Video: Multiplying Complex Numbers in Polar Form

Given that 𝑧₁ = 5(cos (5πœ‹/6) + 𝑖 sin (5πœ‹/6)) and 𝑧₂ = 4(cos 180Β° + 𝑖 sin180Β°), determine 𝑧₁𝑧₂.

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Video Transcript

Given that 𝑧 one is equal to five multiplied by cos of five πœ‹ by six plus 𝑖 sin of five πœ‹ by six and 𝑧 two is equal to four multiplied by cos of 180 degrees plus 𝑖 sin of 180 degrees, determine 𝑧 one 𝑧 two.

Here, we recall that, to multiply two complex numbers in polar form, we multiply their moduli and we add their arguments. The modulus of our first complex number is five. And the modulus of our second complex number is four. And the argument of our first complex number is five πœ‹ by six. And the argument of our second complex number is 180 degrees.

It should be fairly clear that multiplying the moduli of these two complex numbers is not particularly tricky. But adding their arguments is. And that’s because the argument of 𝑧 one is in terms of radians. And the argument of 𝑧 two is in degrees. So let’s convert the argument for 𝑧 one into degrees.

To do this, we recall the fact that two πœ‹ radians is equal to 360 degrees. To find the value of one radian, we’re going to divide through by two πœ‹. And we see that one radian is equal to 360 over two πœ‹ degrees. This simplifies to 180 over πœ‹. And this means we can convert the argument for 𝑧 one from radians into degrees by multiplying it by 180 over πœ‹. Now doing this, we can see that the πœ‹s cancel. And we can also divide through by a factor of six. We’re left with five multiplied by 30, which is 150 degrees.

So to find the product of 𝑧 one and 𝑧 two, we multiply their moduli. That’s five multiplied by four, which is 20. We’re then going to add their arguments, this time dealing with them in degrees. That’s 150 plus 180. 150 plus 180 is 330.

And we can now see that the product of our two complex numbers is 20 multiplied by cos of 330 degrees plus 𝑖 sin of 330 degrees.

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