 1 𝑖
  • C 𝑍 = βˆ’ 5 . 9 4 + 5 . 9 4 𝑖
  • D 𝑍 = 3 . 7 1 βˆ’ 5 . 9 4 𝑖

Q23:

Given that , find the principal argument of , where .

  • A
  • B
  • C
  • D
  • E

Q24:

Find c o s πœ‹ 6 .

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

Find s i n πœ‹ 6 .

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

Hence, express the complex number 1 0 ο€» πœ‹ 6 + 𝑖 πœ‹ 6  c o s s i n in rectangular form.

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

Q25:

Given that 𝑧 = 1 3 ( 3 0 + 𝑖 3 0 ) c o s s i n ∘ ∘ , find 1 𝑧 .

  • A 1 3 ( 3 3 0 + 𝑖 3 3 0 ) c o s s i n ∘ ∘
  • B 3 ( 3 0 + 𝑖 3 0 ) c o s s i n ∘ ∘
  • C 3 ( 2 1 0 + 𝑖 2 1 0 ) c o s s i n ∘ ∘
  • D 3 ( 3 3 0 + 𝑖 3 3 0 ) c o s s i n ∘ ∘
  • E 1 3 ( 2 1 0 + 𝑖 2 1 0 ) c o s s i n ∘ ∘