KKhaemenes PrecalculusTeacher Answer Key
Student Lesson

Teacher Key · Unit 08 · Lesson 04

Complex Numbers in Rectangular and Polar Form

Use this key to verify method, notation, domain, units, and interpretation. Accept equivalent exact forms when mathematically valid.

Khaemenes Academy · Teacher Key

Complex Numbers in Rectangular and Polar Form

U08 · L04
1

Write 3+3i in polar form.

Complex visualInterpret complex numbers geometrically and use polar form, multiplication, division, powers, and roots to model rotation and scaling.Pθz=r cis θ
Answer: 3√2 cis(π/4)
Reasoning: Magnitude is 3√2 and argument is π/4.
2

Multiply 2cis(π/6) by 3cis(π/4).

Complex visualInterpret complex numbers geometrically and use polar form, multiplication, division, powers, and roots to model rotation and scaling.Pθz=r cis θ
Answer: 6cis(5π/12)
Reasoning: Multiply magnitudes and add angles.
3

Divide 8cis(2π/3) by 2cis(π/6).

Complex visualInterpret complex numbers geometrically and use polar form, multiplication, division, powers, and roots to model rotation and scaling.Pθz=r cis θ
Answer: 4cis(π/2)
Reasoning: Divide magnitudes and subtract angles.
4

Use De Moivre to square 2cis(π/3).

Complex visualInterpret complex numbers geometrically and use polar form, multiplication, division, powers, and roots to model rotation and scaling.Pθz=r cis θ
Answer: 4cis(2π/3)
Reasoning: Square magnitude and double angle.
5

How many distinct complex fourth roots does a nonzero number have?

Complex visualInterpret complex numbers geometrically and use polar form, multiplication, division, powers, and roots to model rotation and scaling.Pθz=r cis θ
Answer: 4
Reasoning: The roots are equally spaced around a circle.
6

What geometric action is multiplication by i?

Complex visualInterpret complex numbers geometrically and use polar form, multiplication, division, powers, and roots to model rotation and scaling.Pθz=r cis θ
Answer: Rotate 90° counterclockwise
Reasoning: i has magnitude 1 and argument π/2.
β Beta Program