Question

Multiplying Complex Numbers: Find the product z₁z₂ of the complex numbers z₁=2(cos2π/3+isin2π/3) and z₂=8(cos11π/6+isin11π/6).

a) 16(cos7π/6+isin7π/6)
b) 16(cos13π/6+isin13π/6)
c) 16(cos5π/6+isin5π/6)
d) 16(cosπ/6+isinπ/6)

Answer 1

The product of the two complex numbers is 16(cos π/2 + i sin π/2), which corresponds to answer option d).

Step-by-step explanation:

The product of two complex numbers z₁z₂ can be found using the formula z₁z₂ = r₁r₂(cos(θ₁ + θ₂) + i sin(θ₁ + θ₂)), where z₁ = r₁(cos θ1 + i sin θ1) and z₂ = r₂(cos θ2 + i sin θ2). For z₁ = 2(cos 2π/3 + i sin 2π/3) and z₂ = 8(cos 11π/6 + i sin 11π/6), we multiply the moduli and add the angles: r₁r₂ = 2 * 8 = 16 and θ₁ + θ2 = 2π/3 + 11π/6. By converting angles to a common denominator and simplifying, we obtain θ₁ + θ2 = 4π/6 + 11π/6 = 15π/6, which simplifies to 5π/2. However, since angles in trigonometric functions are periodic with period 2π, we can subtract 2π to get θ = 5π/2 - 4π/2 = π/2. Thus, the product in trigonometric form is 16(cos π/2 + i sin π/2), which corresponds to answer option d).