Chapter 11: Bankruptcy restructuring

WEBVTT
Kind: captions
Language: en

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In the last video we talked
about the scenario where a

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company, for whatever reason,
it just couldn't

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pay it's debt holders.

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So let's say these are debt
holders right here.

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This is the debt, or
the liabilities.

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It couldn't pay it's
debt holders.

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It went into bankruptcy, and it
was determined that these

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assets that it had right here,
that it made no sense

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operating them as a company.

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And then the bankruptcy court
essentially just decided to

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liquidate it.

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And we learned that the debt
holders were actually more

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senior to the equity holders.

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And they get paid first. And if
there wasn't enough money

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to pay all of the debt holders,
then the equity

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holders got nothing.

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And that was called
a Chapter 7.

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We're just focusing on the
corporate world right now.

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Maybe we'll do personal soon.

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So that's Chapter
7 liquidation.

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That was the last video.

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And in that case, and I think
that's what most people

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associate when you say that a
company has gone bankrupt.

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That it'll just disappear.

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That people just say,
OK, these assets

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don't make any sense.

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They can't pay these guys.

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We're just going to take these
into possession by the courts

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and then just liquidate
the assets.

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But that raises kind of an
obvious question of, well,

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what if these assets are
worth something?

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What if I sell a socks website,
and socks have gotten

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even more popular.

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And the only problem is I just
can't pay all of the interest

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that I owe on the debt.

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Right?

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Maybe, for whatever reason, I
took out a really crazy loan

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that was variable rate.

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Or for some reason, I have to
pay back some loans because I

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messed-- and I'll talk
more about covenants

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and things like that.

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Covenants are pretty much a
bunch of rules that the debt

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holders say, look, you're good,
but if any of these x,

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y, or z things happen, we can
take you into bankruptcy.

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And we could force you
into bankruptcy.

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So maybe because of that,
I'm in bankruptcy.

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But it's determined that these
assets, right here, are

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actually worth more as an
operating entity than they are

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if you were to liquidate them.

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A good example might be, I don't
know, a car company.

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Right?

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Let's actually take this example
as a car company,

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because it's very salient to
our, at least it was-- I've

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heard a lot less about the auto
bailouts, but it was very

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salient at the end
of last year.

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So let's say that these are
car factories and land and

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whatever else.

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And if we're the debt holders,
and let's say it goes into

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bankruptcy.

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Let's say this is
generating cash.

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And I'll teach you in a future
video how do you see what is

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the cash being generated
by the assets.

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And then you have to subtract
out the cash that has to be

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used to pay the debt holders,
because you're paying

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interest, and then what's
left over for equity.

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And I'll show you how to do that
on an income statement.

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But let's say this is generating
a lot of cash.

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Right?

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It's generating a good bit of
cash, but let's say, these

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guys eat up interest. Right?

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So some of the cash will go to
the debt holders as interest.

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And let's say, for whatever
reason, either interest rates

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went up, or they had a bad
quarter or a bad year, and

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they just didn't generate enough
cash, let's say they

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couldn't pay off one of
the debt holders.

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And that debt holder says,
hey, you couldn't pay my

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interest payment, or
you couldn't pay

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the principal payment.

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I'm taking you into
bankruptcy.

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Right?

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I'm taking you into
bankruptcy.

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So it goes into bankruptcy.

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And in this situation,
immediately we realize it

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makes no sense to shutter
this asset.

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If we were just to shut down
the factory and lay off the

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employees, we're going to get
nothing for these assets.

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Because the land is in a part of
the country where'd there's

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no obvious buyer for the land.

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An empty car factory is pretty
much useless, especially when

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the other people in the industry
are in no mood to buy

00:03:28.380 --> 00:03:29.530
the factories from you.

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So everyone decides that it's
in their best interests to

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keep this thing running.

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So what happens is that the
debtor stays in possession of

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the assets.

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So you can kind of view the
debtor as the equity holders

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and the management
of the company.

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So they stay in possession
of the assets.

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And actually what happens is--
because these guys didn't have

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enough cash to pay off their
debt holders-- what happens is

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that they take on a
new loan, called a

00:03:51.680 --> 00:03:53.270
debtor-in-possession loan.

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And this new loan is the
most senior loan.

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It's called DIP financing.

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It's actually a great business,
although it's become

00:03:59.520 --> 00:04:00.770
scarce recently.

