Kenneth ChengMathematical modeling
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Trigonometric representation of complex numbers (starred)

Read the idea, work independently, then explain what changed.

高一必修 第二册(A版).pdf · 7.3 · PDF 90 / printed page 83

Revisit first: Arithmetic of complex numbers

TOPIC 01

Trigonometric representation of complex numbers (starred)

Build understanding of trigonometric representation of complex numbers (starred) through definitions, contrasting cases and justified applications.

What you will be able to explain

  • Connect geometric and algebraic forms of trigonometric representation of complex numbers (starred).
  • Connect representations and justify the steps, including boundary cases.
  • Explain a solution and apply the idea to a changed situation.

Polar form

A nonzero complex number combines positive modulus with an argument modulo 2π.

z=r(cos⁡θ+isin⁡θ)z=r(\cos\theta+i\sin\theta)

Products and powers

Multiply moduli and add arguments; powers multiply arguments.

PREDICT → EXPLORE → EXPLAIN → TRANSFER

Make a prediction, then explore the relationship.

Lesson question: Before calculating, predict how the conclusion changes when one defining condition in trigonometric representation of complex numbers (starred) changes. Record a reason.

Model exploration: predict which displayed result changes with the parameters, check the values, and compare the observation with the lesson question.

z=2+(1)i; |z|=2.2361; argument=0.4636 rad (undefined at zero).

z=2+(1)i; |z|=2.2361; argument=0.4636 rad (undefined at zero).

Explain: Compare two admissible cases and one boundary or invalid case. Explain the observed difference using the stated definition.

Transfer: Construct a new example and a tempting incorrect solution. Repair the solution by naming the missing condition.

Try it. Leave your reasoning visible.

Use one hint at a time. A correction explains what changed, not just the final answer.

01 / Foundation#Worked example

Give a polar representation.

z=2iz=2i
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Use modulus and argument, remembering that arguments differ by full turns.
Hint 2
The point lies on the positive imaginary axis.
Worked solution

  1. Use modulus and argument, remembering that arguments differ by full turns.

  2. Calculate or simplify this relation.

    ∣z∣=2,arg⁡z=π/2+2nπ|z|=2,\quad\arg z=\pi/2+2n\pi

  3. A chosen argument is one representative from infinitely many.

2(cos(π/2)+i sin(π/2)).

Checks and common pitfalls: A chosen argument is one representative from infinitely many.

Reasoning checklist · self / teacher assessment
  • State the relevant definition, condition or model.
  • Show a valid calculation, proof or counterexample.
  • Interpret the conclusion with its restrictions.

Think first. Reveal a hint when the class is ready.

02 / Standard#Worked example

Find the modulus of the product.

∣z1∣=2,∣z2∣=3|z_1|=2,\quad|z_2|=3
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Use modulus and argument, remembering that arguments differ by full turns.
Hint 2
Multiplication multiplies moduli.
Worked solution

  1. Use modulus and argument, remembering that arguments differ by full turns.

  2. Calculate or simplify this relation.

    ∣z1z2∣=∣z1∣∣z2∣=6|z_1z_2|=|z_1||z_2|=6

  3. Arguments add, but lengths multiply.

The requested value is 6.

Checks and common pitfalls: Arguments add, but lengths multiply.

Think first. Reveal a hint when the class is ready.

03 / Transfer#Worked example

Why does zero have no uniquely defined argument?

0=0(cos⁡θ+isin⁡θ)0=0(\cos\theta+i\sin\theta)
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Use modulus and argument, remembering that arguments differ by full turns.
Hint 2
Look at the position of the origin relative to rays.
Worked solution

  1. Use modulus and argument, remembering that arguments differ by full turns.

  2. Calculate or simplify this relation.

    0cos⁡θ=0,0sin⁡θ=00\cos\theta=0,\quad0\sin\theta=0

  3. Argument describes a direction and requires a nonzero complex number.

Every θ gives zero because the modulus is zero.

Checks and common pitfalls: Argument describes a direction and requires a nonzero complex number.

Reasoning checklist · self / teacher assessment
  • State the relevant definition, condition or model.
  • Show a valid calculation, proof or counterexample.
  • Interpret the conclusion with its restrictions.

Think first. Reveal a hint when the class is ready.

