Hello everyone. In this video, we are going to
to convert a complex number in standard form into polar coordinates.
So the question here will be: using the fact that
tan of pi over 12 is equal to 2 minus the square root of 3,
find the polar coordinates of the following complex number,
negative square root of 3 minus
2 times the square root of 3 minus 3 times i.
So the solution, in order to do this, we need to compute two facts, right? We need to compute
the angle with respect to the x axis, and we need to compute the length
of our vector, the radius,
or the modulus of this complex number.
So that's what we're going to do first. So it's usually easier to compute the length
of your complex number, so we're going to begin with that.
Solution, first we compute
the length of the given complex number.
So how do we do that?
Well we're going to just take the modulus, and we're going to set this
to equal r, okay, r for radius.
Now what
is the length of this? Well we're going to take negative the square root of 3 and square it,
and we're going to add to that
negative
2 times the square root of 3,
minus 3,
all squared.
Let's see if I can fix that.
There we go.
Great, okay. So we're going to compute this number, and simplify it, okay?
So both the negatives are going to go away, which is nice.
So we're going to get
3 plus…
so we're just going to expand the second term out
as well. So let's see what this looks like.
So negative root 3 squared, that's 3.
2 times the square root of 3 if we -
so we're going to square this whole thing, we square the first term,
4 times square root of 3 squared which is 3.
Square the last term, that's the 9,
and then take 2 times the cross term, which is negative 12 times the square root of 3.
Now we add up all these terms,
so 12, 9, 4, that’s 24,
minus 12 square root 3.
We should probably factor out the 4,
just to make it look a little bit nicer, or a little bit smaller.
2 square root of 8 minus 3 square root of
3.
Now it's always good to do a little bit of a sanity check.
Root 3 is less than 2, this number is positive so we didn't
make any obvious blunders.
Okay that's great. So that's going to be the radius of our complex number up here.
Now we want to compute the angle,
and so hopefully we're going to need this fact that
tan of pi over 12 is equal to 2 minus the square root of 3.
I made this problem, so there's some
chance that there's a mistake, but I'm hoping that this is going to work out,
we need this fact that we're given, okay?
So that's great. So now with the radius, what I'm going to do is I'm going to try to compute
the arctan.
The…sorry the theta, the theta value, the angle
with respect to the x-axis.
So now, to compute…
to compute theta, the angle…
the angle of rotation with respect to the
x-axis, we
perform...
so we're going to take
the y value and divide it by the x value,
and take the arctan of that.
So let's do that. So we're going to take the arc-
-tan of
y minus - or y over x.
So maybe I'll just do this.
So we're going to compute the arctan of y over x, that's really what we want to do here,
and I'm going to plug in the y and x there.
What are we going to get? What are we going to get?
We’re going to get a mess, okay, let's see if we can
simplify this mess. 2 times the square root of
3 minus 3…
well I guess it's all negation, so the negative of all of that,
and then this is going to be negative square root of 3.
Alright.
I think that's all the information we need from the question.
So those will cancel, the root 3’s…
you know what? There's there's a couple of ways to do this. I'm going to
factor out a root 3 from the top.
So let's see what happens when I do that.
So I'm going to cancel the negatives,
that's easy, and I'm going to factor out a square root of 3 from the top, so I'm going to get…
should get this, right? So if I factor out a root 3,
the first term becomes 2, and the second term becomes root 3.
Well, okay I guess I can see that over here.
So I bring this back in, I get 2 root 3 minus 3,
you could also multiply top and bottom by the square root of 3. I just
feel like this was a little bit easier.
Okay, now
slightly miraculously, the root 3’s will cancel,
and I'm going to be left with
the following. Let's change the…
let's make it look a little bit nicer now.
And…
okay.
So we cancel out those, we get arctan of 2 minus the square root of 3.
2 minus the square root of 3, that number should look familiar.
Of course it happened to be the one from
the original question, so arctan of 2 minus root 3 is going to be pi over 12.
So this value should equal
pi over 12.
Now you have to be a little bit careful at this point.