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It's a great business
because you're at

00:04:04.105 --> 00:04:06.340
the top of the stack.

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You're more senior than
even the senior guys.

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And it's called DIP financing.

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Debtor-in-possession
financing.

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And what this provides is a
company with some kind of

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cushion cash so that it can keep
operating, so it can keep

00:04:18.630 --> 00:04:21.100
the lights on.

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So it's essentially a debt.

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It's just a very senior
type of debt.

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And it happens once a company
has entered bankruptcy.

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Right?

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And this bankruptcy that
we're going to talk

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about is Chapter 11.

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Chapter 11 restructuring.

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And in Chapter 11 restructuring,
you keep

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operating the company.

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You might do some things on the

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left-hand side of the equation.

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You might want to sell off some
of the assets and all of

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that, but we won't
go into that.

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Most of what you do is you
rearrange this side of the

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balance sheet.

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And this is why, you probably--
every airline has,

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some of them, have gone into
bankruptcy multiple times, but

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they still exist. It's not
like when you go into

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bankruptcy the company
just disappears.

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The assets will persist and all
of this gets reorganized

00:05:07.170 --> 00:05:08.100
on this side.

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A lot of times when someone goes
into Chapter 11 and then

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they come out of it and they
go back into it, they call

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that Chapter 22, and
then Chapter 33.

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I think you get the idea.

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So anyway, what happens
in Chapter 11?

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So the assets-- essentially it
becomes kind of the bankruptcy

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court takes over, and they
hire some investments.

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They'll get the
debtor-in-possession financing

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so that the company has some
cash to operate, pay the

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bills, and pay the employees
and whatever else.

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The company keeps operating as
it always would so it can pay

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its suppliers and operate
as a regular business.

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And then all of these guys
hire a bunch of lawyers.

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And they start negotiating
with each other.

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And essentially there will be
a bank associated with the

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bankruptcy court whose whole
job-- and it's all part of a

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negotiation-- is
to value this.

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And it's often, maybe this
debtor right here,

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he'll hire one bank.

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This debtor will
hire one bank.

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Maybe the management will
hire another bank.

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And everyone's going to come
up with bankruptcy plans.

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But bankruptcy plans
are usually

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of one or more varieties.

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It's essentially just saying,
well, we need to value these

00:06:24.370 --> 00:06:25.060
assets, right?

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We're not selling it.

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So we're not just going
to get cash.

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We're going to hire
some bankers.

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And we'll do a lot of videos
on that in the future.

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And they're just going to say--
based on the prospects

00:06:32.640 --> 00:06:35.080
of this company, how fast it's
growing or how fast it's not

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growing, or how much cash it's
generating in a year-- they're

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going to assign a value to it.

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So let's say that this guy up
here, he hires a banker.

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And this banker says-- Let's
say this was originally the

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same situation.

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This was $10 million.

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Let's say that the liabilities
were $6 million.

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And that the original equity
was $4 million.

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Right?

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And let's say these bankers
evaluate the business.

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They make detailed models.

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They take it in the context of
the current macro environment.

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And they say, you know what?

00:07:06.290 --> 00:07:08.520
I think this company
is actually

00:07:08.520 --> 00:07:09.780
only worth $5 million.

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And given that it's worth $5
million, and we think that it

00:07:17.960 --> 00:07:24.120
can sustain-- it's only worth $5
million and there's no way

00:07:24.120 --> 00:07:27.360
that it can pay interest
on $6 million of debt.

00:07:27.360 --> 00:07:27.590
Right?

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It doesn't have enough cash to
generate $6 million of debt.

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We think it can afford
$2 million of debt.

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Right?

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So what will happen is,
the new company--

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And this is just a plan.

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And then once you have a plan,
then everyone has to vote on

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it, and there are things called
cram downs-- and we''l

00:07:42.470 --> 00:07:44.460
do that in more detail--
but the plan will

00:07:44.460 --> 00:07:45.760
say, you know what?

00:07:45.760 --> 00:07:48.480
The assets are worth
$5 million.

00:07:48.480 --> 00:07:51.770
I thought I was using
the square tool.

00:07:51.770 --> 00:07:53.590
Undo.

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This plan might say, you
know, those assets

00:07:55.270 --> 00:07:56.520
are worth $5 million.

