LEARN · EXPLAIN · REVISE

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

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

TOPIC 01

Trigonometric representation of complex numbers (starred)

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

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.

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

01 / Standard#Your turn

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

Use modulus and argument, remembering that arguments differ by full turns.
Calculate or simplify this relation.
$r(\cos(\theta+2\pi)+i\sin(\theta+2\pi))=r(\cos\theta+i\sin\theta)$
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

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

Teacher preparation and assessment

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?

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.
$z=r(\cos\theta+i\sin\theta)$
Products and powers: Multiply moduli and add arguments; powers multiply arguments.
Close with: conditions → representation → reasoning → check.

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.