Lesson 4: Complex Numbers in Trig Form and DeMoivre's Theorem
Estimated time: 35-40 minutes
Learning Objectives
- Write complex numbers in trigonometric (polar) form: z = r(cosθ + i sinθ)
- Multiply and divide complex numbers in trig form
- Apply DeMoivre's Theorem to find powers of complex numbers
- Find nth roots of complex numbers
Trig Form of a Complex Number
Every complex number z = a + bi can be written as:
z = r(cos θ + i sin θ)
where r = |z| = √(a² + b²) is the modulus and θ = arctan(b/a) is the argument.
Example 1: Convert to Trig Form
Write z = 1 + i in trig form.
r = √(1+1) = √2. θ = arctan(1/1) = π/4.
z = √2(cos π/4 + i sin π/4)
Example 2: Convert to Trig Form
Write z = −3 in trig form.
r = 3. θ = π (on the negative real axis).
z = 3(cos π + i sin π)
Multiplying and Dividing in Trig Form
Multiplication: z₁z₂ = r₁r₂[cos(θ₁+θ₂) + i sin(θ₁+θ₂)]
Division: z₁/z₂ = (r₁/r₂)[cos(θ₁−θ₂) + i sin(θ₁−θ₂)]
Multiply the moduli and add the arguments. Divide the moduli and subtract the arguments.
Example 3: Multiplication
z₁ = 2(cos30+isin30), z₂ = 3(cos60+isin60). Find z₁z₂.
r = 2·3 = 6. θ = 30+60 = 90.
z₁z₂ = 6(cos90+isin90) = 6i
DeMoivre's Theorem
DeMoivre's Theorem:
[r(cosθ + i sinθ)]n = rn(cos nθ + i sin nθ)
Example 4: Finding a Power
Find (1+i)²⁰ using DeMoivre's Theorem.
First write 1+i in trig form: √2(cos45+isin45)
[√2]²⁰ (cos(20·45)+isin(20·45)) = 2¹⁰(cos900+isin900)
900 = 2(360)+180, so cos900=cos180=−1, sin900=0.
(1+i)²⁰ = 1024(−1) = −1024
Finding nth Roots
The n distinct nth roots of z = r(cosθ+isinθ) are:
wk = r1/n[cos((θ+2kπ)/n) + i sin((θ+2kπ)/n)]
for k = 0, 1, 2, ..., n−1.
Example 5: Cube Roots of 8
8 = 8(cos0+isin0). The three cube roots are:
wk = 2[cos(2kπ/3)+isin(2kπ/3)] for k=0,1,2.
w₀ = 2(cos0+isin0) = 2
w₁ = 2(cos(2π/3)+isin(2π/3)) = −1+i√3
w₂ = 2(cos(4π/3)+isin(4π/3)) = −1−i√3
Check Your Understanding
1. Write z = −2i in trig form.
2. Use DeMoivre's to find [2(cos60+isin60)]³.
3. How many 4th roots does any nonzero complex number have?
Key Takeaways
- Trig form: z = r(cosθ + i sinθ) where r = |z| and θ = arg(z).
- Multiply: multiply moduli, add arguments. Divide: divide moduli, subtract arguments.
- DeMoivre's Theorem: raise modulus to nth power, multiply argument by n.
- Every nonzero complex number has exactly n distinct nth roots, equally spaced around a circle.
Course Complete!
Congratulations on completing the Trigonometry course! Review any modules you found challenging and continue practicing.