Pure Mathematics · 3rd secondary · First Term

Algebra - Complex numbers — The trigonometric form of a complex number

About this lesson

This unit focuses on the trigonometric (polar) and exponential (Euler) forms of complex numbers. It begins by reviewing the algebraic form z = x + yi and introduces the Argand diagram for graphical representation. Students learn to convert between Cartesian and polar coordinates, and to find the modulus and principle amplitude (argument) of a complex number, with special attention to the quadrant in which the number lies. The unit then presents the trigonometric form z = r(cosθ + i sinθ) and demonstrates how to multiply and divide complex numbers in this form, including the use of De Moivre's theorem for powers. Later, the exponential form z = re^(iθ) is introduced via Taylor series, and students practice converting between algebraic, trigonometric, and exponential forms, as well as performing multiplication and division in exponential form. The unit includes numerous worked examples and exercises, often involving angles in both degrees and radians, and emphasizes the importance of the principle amplitude being in the interval (-π, π]. By the end, students should be able to represent complex numbers in all three forms, convert between them, and perform arithmetic operations using the trigonometric and exponential forms.

Main topics in this lesson

  • Trigonometric form of complex numbers
  • Exponential (Euler) form of complex numbers
  • Modulus and amplitude (argument) of complex numbers
  • Argand diagram representation
  • Multiplication and division of complex numbers in trigonometric and exponential forms
  • De Moivre's theorem for powers

Key terms and vocabulary

  • complex number
  • algebraic form
  • trigonometric form
  • exponential form
  • Euler form
  • Argand's plane
  • modulus
  • amplitude
  • principle amplitude
  • argument
  • conjugate
  • polar coordinates
  • Cartesian coordinates
  • De Moivre's theorem
  • Taylor series

Questions and answers on Algebra - Complex numbers — The trigonometric form of a complex number - practice and revision

Multiple choice questions on this lesson in Pure Mathematics for 3rd secondary, First Term, with practice exercises for revision and exam preparation.

1

باستخدام نظرية ديموافر، فإن (cos π/6 + i sin π/6)⁴ يساوي:

  • 1 cos 2π/3 + i sin 2π/3
  • 2 cos π/24 + i sin π/24
  • 3 4cos 2π/3 + i 4sin 2π/3
  • 4 cos π/6 + i sin π/6
2

إذا كان z₁ = 6(cos 50° + i sin 50°) و z₂ = 2(cos 20° + i sin 20°)، فإن z₁ ÷ z₂ يساوي:

  • 1 3(cos 30° + i sin 30°)
  • 2 3(cos 70° + i sin 70°)
  • 3 12(cos 30° + i sin 30°)
  • 4 4(cos 30° + i sin 30°)
3

الصورة المثلثية للعدد المركب z = -1 - i هي:

  • 1 √2(cos 225° + i sin 225°)
  • 2 √2(cos 45° + i sin 45°)
  • 3 2(cos 225° + i sin 225°)
  • 4 √2(cos 135° + i sin 135°)
4

إذا كان z = -2i، فإن السعة الأساسية للعدد z تساوي:

  • 1 -π/2
  • 2 π/2
  • 3 π
  • 4 0
5

حاصل ضرب العددين z₁ = 3e^(iπ/6) و z₂ = 4e^(iπ/3) في الصورة الأسية يساوي:

  • 1 12e^(iπ/2)
  • 2 7e^(iπ/2)
  • 3 12e^(iπ/9)
  • 4 12e^(iπ/6)
6

إذا كان z = 5(cos 200° + i sin 200°)، فإن السعة الأساسية للعدد z هي:

  • 1 -160°
  • 2 200°
  • 3 160°
  • 4 -200°

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