Video: Multiplying Complex Numbers in Polar Form

Given that 𝑧₁ = 2(cos (5π‘Ž βˆ’ 2𝑏) + 𝑖 sin (5π‘Ž βˆ’ 2𝑏)) and 𝑧₂ = 4(cos(4π‘Ž βˆ’ 3𝑏) + 𝑖 sin(4π‘Ž βˆ’ 3𝑏)), find 𝑧₁𝑧₂.

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Video Transcript

Given that 𝑧 one is equal to two cos of five π‘Ž minus two 𝑏 plus 𝑖 sin of five π‘Ž minus two 𝑏 and 𝑧 two equals four cos of four π‘Ž minus three 𝑏 plus 𝑖 sin of four π‘Ž minus three 𝑏, find 𝑧 one multiplied by 𝑧 two.

We have been given two complex numbers represented in polar or trigonometric form. And we’re looking to find their product. Remember, to multiply complex numbers in polar form, we multiply their moduli. And we add their arguments. And the general form of a complex number in polar form is π‘Ÿ cos πœƒ plus 𝑖 sin πœƒ, where π‘Ÿ is the modulus and πœƒ is the argument. We’ll compare this general form to the complex numbers in our question.

The modulus of our first complex number 𝑧 one is two. And the modulus of our second complex number is four. The argument of our first complex number is five π‘Ž minus two 𝑏. And the argument of our second complex number is four π‘Ž minus three 𝑏.

We said that, to find the modulus of the product of these two complex numbers, we need to find the product of their moduli. That’s two multiplied by four, which is of course eight. And we said that, to find the argument of 𝑧 one multiplied by 𝑧 two, we add their respective arguments. That’s five π‘Ž minus two 𝑏 plus four π‘Ž minus three 𝑏.

We can collect like terms. And we see that the argument of the product of 𝑧 one and 𝑧 two is nine π‘Ž minus five 𝑏. And all that’s left is to substitute these values into the general form of a complex number in polar form.

It’s eight cos of nine π‘Ž minus five 𝑏 plus 𝑖 sin of nine π‘Ž minus five 𝑏.

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