Unit 08 · Lesson 06 · Week 25 · Polar Coordinates & Complex Numbers
De Moivre's Theorem and Powers
Interpret complex numbers geometrically and use polar form, multiplication, division, powers, and roots to model rotation and scaling. In this lesson, the central focus is de moivre's theorem and powers. Learners move among symbolic, graphical, numerical, and contextual evidence rather than relying on one representation alone.
Learning intentions
- Interpret the defining quantities and notation used in de moivre's theorem and powers.
- Construct or analyze at least two representations related to de moivre's theorem and powers.
- Show symbolic work with correct domains, units, intervals, or angle measures.
- Explain why the selected method is valid and check the result against another representation.
Essential question
How do polar coordinates and complex-number forms reveal rotation, magnitude, symmetry, and curves that Cartesian form can obscure?
Multiple-representation model
Interpret complex numbers geometrically and use polar form, multiplication, division, powers, and roots to model rotation and scaling.
z=a+bi=r(\cos\theta+i\sin\theta)z_1z_2=r_1r_2\operatorname{cis}(\theta_1+\theta_2)z^n=r^n\operatorname{cis}(n\theta)
Worked reasoning
Misconception check: When multiplying complex numbers in polar form, multiply magnitudes and add angles; do not multiply the angles.
Communication and reasoning
Write a complete response that identifies the representation used, shows the symbolic steps, interprets parameters or units, and verifies the conclusion with a second representation or reasonableness check.
Khaemenes Academy · Open-Age Precalculus
De Moivre's Theorem and Powers
Analyze, calculate, graph, or justify each result. Show reasoning, intervals, units, and calculator evidence when used.
Name
Date
Write 3+3i in polar form.
Multiply 2cis(π/6) by 3cis(π/4).
Divide 8cis(2π/3) by 2cis(π/6).
Use De Moivre to square 2cis(π/3).
How many distinct complex fourth roots does a nonzero number have?
What geometric action is multiplication by i?