Question

For the following expressions, it is better to change to polar form before solving.

a. Multiply the following: z₁ = A cos(θ) + jA sin(θ) and z₂ = B cos (∅) + jB sin(∅)

Answer 1

It is advisable to change the expressions to polar form before multiplication. The product of z₁ = A cos(θ) + jA sin(θ) and z₂ = B cos (∅) + jB sin(∅) yields z₃ = (AB) cos(θ + ∅) + j(AB) sin(θ + ∅) in polar form.

Step-by-step explanation:

To begin, let's express the given complex numbers in polar form. The first complex number z₁ = A cos(θ) + jA sin(θ) can be represented as z₁ = A∠θ, where A is the magnitude and θ is the angle. Similarly, the second complex number z₂ = B cos (∅) + jB sin(∅) can be expressed as z₂ = B∠∅.

Now, to multiply z₁ and z₂, we multiply their magnitudes and add their angles. The product z₃ can be obtained as follows:

z₃ = (A∠θ) x (B∠∅) = (AB)∠(θ + ∅).

The final result is in polar form, where the magnitude is AB, and the angle is the sum of the angles of the original numbers. This form is particularly advantageous when dealing with complex number multiplication because it simplifies the process by combining the magnitudes and angles directly.

In summary, converting the expressions to polar form before multiplication allows us to leverage the simplicity of polar multiplication rules, providing a clearer and more manageable solution. The polar form not only streamlines calculations but also offers insights into the geometric interpretation of complex number operations.