Lesson Worksheet: Polar Form of Complex Numbers Mathematics

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.

  • A4𝜋3+𝑖𝜋3cossin
  • B4𝜋3+𝑖𝜋3cossin
  • C4𝜋3+𝑖𝜋3cossin
  • D4𝜋3+𝑖𝜋3cossin

Q2:

Express 125𝜋6+𝑖5𝜋6cossin in algebraic form.

  • A636𝑖
  • B63+6𝑖
  • C663𝑖
  • D663𝑖

Q3:

Find the modulus of the complex number 1+𝑖.

  • A1
  • B3
  • C2
  • D2
  • E4

Find the argument of the complex number 1+𝑖.

  • A𝜋4
  • B𝜋
  • C𝜋4
  • D𝜋2
  • E𝜋2

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

  • A2(𝜋+𝑖𝜋)cossin
  • B2𝜋2+𝑖𝜋2cossin
  • C2𝜋2+𝑖𝜋2cossin
  • D2𝜋4+𝑖𝜋4cossin
  • E2𝜋4+𝑖𝜋4cossin

Q4:

Consider the diagram.

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

  • A𝑎=𝑟𝜃sin
  • B𝑎=𝑟𝜃tan
  • C𝑎=𝑟𝜃cos
  • D𝑎=𝜃𝑟cos
  • E𝑎=𝜃𝑟sin

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

  • A𝑏=𝑟𝜃tan
  • B𝑏=𝜃𝑟cos
  • C𝑏=𝜃𝑟sin
  • D𝑏=𝑟𝜃sin
  • E𝑏=𝑟𝜃cos

Hence, express 𝑧 in terms of 𝑟 and 𝜃.

  • A𝑧=𝑟𝜃+𝑟𝑖𝜃cossin
  • B𝑧=𝜃𝑟+𝑖𝜃𝑟sincos
  • C𝑧=𝑟𝜃+𝑖𝜃𝑟cossin
  • D𝑧=𝑟𝜃+𝑟𝑖𝜃sincos
  • E𝑧=𝜃𝑟+𝑖𝜃𝑟cossin

Q5:

The Argand diagram shows the complex number 𝑧.

Write 𝑧 in rectangular form.

  • A5+3𝑖
  • B(3+5𝑖)
  • C35𝑖
  • D3+5𝑖
  • E53𝑖

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

  • A34(1.03𝑖1.03)cossin
  • B34(1.03+𝑖1.03)cossin
  • C8(1.03+𝑖1.03)cossin
  • D34(1.03+𝑖1.03)cossin
  • E8(1.03𝑖1.03)cossin

Q6:

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

  • A𝑍=4𝜋2+𝑖𝜋2cossin
  • B𝑍=4𝜋2𝑖𝜋2cossin
  • C𝑍=4𝜋2𝑖𝜋2cossin
  • D𝑍=4𝜋2+𝑖𝜋2cossin

Q7:

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

  • A1311𝜋6+𝑖11𝜋6cossin
  • B217𝜋6+𝑖17𝜋6cossin
  • C211𝜋6+𝑖11𝜋6cossin
  • D27𝜋3+𝑖7𝜋3cossin
  • E211𝜋6𝑖11𝜋6cossin

Q8:

Simplify 66𝑖2𝑖, giving your answer in both algebraic and trigonometric form.

  • A33𝑖, 323𝜋4+𝑖3𝜋4cossin
  • B3+3𝑖, 32𝜋4+𝑖𝜋4cossin
  • C33𝑖, 32𝜋4+𝑖𝜋4cossin
  • D3+3𝑖, 323𝜋4+𝑖3𝜋4cossin

Q9:

Simplify 5+53𝑖3𝑖, giving your answer in both algebraic and trigonometric form.

  • A5𝑖, 5𝜋2+𝑖𝜋2cossin
  • B5𝑖, 5(0+𝑖0)cossin
  • C5𝑖, 5(𝜋+𝑖𝜋)cossin
  • D5𝑖, 5𝜋2+𝑖𝜋2cossin

Q10:

Given that 𝑍2=(𝑍+2)𝑖, determine the trigonometric form of the complex number 𝑍.

  • A2𝜋2+𝑖𝜋2cossin
  • B2(𝜋+𝑖𝜋)cossin
  • C2𝜋2+𝑖𝜋2cossin
  • D2(0+𝑖0)cossin

Q11:

Given that 𝑍=(6𝑖6)(4+3𝑖)(1+2𝑖), express the complex number 𝑍 in the form of 𝑥+𝑦𝑖, and then determine its trigonometric form.

  • A66𝑖, 𝜋4+𝑖𝜋4cossin
  • B66𝑖, 62𝜋4+𝑖𝜋4cossin
  • C6+6𝑖, 62𝜋4+𝑖𝜋4cossin
  • D6+6𝑖, 𝜋4+𝑖𝜋4cossin

Q12:

Simplify 7+43+734𝑖7+4𝑖, giving your answer in both algebraic and trigonometric form.

