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Worksheet: Converting between Different Forms of Complex Numbers

Q1:

Put 𝑧 = 4 √ 3 ο€Ό 5 πœ‹ 6 βˆ’ 𝑖 5 πœ‹ 6  c o s s i n in exponential form.

  • A 𝑒 7 πœ‹ 6 𝑖
  • B 4 √ 3 𝑒 5 πœ‹ 6 𝑖
  • C 𝑒 5 πœ‹ 6 𝑖
  • D 4 √ 3 𝑒 7 πœ‹ 6 𝑖
  • E √ 3 1 2 𝑒 7 πœ‹ 6 𝑖

Q2:

Put 𝑧 = βˆ’ 4 √ 3 ο€Ό 5 πœ‹ 6 + 𝑖 5 πœ‹ 6  s i n c o s in exponential form.

  • A 𝑒 2 πœ‹ 3 𝑖
  • B 4 √ 3 𝑒 5 πœ‹ 6 𝑖
  • C βˆ’ 4 √ 3 𝑒 2 πœ‹ 3 𝑖
  • D 4 √ 3 𝑒 2 πœ‹ 3 𝑖
  • E 1 2 𝑒 2 πœ‹ 3 𝑖

Q3:

Express the complex number 𝑍 = 𝑒 βˆ’ 4 βˆ’ 𝑖 2 3 πœ‹ 1 2 in exponential form.

  • A 𝑒 βˆ’ 𝑒 βˆ’ 4 𝑖 πœ‹ 1 2
  • B 𝑒 β‹… 𝑒 4 𝑖 πœ‹ 1 2
  • C 𝑒 βˆ’ 𝑖 7 1 πœ‹ 1 2
  • D 𝑒 β‹… 𝑒 βˆ’ 4 𝑖 πœ‹ 1 2

Q4:

Given that 𝑧 = √ 3 2 βˆ’ 3 2 𝑖 , find 𝑧 5 , giving your answer in exponential form.

  • A 5 √ 3 𝑒 πœ‹ 3 𝑖
  • B √ 3 𝑒 πœ‹ 3 𝑖
  • C 9 √ 3 𝑒 πœ‹ 6 𝑖
  • D 9 √ 3 𝑒 πœ‹ 3 𝑖

Q5:

Given that 𝑍 = 1 1 ( 3 1 5 + 𝑖 3 1 5 ) c o s s i n ∘ ∘ , express 𝑍 in algebraic form.

  • A 𝑍 = βˆ’ 1 1 √ 2 2 + 1 1 √ 2 2 𝑖
  • B 𝑍 = 1 1 √ 2 2 + 1 1 √ 2 2 𝑖
  • C 𝑍 = βˆ’ 1 1 √ 2 2 βˆ’ 1 1 √ 2 2 𝑖
  • D 𝑍 = 1 1 √ 2 2 βˆ’ 1 1 √ 2 2 𝑖

Q6:

Given that 𝑍 = 5 [ ( βˆ’ 3 3 0 ) + 𝑖 ( βˆ’ 3 3 0 ) ] c o s s i n ∘ ∘ , express 𝑍 in algebraic form.

  • A 𝑍 = βˆ’ 5 2 + 5 √ 3 2 𝑖
  • B 𝑍 = 5 2 + 5 √ 3 2 𝑖
  • C 𝑍 = βˆ’ 5 √ 3 2 βˆ’ 5 2 𝑖
  • D 𝑍 = 5 √ 3 2 + 5 2 𝑖
  • E 𝑍 = 5 √ 3 2 βˆ’ 5 2 𝑖

Q7:

Put 𝑧 = 6 ο€» βˆ’ πœ‹ 4 + 𝑖 πœ‹ 4  c o s s i n in exponential form.

  • A 𝑒 3 πœ‹ 4 𝑖
  • B 6 𝑒 πœ‹ 4 𝑖
  • C 𝑒 πœ‹ 4 𝑖
  • D 6 𝑒 3 πœ‹ 4 𝑖
  • E √ 2 2 𝑒 3 πœ‹ 4 𝑖

Q8:

Given that 𝑍 = 6 [ πœ‹ + 𝑖 πœ‹ ] c o s s i n , find the algebraic form of 𝑍 .

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

Q9:

Given that 𝑍 = 6 √ 2 βˆ’ 6 √ 2 𝑖 , write 𝑍 in trigonometric form.

  • A 𝑍 = 1 2  1 1 πœ‹ 4 + 𝑖 1 1 πœ‹ 4  c o s s i n
  • B 𝑍 = 1 2  9 πœ‹ 4 + 𝑖 9 πœ‹ 4  c o s s i n
  • C 𝑍 = 1 2  7 πœ‹ 4 βˆ’ 𝑖 7 πœ‹ 4  c o s s i n
  • D 𝑍 = 1 2  7 πœ‹ 4 + 𝑖 7 πœ‹ 4  c o s s i n
  • E 𝑍 = 3  7 πœ‹ 4 + 𝑖 7 πœ‹ 4  c o s s i n

Q10:

Express the number √ 3 𝑖 in trigonometric form.

  • A √ 3 ( 9 0 + 𝑖 9 0 ) s i n c o s ∘ ∘
  • B βˆ’ √ 3 ( 9 0 + 𝑖 9 0 ) c o s s i n ∘ ∘
  • C √ 3 ( 0 + 𝑖 0 ) s i n c o s ∘ ∘
  • D √ 3 ( 9 0 + 𝑖 9 0 ) c o s s i n ∘ ∘
  • E √ 3 ( 1 8 0 + 𝑖 1 8 0 ) c o s s i n ∘ ∘

Q11:

Express the number βˆ’ 1 + 𝑖 in trigonometric form.

