00:01
So in this problem, you're given these two numbers in polar form.
00:03
So you have z, which is 5 times the cosine of 35 plus i sine 35, and you have w, which is 2 times the cosine of 40 plus i sine 40.
00:12
So the first thing we want to find is z times w.
00:15
Well, if you recall that we have a formula to help us to multiply complex numbers that are already given to us in polar form.
00:21
So here's how it works.
00:23
It says if we have a complex number in polar form, let's call it z1, that's equal to r sub 1 times cosine of theta sub 1 plus i sine of theta sub 1.
00:37
And we also have this other complex number z2, which is r sub 2 times the cosine of theta sub 2 plus i sine of theta sub 2.
00:48
Then when we multiply z times w, the result ends up, or actually we'll do z1 times z2.
00:54
So z1 times z2, the result is r1 times r2 times the cosine of theta 1 plus theta 2 plus i sine of theta 1 plus theta 2.
01:10
So now we can correlate this to our z and w complex numbers that we were given.
01:15
So in this case for our z, we'll call that our z sub 1, where 5 would be r sub 1, 35 would be theta sub 1, and for w, 2 would be our r sub 2, and 40 would be our theta sub 2.
01:28
So to find z times w, we can now substitute these values in.
01:32
So we're going to have 5 times 2, that's r sub 1 times r sub 2, times the cosine of theta sub 1, which is 35, plus theta sub 2, which is 40, plus i sine of theta sub 1, which is 35, plus theta sub 2, which is 40.
01:49
And now we can simplify this a little bit, because we have 5 times 2, which is 10, and then we have 35 plus 40, which is 75.
01:57
So we'll have the cosine of 75, plus i sine of 75.
02:02
Perfect...