Question Video: The Argument of the Product of Complex Numbers in Polar Form Mathematics

What is the argument of the product of 𝑍₁ = π‘Ÿ(cos πœƒ + 𝑖 sin πœƒ) and 𝑍₂ = 𝑠(cos πœ‘ + 𝑖 sin πœ‘)?

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Video Transcript

What is the argument of the product of 𝑍 sub one is equal to π‘Ÿ multiplied by the cos of πœƒ plus 𝑖 sin πœƒ and 𝑍 sub two is 𝑠 multiplied by cos of πœ‘ plus 𝑖 sin πœ‘?

In this question, we’re asked to determine the argument of the products of two complex numbers. And we can see that the two complex numbers 𝑍 sub one and 𝑍 sub two are given in polar form. And we recall when a complex number is given in polar form, we can read off the value of its modulus and its argument. For example, the modulus of 𝑍 sub one is π‘Ÿ and its argument is πœƒ. Similarly, the modulus of 𝑍 sub two is 𝑠 and its argument is πœ‘.

We want to use this to determine the argument of the product of these two numbers. And to do this, let’s start by recalling a fact about multiplying complex numbers given in polar form. We recall we can multiply two complex numbers given in polar form by multiplying the moduli of both number and adding the arguments. So 𝑍 sub one times 𝑍 sub two is π‘Ÿ multiplied by 𝑠 times the cos of πœƒ plus πœ‘ plus 𝑖 sin of πœƒ plus πœ‘. This then gives us two useful results. The modulus of 𝑍 sub one times 𝑍 sub two is the product of the moduli of 𝑍 sub one and 𝑍 sub two. That’s π‘Ÿ multiplied by 𝑠. And the argument of 𝑍 sub one times 𝑍 sub two is the sum of the arguments of 𝑍 sub one and 𝑍 sub two. That’s πœƒ plus πœ‘. And this is what we were asked to find.

Therefore, we were able to show the argument of 𝑍 sub one times 𝑍 sub two is the sum of the arguments of 𝑍 sub one and 𝑍 sub two. That’s πœƒ plus πœ‘.

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