  • A √ 2 ( 1 3 5 + 𝑖 1 3 5 ) s i n c o s ∘ ∘
  • B βˆ’ √ 2 ( 1 3 5 + 𝑖 1 3 5 ) c o s s i n ∘ ∘
  • C βˆ’ √ 2 ( 1 3 5 + 𝑖 1 3 5 ) s i n c o s ∘ ∘
  • D √ 2 ( 1 3 5 + 𝑖 1 3 5 ) c o s s i n ∘ ∘
  • E √ 2 ( 2 2 5 + 𝑖 2 2 5 ) c o s s i n ∘ ∘

Q12:

Simplify 𝑧 = 4 √ 3 πœ” + 4 √ 3 πœ” + 3 √ 3 + ο€Ή 5 πœ” + 6 πœ” + 6 πœ”  𝑖 4 5 3 2 , giving your answer in algebraic form, and hence express the square roots of 𝑧 in exponential form.

  • A √ 3 + 𝑖 , √ 2 𝑒 πœ‹ 1 2 𝑖 , √ 2 𝑒 βˆ’ 𝑖 1 1 πœ‹ 1 2
  • B √ 3 βˆ’ 𝑖 , √ 2 𝑒 1 1 πœ‹ 1 2 𝑖 , √ 2 𝑒 βˆ’ 𝑖 πœ‹ 1 2
  • C βˆ’ √ 3 + 𝑖 , √ 2 𝑒 5 πœ‹ 1 2 𝑖 , √ 2 𝑒 βˆ’ 𝑖 7 πœ‹ 1 2
  • D βˆ’ √ 3 βˆ’ 𝑖 , √ 2 𝑒 7 πœ‹ 1 2 𝑖 , √ 2 𝑒 βˆ’ 𝑖 5 πœ‹ 1 2

Q13:

Put 𝑧 = 3 √ 2 ο€» βˆ’ πœ‹ 4 + 𝑖 πœ‹ 4  s i n c o s in exponential form.

  • A 𝑒 3 πœ‹ 4 𝑖
  • B 3 √ 2 𝑒 πœ‹ 4 𝑖
  • C βˆ’ 3 √ 2 𝑒 3 πœ‹ 4 𝑖
  • D 3 √ 2 𝑒 3 πœ‹ 4 𝑖
  • E √ 2 2 𝑒 3 πœ‹ 4 𝑖

Q14:

Which of the following expresses the complex number 𝑖 in polar coordinates?

  • A ( 1 , πœ‹ )
  • B ο€» βˆ’ 1 , πœ‹ 2 
  • C ( βˆ’ 1 , πœ‹ )
  • D ο€» 1 , πœ‹ 2 

Q15:

Given that 𝑉 = 5 √ 2 2 βˆ’ 5 √ 2 2 𝑖 , find the trigonometric form of 1 𝑉 .

  • A 1 𝑉 = 1 2 5  ο€Ό 1 1 πœ‹ 4  + 𝑖 ο€Ό 1 1 πœ‹ 4   c o s s i n
  • B 1 𝑉 = 1 5  ο€Ό 9 πœ‹ 4  + 𝑖 ο€Ό 9 πœ‹ 4   c o s s i n
  • C 1 𝑉 = 5  ο€Ό 7 πœ‹ 4  + 𝑖 ο€Ό 7 πœ‹ 4   c o s s i n
  • D 1 𝑉 = 1 5  ο€Ό 7 πœ‹ 4  + 𝑖 ο€Ό 7 πœ‹ 4   c o s s i n
  • E 1 𝑉 = 1 2 5  ο€Ό 7 πœ‹ 4  + 𝑖 ο€Ό 7 πœ‹ 4   c o s s i n

Q16:

Simplify ( βˆ’ 1 βˆ’ 𝑖 ) 6 , giving your answer in trigonometric form.

  • A 8 ( 2 7 0 + 𝑖 2 7 0 ) s i n c o s ∘ ∘
  • B βˆ’ 8 ( 2 7 0 + 𝑖 2 7 0 ) c o s s i n ∘ ∘
  • C 8 ( 1 8 0 + 𝑖 1 8 0 ) s i n c o s ∘ ∘
  • D 8 ( 2 7 0 + 𝑖 2 7 0 ) c o s s i n ∘ ∘
  • E 8 ( 9 0 + 𝑖 9 0 ) c o s s i n ∘ ∘

Q17:

Put βˆ’ 8 2 + 2 √ 3 𝑖 in the form π‘₯ + 𝑦 𝑖 , where π‘₯ , 𝑦 ∈ ℝ , and then write it in trigonometric form.

  • A 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
  • 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 + √ 3 𝑖 , 2 ο€Ό ο€Ό 2 πœ‹ 3  + 𝑖 ο€Ό 2 πœ‹ 3   c o s s i n

Q18:

Put 𝑧 = βˆ’ 7 ο€» πœ‹ 4 + 𝑖 πœ‹ 4  c o s s i n in exponential form.

  • A 𝑒 5 πœ‹ 4 𝑖
  • B 7 𝑒 πœ‹ 4 𝑖
  • C βˆ’ 7 𝑒 5 πœ‹ 4 𝑖
  • D 7 𝑒 5 πœ‹ 4 𝑖

Q19:

Express the complex number 𝑍 = 1 4 4 3 √ 3 + 3 𝑖 in trigonometric form.

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

Q20:

Simplify βˆ’ 8 + 4 √ 3 𝑖 5 + √ 3 𝑖 , giving your answer in both algebraic and trigonometric form.

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

Q21:

Determine the real part of a complex number whose modulus is 4 √ 3 and principal amplitude is 5 πœ‹ 6 .

  • A6
  • B 2 √ 3
  • C βˆ’ 2 √ 3
  • D βˆ’ 6