Question

For two complex numbers z₁=r₁(cosθ₁+isinθ₁) and z₂=r₂(cosθ₂+isinθ₂), the product of z₁ and z₂ is...?

a) r₁r₂(cos(θ₁+θ₂)+isin(θ₁+θ₂))
b) r₁r₂(cos(θ₁-θ₂)+isin(θ₁-θ₂))
c) r₁r₂(cosθ₁cosθ₂−sinθ₁sinθ₂+isinθ₁cosθ₂+cosθ₁sinθ₂)
d) r₁r₂(cosθ₁cosθ₂−sinθ₁sinθ₂+isinθ₁sinθ₂+cosθ₁cosθ₂)

Answer 1

The product of the two complex numbers z₁ and z₂ in polar form is r₁r₂(cos(θ₁+θ₂)+isin(θ₁+θ₂)).

Step-by-step explanation:

For two complex numbers z₁=r₁(cosθ₁+isinθ₁) and z₂=r₂(cosθ₂+isinθ₂), when you multiply them together, you use the properties of trigonometric functions to combine the angles. The formulaic expression of the multiplication of two complex numbers in polar form leads to z₁z₂ = r₁r₂(cos(θ₁+θ₂)+isin(θ₁+θ₂)). This result is derived using trigonometric identities such as cos(A)cos(B) - sin(A)sin(B) = cos(A+B) and sin(A)cos(B) + cos(A)sin(B) = sin(A+B).