Question

Let z₁=r₁(cosθ₁+isinθ₁) and z₂=r₂(cosθ₂+isinθ₂) be complex numbers. z₁/z₂=...?

a) r₁/r₂(cos(θ₁+θ₂)+isin(θ₁+θ₂))
b) r₁/r₂(cos(θ₁-θ₂)+isin(θ₁-θ₂))
c) r₁/r₂(cosθ₁cosθ₂−sinθ₁sinθ₂+isinθ₁cosθ₂+cosθ₁sinθ₂)
d) r₁/r₂(cosθ₁cosθ₂−sinθ₁sinθ₂+isinθ₁sinθ₂+cosθ₁cosθ₂)

Answer 1

When dividing two complex numbers in trigonometric form, the correct result is the division of the moduli and the subtraction of the angles, which corresponds to option b) r₁/r₂(cos(θ₁-θ₂) + isin(θ₁-θ₂)).

Step-by-step explanation:

The division of two complex numbers z₁ and z₂ represented in trigonometric form is given by:

z₁ = r₁(cosθ₁ + isinθ₁) and z₂ = r₂(cosθ₂ + isinθ₂)

To find z₁/z₂, you divide the moduli and subtract the angles, resulting in the formula:

z₁/z₂ = r₁/r₂(cos(θ₁-θ₂) + isin(θ₁-θ₂))

This means the correct answer is option b) r₁/r₂(cos(θ₁-θ₂) + isin(θ₁-θ₂)).