BAM! Mr. Tarrou!
Okay, I'm doing a quick video here, showing you
how to multiply complex numbers in polar form
and divide. It's a very, very straightforward process as long as you
have these formulas.
either available to you or through memory
In the next video, introducing the de Moivre's Theorem
and using that to find powers and roots
we're going to work on the product rule here to
prove where that Rule comes from
at least for the products
excuse me, the powers
But, for now, we're just going to work on a couple examples and call it a day.
Soo... I want to do the product one here
and, to multiply two complex numbers in polar form
all you need to do is multiply the two radiuses together
and then multiply that by the two angles being added
to "i" times the sin
of the two angles again being added
if the angles are the same angles it will of course be the same answer
z sub 1 is equal to two times cosine
of 20 plus i sin 20 degrees
and then, z sub 2
is equal to 3 times the cosine of
190 degrees plus i times sine of 190 degrees.
And like this formula shows, if you want to find what Z1 times Z2 is just take the two radiuses
multiply them together... so 3 times 2 is 6
Now times cosine..
of the two angles being added.
So that is going to be 20 plus 190.
... plus i sine of those same two angles being added again.
I am running out of room here.
So 20 plus 190, we will just squeeze that in there.
And 3 times 2 is 6, times the cosine of 20 and 190 is 210
plus i sine of 210.
And uh. I don't know why I wrote that one in parenthesis.
So that is pretty much it. Just if you want to multiply two complex numbers in polar form
multiply the radiuses and add those angle measures.
That is it. If you want to put this back into for some reason rectangular form
distribute the 6 and put 6 cosine of 210 into your calculator and 6 times sine of 210.
I said just put that into your calculator. 210 degrees is on the unit circle:/
So... Well lets just do that.
There is not much to these problems anyway.
6 cosine of 210 degrees plus i 6 sine of 210 degrees.
The cosine of 210 is negative square root of three over two.
plus i. The sine of 210 is negative 1/2.
So we have
negative 3 square root of 3 minus... because that is negative... half of 6 is 3.
So... I don't know. Just to remind you that if you want to go from polar form of a complex number
into rectangular form just work it out.
You have got to work... work it out...
Let's do a quotient and cut this video off before I really start doing some bad songs.
Actually if I am singing them they will all be bad!
z1... Z sub 1 is equal to 28 times the cosine of 5pi/6 plus i sine of 5pi/6.
Z sub 2 is 7 times the cosine of 3pi/4 plus i sine of 3pi/4.
Yes, polar form of complex numbers can either be degrees or radians.
It is no big deal, either one. Which ever you are using.
I want to find the difference of those two complex numbers in polar form.
So.
We are going to divide those radii and we are going to subtract those angles.
So...
Make sure that you keep this in proper order by the way because when you subtract of course
it is going to make a difference right.
5 minus 3 is 2 and 3 minus 5 is negative two.
So don't... Make sure you don't cause yourself any issues there getting that subtraction backwards.
So, I want to do z1 divided by z2.
That means that we are going to do 28 divided by 7.
... times the cosine
of the angle measure that goes along with the radius that you placed on the top of your quotient
This is r1 and this is theta 1 minus theta 2.
So we are going to 5pi/6
minus 3pi/4
plus i sine of again those same two angles in the proper order.
5pi/6 minus 3pi/4
We get 4
times the cosine of...
well...
Let's just write this step out here.
You cannot subtract fractions unless you have common denominators.
So we are going to multiply this top and bottom with 2.
And then top and bottom by 3.
So we make that common denominator of 12.
We are going to get 5 times 2 is 10pi over 12 minus 3 times 3 is 9pi over 12.
plus i times sine of the same thing.
10pi/12 minus 9pi/12.
Our final answer therefore is 4 times the cosine of
of 10 minus 9 is 1, so pi/12
plus i sine of again pi/12.
I am Mr. Tarrou and that the product and quotient of complex numbers in polar form.
WHOOO!!! All right. BAM!!!
Go do your homework.
Or.. Watch my next video about D'Moires Theorem:)