Unlocking Complex Number Operations: Multiplication and Division in Trigonometric Form

PreAlgebra Grades High School 2:16 Video
Print Lesson Plan Watch Video

Lesson Description

Explore how to multiply and divide complex numbers when expressed in trigonometric form. This lesson covers the formulas, provides examples, and explains the conversion back to standard form.

Video Resource

Multiply and Divide Complex Numbers in Trigonometric Form (Formulas)

Mario's Math Tutoring

Duration: 2:16
Watch on YouTube

Key Concepts

  • Trigonometric form of complex numbers
  • Multiplication of complex numbers in trigonometric form
  • Division of complex numbers in trigonometric form
  • Conversion between trigonometric and standard form of complex numbers

Learning Objectives

  • Students will be able to multiply complex numbers in trigonometric form.
  • Students will be able to divide complex numbers in trigonometric form.
  • Students will be able to convert between trigonometric and standard form of complex numbers after performing operations.

Educator Instructions

  • Introduction (5 mins)
    Begin by reviewing the trigonometric form of complex numbers. Briefly discuss the representation of a complex number as z = r(cosθ + isinθ), where r is the magnitude and θ is the angle.
  • Formula Presentation (5 mins)
    Present the formulas for multiplying and dividing complex numbers in trigonometric form: Multiplication: z1 * z2 = r1 * r2 [cos(θ1 + θ2) + isin(θ1 + θ2)] Division: z1 / z2 = (r1 / r2) [cos(θ1 - θ2) + isin(θ1 - θ2)] Emphasize the importance of order in division.
  • Worked Examples (15 mins)
    Work through the example from the video: Multiply and Divide: z1 = 4(cos60° + isin60°) and z2 = 2(cos30° + isin30°) Multiplication: 4 * 2 [cos(60° + 30°) + isin(60° + 30°)] = 8(cos90° + isin90°) = 8i Division: (4 / 2) [cos(60° - 30°) + isin(60° - 30°)] = 2(cos30° + isin30°) = √3 + i Explain each step clearly, emphasizing the addition/subtraction of angles and multiplication/division of magnitudes.
  • Practice Problems (15 mins)
    Provide students with practice problems to solve in class. Example problems: Multiply: 3(cos45° + isin45°) and 5(cos15° + isin15°) Divide: 10(cos120° + isin120°) by 2(cos30° + isin30°) Have students work individually or in pairs, then review the solutions as a class.
  • Wrap-up (5 mins)
    Summarize the key concepts: multiplying magnitudes and adding angles for multiplication, dividing magnitudes and subtracting angles for division. Reiterate the importance of order in division and the ability to convert back to standard form. Mention the links to DeMoivre's Theorem and Nth Roots of Complex Numbers for further exploration.

Interactive Exercises

  • Complex Number Operation Game
    Create a game where students are given two complex numbers in trigonometric form and are asked to either multiply or divide them. The game can be played individually or in teams, with points awarded for correct answers and speed.

Discussion Questions

  • Why is the order important when dividing complex numbers in trigonometric form?
  • How does converting back to standard form help you visualize the result of the operation?
  • Can you relate these operations to transformations in the complex plane?

Skills Developed

  • Applying trigonometric identities
  • Performing arithmetic operations with complex numbers
  • Converting between different forms of complex numbers
  • Problem-solving

Multiple Choice Questions

Question 1:

What operation is performed on the magnitudes (r values) when multiplying two complex numbers in trigonometric form?

Correct Answer: Multiplication

Question 2:

What operation is performed on the angles (θ values) when dividing two complex numbers in trigonometric form?

Correct Answer: Subtraction

Question 3:

If z1 = 2(cos30° + isin30°) and z2 = 3(cos60° + isin60°), what is the magnitude of z1 * z2?

Correct Answer: 6

Question 4:

If z1 = 8(cos120° + isin120°) and z2 = 4(cos30° + isin30°), what is the angle of z1 / z2?

Correct Answer: 90°

Question 5:

Which of the following is the correct formula for multiplying two complex numbers in trigonometric form?

Correct Answer: z1 * z2 = r1 * r2 [cos(θ1 + θ2) + isin(θ1 + θ2)]

Question 6:

When dividing z1 by z2, which 'r' value should be in the numerator?

Correct Answer: r1

Question 7:

If z1 = 5(cos45° + isin45°) and z2 = 2(cos15° + isin15°), what is the magnitude of z1/z2?

Correct Answer: 2.5

Question 8:

What is the first step to multiplying two complex numbers in trigonometric form?

Correct Answer: Multiply the radii

Question 9:

What is the second step to dividing two complex numbers in trigonometric form?

Correct Answer: Subtract the angles

Question 10:

Which form represents a complex number as z = r(cosθ + isinθ)?

Correct Answer: Trigonometric Form

Fill in the Blank Questions

Question 1:

When multiplying complex numbers in trigonometric form, you _________ the magnitudes.

Correct Answer: multiply

Question 2:

When dividing complex numbers in trigonometric form, you _________ the angles.

Correct Answer: subtract

Question 3:

The formula for multiplying complex numbers z1 and z2 is z1 * z2 = r1 * r2 [cos(θ1 + θ2) + i sin(θ1 + θ2)]. The angles θ1 and θ2 are ________.

Correct Answer: added

Question 4:

The formula for dividing complex numbers z1 and z2 is z1 / z2 = (r1 / r2) [cos(θ1 - θ2) + i sin(θ1 - θ2)]. The magnitudes r1 and r2 are _________.

Correct Answer: divided

Question 5:

In the trigonometric form of a complex number, 'r' represents the _________.

Correct Answer: magnitude

Question 6:

In the trigonometric form of a complex number, 'θ' represents the _________.

Correct Answer: angle

Question 7:

When converting back to standard form, one must evaluate sine and cosine of the resulting ________.

Correct Answer: angle

Question 8:

The result of dividing two complex numbers can be converted back to ________ form.

Correct Answer: standard

Question 9:

When dividing complex numbers, the 'r' value of z1 should be in the _________ of the fraction.

Correct Answer: numerator

Question 10:

Addition is commutative, but _________ is not commutative.

Correct Answer: subtraction

Educational Standards

PC.T.2.1:Create models for situations involving trigonometry. PC.F.1.1:Interpret characteristics of a function defined by an expression in the context of the situation. PC.T.2:Apply trigonometric concepts beyond the right triangle.

Teaching Materials

Download ready-to-use materials for this lesson:

Student Worksheet Classroom Slides Assessment Quiz
Add to Google Classroom

User Actions

Sign in to save this lesson plan to your favorites.

Sign In

Share This Lesson

Related Lesson Plans