04 / Foundation#Your turn

Give a polar representation.

z=3iz=3i
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Use modulus and argument, remembering that arguments differ by full turns.
Hint 2
The point lies on the positive imaginary axis.
Worked solution

  1. Use modulus and argument, remembering that arguments differ by full turns.

  2. Calculate or simplify this relation.

    ∣z∣=3,arg⁡z=π/2+2nπ|z|=3,\quad\arg z=\pi/2+2n\pi

  3. A chosen argument is one representative from infinitely many.

3(cos(π/2)+i sin(π/2)).

Checks and common pitfalls: A chosen argument is one representative from infinitely many.

Reasoning checklist · self / teacher assessment
  • State the relevant definition, condition or model.
  • Show a valid calculation, proof or counterexample.
  • Interpret the conclusion with its restrictions.

Think first. Reveal a hint when the class is ready.

05 / Foundation#Your turn

Find the modulus of the product.

∣z1∣=3,∣z2∣=3|z_1|=3,\quad|z_2|=3
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Use modulus and argument, remembering that arguments differ by full turns.
Hint 2
Multiplication multiplies moduli.
Worked solution

  1. Use modulus and argument, remembering that arguments differ by full turns.

  2. Calculate or simplify this relation.

    ∣z1z2∣=∣z1∣∣z2∣=9|z_1z_2|=|z_1||z_2|=9

  3. Arguments add, but lengths multiply.

The requested value is 9.

Checks and common pitfalls: Arguments add, but lengths multiply.

Think first. Reveal a hint when the class is ready.

06 / Foundation#Your turn

Find the real part using de Moivre’s formula.

[3(cos⁡(π/3)+isin⁡(π/3))]3[3(\cos(\pi/3)+i\sin(\pi/3))]^3
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Use modulus and argument, remembering that arguments differ by full turns.
Hint 2
Cube the modulus and triple the angle.
Worked solution

  1. Use modulus and argument, remembering that arguments differ by full turns.

  2. Calculate or simplify this relation.

    27(cos⁡π+isin⁡π)=−2727(\cos\pi+i\sin\pi)=-27

  3. The angle multiplication must accompany the modulus power.

The requested value is -27.

Checks and common pitfalls: The angle multiplication must accompany the modulus power.

Think first. Reveal a hint when the class is ready.

07 / Foundation#Your turn

List the two square roots of 1 in polar or algebraic form.

z2=1z^2=1
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Use modulus and argument, remembering that arguments differ by full turns.
Hint 2
Halve the arguments, including different full-turn representatives.
Worked solution

  1. Use modulus and argument, remembering that arguments differ by full turns.

  2. Calculate or simplify this relation.

    ∣z∣=1,2θ=2nπ⇒θ=nπ|z|=1,\quad2\theta=2n\pi\Rightarrow\theta=n\pi

  3. Using only argument zero would miss one root.

1 and −1.

Checks and common pitfalls: Using only argument zero would miss one root.

Reasoning checklist · self / teacher assessment
  • State the relevant definition, condition or model.
  • Show a valid calculation, proof or counterexample.
  • Interpret the conclusion with its restrictions.

Think first. Reveal a hint when the class is ready.

08 / Standard#Your turn

Find the modulus of the quotient.

∣z1∣=9,∣z2∣=3|z_1|=9,\quad|z_2|=3
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Use modulus and argument, remembering that arguments differ by full turns.
Hint 2
The divisor must be nonzero.
Worked solution

  1. Use modulus and argument, remembering that arguments differ by full turns.

  2. Calculate or simplify this relation.

    ∣z1/z2∣=∣z1∣/∣z2∣=3|z_1/z_2|=|z_1|/|z_2|=3

  3. Arguments subtract under division.

The requested value is 3.

Checks and common pitfalls: Arguments subtract under division.

Think first. Reveal a hint when the class is ready.

09 / Standard#Your turn

Are arguments θ and θ+2π different complex numbers at the same positive modulus? Explain.

  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Use modulus and argument, remembering that arguments differ by full turns.
Hint 2
Compare both real and imaginary parts.
Worked solution

  1. Use modulus and argument, remembering that arguments differ by full turns.

  2. Calculate or simplify this relation.

    r(cos⁡(θ+2π)+isin⁡(θ+2π))=r(cos⁡θ+isin⁡θ)r(\cos(\theta+2\pi)+i\sin(\theta+2\pi))=r(\cos\theta+i\sin\theta)

  3. A representation is not unique even though the complex number is fixed.

No; sine and cosine are unchanged by one full turn.

Checks and common pitfalls: A representation is not unique even though the complex number is fixed.

Reasoning checklist · self / teacher assessment
  • State the relevant definition, condition or model.
  • Show a valid calculation, proof or counterexample.
  • Interpret the conclusion with its restrictions.