We can't immediately jump the gun and conclude that our angle is pi over 12,
we have to do one last check.
What is that last check?
Well if I take
tan of pi over 12, where does this send me?
This sends me if I look - or if I think about this in the complex plane,
this sends me into quadrant 1, okay?
Now quadrant 1, now it's fine, maybe the answer is in quadrant 1,
but if we look at the actual number here,
the original complex number has a negative x value and a negative y value.
So if I…
let me start a new…
let me start a new paint pad here, or paintbrush here.
Sure. So if we think about this in the complex plane,
right, over and over, right,
the number is negative root 3
minus 2 minus root 3, so it's some
number that lives somewhere over here,
right, where this is I'm going to call negative root 3
and this is negative…
2 minus root 3.
2 minus
root 3.
But pi over 12, the angle pi over 12, is somewhere over here,
right? pi over 12.
It's going to look something over here-ish.
So what's the problem? Did we break math? I mean, we did say that
arctan of y over x should be equal to the angle
that we get in polar coordinates when we convert.
The problem here is that arctan…
arctan doesn’t…. arctan has a range between negative pi over 2 and pi over 2.
It's the easiest way to say this.
So what does that mean? arctan is kind of a silly function, right? It doesn't
determine which theta you want,
and why does it not do that? Well
the main reason is that tan is pi periodic.
So because tan is pi periodic, we're going to
have this issue that we either want this point, or the point where we add an extra copy of pi,
and the…
the functions don't know this. So
arctan doesn't know that I wanted - I didn't want the one in the first [quadrant],
I wanted the one in the third [quadrant].
arctan always returns
points in the region defined
by, let's spray paint it in
orange.
So it picks up
the points in this region. So arctan only knows points between -
or angles between negative pi over 2 and pi over 2.
That’s going to be all this stuff shaded in orange, okay?
So in order to get the point on the other side, you have to think do I have to add…
do I have to add an extra copy of pi or not?
Also note that the imaginary axis, arctan has a lot of problems with because
infinities and things like this, right?
arctan doesn't ever equal a value between
exactly negative pi over 2 and pi over 2. So for those values,
you just have to sort of look at them and make sure you know which one is which, right?
It's very easy to tell when a complex number is on
the complex axis, that's not hard, so you don't need to go through all this rigmarole
if you're dealing something like that.
Okay, with that being said, we know now that our theta should be not pi over 12, but
[pi] plus pi over 12. So
thus since our
original complex point
is in the third quadrant,
we have that theta is equal to
pi plus pi over 12
which is equal to 13 pi over 12.
Hence, in polar coordinates…
hence in polar coordinates our number is…
r e to the power of
i theta.
So
r e to the i theta, or if you want the other way to denote this is r
cos theta
plus i sine theta.
Either way, and then just plug in
the theta that you get, and the r that you get, so maybe I'll…
like yeah I'll try to fit it in.
So if I want to plug this in…
Let’s see if I'm going to fit it.
It's a bit tight, but
it will fit. So hence, in polar coordinates we have
the following. Oh I need an extra bracket.
So in polar coordinates, we get this
messy looking expression, but nonetheless it's in that expression.
So...
you know what? I'm going to I'm going to just make it look a little nicer.
I can't stand when things don't look nice.
Can you tell I'm a mathematician? Okay, give me just a second…
and...
voila?
There we go. Okay.
So this complex number in standard form is equal to
this complex number in polar form.
The important message to take away from this…
from this exercise is the following: even when you do arctan of y over x,
you're not guaranteed to get the correct answer.
You need to double check your final answer,
and make sure that the angle that you get is actually the angle that you want.
Again, I like my angles between 0 and 2 pi,
so I always try to put my angles between 0 and 2 pi,
it's up to you. If you've got a negative number,
that's fine,
just make sure that you get the correct negative angle, okay,
and that you're not off by a pi factor.
So, great. I hope
this gives you a little bit of insight as to converting and some of the pitfalls,
and that's all I have to say. Thank you very much for listening and good luck.