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## Multiplying and dividing complex numbers in polar form

# Multiplying complex numbers in polar form

CCSS.Math:

## Video transcript

- [Instructor] We're given
two different complex numbers here, and we want to figure
out what is the product? Pause this video and see
if you can figure that out. All right, now let's
work on this together. So we know from the form
that it's written here that the modulus of w sub
one is equal to three. And we know that the argument of w sub one is equal to 330 degrees. And by the same line of reasoning, we know that the modulus of w
sub two is equal to two. And that the argument of w sub two is going to be equal to, we can see that right
over here, 120 degrees. Now, when you multiply complex numbers you could view as one
transforming the other. We've seen this in multiple examples. So let's imagine that we
are transforming w two by multiplying it by w one. So what is going to happen? Well, let me write it here. So what's the resulting
modulus of w one times w two? Well, we're just going to
scale up w two's modulus by w one's modulus. Or essentially we're just
going to multiply the two. So this is going to be equal
to six, three times two. And then the argument of
w sub one times w sub two, if we start at w sub two's
argument, 120 degrees and then we rotate it
by w sub one's argument, well then you're going
to add these two angles, that gets you to 450 degrees. So this is equal to 450 degrees, which is more than a complete rotation. And so if we wanted to give it an angle between zero and 360 degrees, if we just subtract 360 from that, that is going to be equal to 90 degrees. And so we can rewrite this here, or we can rewrite the product as w sub one times w sub two is equal to its modulus six times cosine
of its argument, 90 degrees. Plus i times sine of its argument. Now we know what the cosine
and sine of 90 degrees is. Cosine of 90 degrees is equal to zero and sine of 90 degrees is equal to one. So all of the simplifies quite nicely. All you're left with is a six times I. So this is equal to
six i, and we are done.