Think first. Reveal a hint when the class is ready.

10 / Standard#Your turn

List all cube roots of unity.

z3=1z^3=1
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Use modulus one and divide three distinct full-turn arguments by three.
Hint 2
Arguments repeat modulo 2π.
Worked solution

  1. Use modulus one and divide three distinct full-turn arguments by three.

  2. Calculate or simplify this relation.

    ∣z∣=1;θ=0,2π/3,4π/3|z|=1;\quad\theta=0,2\pi/3,4\pi/3

  3. The three resulting points are distinct and exhaust the degree-three equation.

1, −1/2+i√3/2, −1/2−i√3/2.

Checks and common pitfalls: The three resulting points are distinct and exhaust the degree-three equation.

Reasoning checklist · self / teacher assessment
  • State the relevant definition, condition or model.
  • Show a valid calculation, proof or counterexample.
  • Interpret the conclusion with its restrictions.

Think first. Reveal a hint when the class is ready.

11 / Standard#Your turn

Give a polar representation.

z=4iz=4i
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Use modulus and argument, remembering that arguments differ by full turns.
Hint 2
The point lies on the positive imaginary axis.
Worked solution

  1. Use modulus and argument, remembering that arguments differ by full turns.

  2. Calculate or simplify this relation.

    ∣z∣=4,arg⁡z=π/2+2nπ|z|=4,\quad\arg z=\pi/2+2n\pi

  3. A chosen argument is one representative from infinitely many.

4(cos(π/2)+i sin(π/2)).

Checks and common pitfalls: A chosen argument is one representative from infinitely many.

Reasoning checklist · self / teacher assessment
  • State the relevant definition, condition or model.
  • Show a valid calculation, proof or counterexample.
  • Interpret the conclusion with its restrictions.

Think first. Reveal a hint when the class is ready.

12 / Transfer#Your turn

Find the modulus of the product.

∣z1∣=4,∣z2∣=3|z_1|=4,\quad|z_2|=3
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Use modulus and argument, remembering that arguments differ by full turns.
Hint 2
Multiplication multiplies moduli.
Worked solution

  1. Use modulus and argument, remembering that arguments differ by full turns.

  2. Calculate or simplify this relation.

    ∣z1z2∣=∣z1∣∣z2∣=12|z_1z_2|=|z_1||z_2|=12

  3. Arguments add, but lengths multiply.

The requested value is 12.

Checks and common pitfalls: Arguments add, but lengths multiply.

Think first. Reveal a hint when the class is ready.

13 / Transfer#Your turn

Find the real part using de Moivre’s formula.

[4(cos⁡(π/3)+isin⁡(π/3))]3[4(\cos(\pi/3)+i\sin(\pi/3))]^3
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Use modulus and argument, remembering that arguments differ by full turns.
Hint 2
Cube the modulus and triple the angle.
Worked solution

  1. Use modulus and argument, remembering that arguments differ by full turns.

  2. Calculate or simplify this relation.

    64(cos⁡π+isin⁡π)=−6464(\cos\pi+i\sin\pi)=-64

  3. The angle multiplication must accompany the modulus power.

The requested value is -64.

Checks and common pitfalls: The angle multiplication must accompany the modulus power.

Think first. Reveal a hint when the class is ready.

Focus on one question

End-of-lesson check and correction

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    Teacher preparation and assessment

    Question sequence

    • What must be true before using the main rule for trigonometric representation of complex numbers (starred)?
    • Which representation makes this task easier, and why?
    • Change one assumption. Does the conclusion survive?

    Board plan

    • Connect geometric and algebraic forms of trigonometric representation of complex numbers (starred).
    • Polar form: A nonzero complex number combines positive modulus with an argument modulo 2π.
      z=r(cos⁡θ+isin⁡θ)z=r(\cos\theta+i\sin\theta)
    • Products and powers: Multiply moduli and add arguments; powers multiply arguments.
    • Close with: conditions → representation → reasoning → check.

    Anticipated thinking

    • Expected reasoning: A nonzero complex number combines positive modulus with an argument modulo 2π.
    • Expected reasoning: Multiply moduli and add arguments; powers multiply arguments.
    • Expected correction: Argument is not unique, and zero has no unique direction.

    Assessment checklist

    • 1: identify the givens and required quantity.
    • 1: choose a valid definition, representation or method.
    • 1: present connected, correct reasoning.
    • 1: check conditions and explain the result.

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    Curriculum and source notes ↗