Cogito
Precalculus · Chapter 5 · Lesson 3
Complex Numbers in Polar Form
Multiplying becomes rotating.
12 problems · about 22 minutes · TEKS P.4.F
What this lesson teaches
The student writes complex numbers in trigonometric form and applies De Moivre’s Theorem.
- Trigonometric form is r(cos θ + i sin θ), with modulus r and argument θ.
- Multiplying multiplies moduli and adds arguments.
- De Moivre’s Theorem extends that to any power.
Warm Up
Straightforward practice. Get the method working first.
5 problemsWhat is the modulus of 8 + 6i?
Answer 10
Why 10.
What happens to the arguments when two complex numbers are multiplied?
Answer They add.
Why Arguments add while moduli multiply.
Modulus of 3 + 4i?
Answer 5
Why √25.
Modulus of 5i?
Answer 5
Why √(0 + 25).
Argument of 4i, in degrees?
Answer 90
Why Straight up the imaginary axis.
Build It Up
The same ideas with more to keep track of.
3 problemsModuli 3 and 4 multiplied. What is the modulus of the product?
Answer 12
Why Multiply the moduli.
Arguments 40° and 25°, multiplied. What is the argument of the product, in degrees?
Answer 65
Why Add the arguments.
Modulus 2, argument 20°, raised to the 4th power. What is the new argument, in degrees?
Answer 80
Why Multiply the argument by 4.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
4 problemsSort each operation by what happens to the modulus and the argument.
Answer Moduli multiply: r₁ × r₂ when multiplying, rⁿ when taking a power · Arguments add: θ₁ + θ₂ when multiplying, nθ when taking a power
Why Lengths multiply; angles add.
Modulus 3, argument 10°, cubed. What is the new modulus?
Answer 27
Why 3³.
The Modulus: What is the modulus of 6 + 8i?
Answer 10
Why 10.
The Power: A complex number has modulus 2 and argument 30°. What is the modulus of its cube?
Answer 8
Why 8.