  • A13𝑖, 2𝜋3+𝑖𝜋3cossin
  • B1+3𝑖, 2𝜋3+𝑖𝜋3cossin
  • C1+3𝑖, 22𝜋3+𝑖2𝜋3cossin
  • D13𝑖, 22𝜋3+𝑖2𝜋3cossin

Q13:

Given that |𝑍|=9 and the argument of 𝑍 is 𝜃=𝜋6, find 𝑍, giving your answer in trigonometric form.

  • A𝑍=9𝜋6+𝑖𝜋6sincos
  • B𝑍=9𝜋6+𝑖𝜋6sincos
  • C𝑍=9𝜋6+𝑖𝜋6cossin
  • D𝑍=9𝜋6𝑖𝜋6cossin
  • E𝑍=9𝜋6+𝑖𝜋6cossin

Q14:

Given that |𝑍|=8 and the argument of 𝑍 is 𝜃=360, find 𝑍, giving your answer in trigonometric form.

  • A𝑍=8𝜋+𝑖𝜋cossin
  • B𝑍=8[2𝜋+𝑖2𝜋]sincos
  • C𝑍=8[𝜋+𝑖𝜋]cossin
  • D𝑍=82𝜋+𝑖2𝜋cossin
  • E𝑍=8[2𝜋+𝑖2𝜋]cossin

Q15:

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

  • A𝑍=5(4𝜋+𝑖4𝜋)cossin
  • B𝑍=5(2𝜋+𝑖2𝜋)sincos
  • C𝑍=5(2𝜋+𝑖2𝜋)cossin
  • D𝑍=10(2𝜋+𝑖2𝜋)sincos
  • E𝑍=10(2𝜋+𝑖2𝜋)cossin

Q16:

Given that |𝑍|=3 and the argument of 𝑍 is 𝜃=𝜋3, find 𝑍, giving your answer in algebraic form.

  • A𝑍=32332𝑖
  • B𝑍=32+332𝑖
  • C𝑍=332+32𝑖
  • D𝑍=32332𝑖
  • E𝑍=332+32𝑖

Q17:

Given that |𝑍|=12 and the argument of 𝑍 is 𝜃=120, find 𝑍, giving your answer in algebraic form.

  • A𝑍=663𝑖
  • B𝑍=636𝑖
  • C𝑍=663𝑖
  • D𝑍=6+63𝑖
  • E𝑍=636𝑖

Q18:

Given that |𝑧|=5 and the argument of 𝑧 is 𝜃=270, find 𝑧, giving your answer in algebraic form.

  • A𝑧=5𝑖
  • B𝑧=5
  • C𝑧=5+5𝑖
  • D𝑧=5𝑖
  • E𝑧=5

Q19:

Given that 𝑍=7[(58)+𝑖(58)]cossin, determine the algebraic form of 𝑍, approximating the real and imaginary parts to the nearest two decimal places.

  • A𝑍=3.71+5.94𝑖
  • B𝑍=5.94+5.94𝑖
  • C𝑍=3.715.94𝑖
  • D𝑍=5.94+3.71𝑖

Q20:

Given that 𝑍=𝜃𝑖𝜃sincos, find the principal argument of 𝑍, where 𝜃0,𝜋2.

  • A𝜃𝜋2
  • B2𝜋𝜃
  • C𝜋+𝜃
  • D𝜋𝜃
  • E𝜃

Q21:

Find cos𝜋6.

  • A33
  • B333
  • C32
  • D12
  • E232

Find sin𝜋6.

  • A232
  • B33
  • C12
  • D32
  • E333

Hence, express the complex number 10𝜋6+𝑖𝜋6cossin in rectangular form.

  • A1033+5𝑖
  • B5+5𝑖
  • C5+1033𝑖
  • D5+53𝑖
  • E53+5𝑖

Q22:

Given that 𝑍=63𝜋4+𝑖3𝜋4cossin, find |𝑍|.

Q23:

Find the modulus and the principal amplitude of the number 𝑍=41(30+𝑖30)cossin.

  • A|𝑍|=41, principal amplitude 150
  • B|𝑍|=41, principal amplitude 150
  • C|𝑍|=41, principal amplitude 150
  • D|𝑍|=41, principal amplitude 150

Q24:

Find the modulus and the principal amplitude of the number 𝑍=375𝜋3𝑖5𝜋3sincos.

  • A|𝑍|=37, principal amplitude 𝜃=𝜋6
  • B|𝑍|=37, principal amplitude 𝜃=𝜋6
  • C|𝑍|=37, principal amplitude 𝜃=𝜋6
  • D|𝑍|=37, principal amplitude 𝜃=𝜋6

Q25:

Find the modulus and the principal amplitude of the number 𝑍=16+16𝑖305tan.

  • A|𝑍|=16305csc, principal amplitude 𝜃=235
  • B|𝑍|=16305cos, principal amplitude 𝜃=55
  • C|𝑍|=16305sec, principal amplitude 𝜃=235
  • D|𝑍|=16305sec, principal amplitude 𝜃=55

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