00:07:59.510 --> 00:08:02.280
And the company can only handle
$2 million of debt, not

00:08:02.280 --> 00:08:03.530
$6 million of debt.

00:08:06.650 --> 00:08:14.180
So now, it can only handle $2
million of debt, and then

00:08:14.180 --> 00:08:20.360
there will be $3 million
left of equity.

00:08:20.360 --> 00:08:21.200
Right?

00:08:21.200 --> 00:08:24.270
And I'll call this
the new equity.

00:08:24.270 --> 00:08:25.990
Because sometimes this
can get confusing.

00:08:25.990 --> 00:08:28.490
So let's just say for a second--
and I want you to

00:08:28.490 --> 00:08:30.080
think about it-- what is
everyone's incentive?

00:08:30.080 --> 00:08:33.440
This guy up here, his incentive
is to value the

00:08:33.440 --> 00:08:36.510
company as lowly as
possible, right?

00:08:36.510 --> 00:08:38.190
Because then he gets more
of the company.

00:08:38.190 --> 00:08:39.530
I think that'll be clear
to you in a second.

00:08:39.530 --> 00:08:41.130
This guy's incentive
is to say, no, this

00:08:41.130 --> 00:08:42.380
company is worth a lot.

00:08:42.380 --> 00:08:44.520
So all of you guys are going
to get paid back and then I

00:08:44.520 --> 00:08:46.610
get what's left over.

00:08:46.610 --> 00:08:49.740
And you're probably asking, what
do you get paid back for

00:08:49.740 --> 00:08:50.850
not liquidating it?

00:08:50.850 --> 00:08:53.670
And the answer is the new
shares of the company.

00:08:53.670 --> 00:08:59.780
So what happens is that this
stock-- let's say this plan

00:08:59.780 --> 00:09:01.270
gets passed.

00:09:01.270 --> 00:09:03.100
This plan right here.

00:09:03.100 --> 00:09:06.370
In this situation, these
guys up here were the

00:09:06.370 --> 00:09:07.840
most senior, right?

00:09:07.840 --> 00:09:12.180
Let's say there was $2 million
of senior debt up here.

00:09:12.180 --> 00:09:14.700
Let me write that in
a different color.

00:09:14.700 --> 00:09:17.620
There's $2 million of
senior debt up here.

00:09:17.620 --> 00:09:19.180
So what they'll do is they'll
actually get $2

00:09:19.180 --> 00:09:21.210
million of the new debt.

00:09:21.210 --> 00:09:23.880
They're most senior.

00:09:23.880 --> 00:09:28.730
And then all of these other $4
million, who are more junior--

00:09:28.730 --> 00:09:30.060
let me see if I can
color it in.

00:09:30.060 --> 00:09:34.930
I know it's hard to read-- these
other $4 million guys,

00:09:34.930 --> 00:09:38.370
instead of getting any kind of
cash or any kind of debt

00:09:38.370 --> 00:09:42.370
securities for having been owed
this money, they'll get

00:09:42.370 --> 00:09:43.390
the new stock.

00:09:43.390 --> 00:09:45.890
So they'll get $3 million
of new stock.

00:09:45.890 --> 00:09:47.550
Let me see if I can
draw that in.

00:09:51.020 --> 00:09:55.930
So this $3 million of new equity
will go to these guys.

00:09:55.930 --> 00:09:58.150
And this unsecured guy down
here, he's not going to get as

00:09:58.150 --> 00:09:59.290
much equity.

00:09:59.290 --> 00:10:00.750
He'll be impaired
a little bit.

00:10:00.750 --> 00:10:03.370
And the old equity guys, the
stock's going to go to 0.

00:10:03.370 --> 00:10:05.490
They're not going
to get anything.

00:10:05.490 --> 00:10:08.480
So the old shareholders of the
company are wiped out.

00:10:08.480 --> 00:10:09.150
They go to 0.

00:10:09.150 --> 00:10:14.280
And essentially, the debt
holders become the new

00:10:14.280 --> 00:10:18.000
shareholders of the company.

00:10:18.000 --> 00:10:21.070
You'll often see when a company
goes into bankruptcy

00:10:21.070 --> 00:10:23.960
but it's getting reorganized,
you'll often see some people

00:10:23.960 --> 00:10:30.180
start to buy up this debt or
these bonds, right here,

00:10:30.180 --> 00:10:33.820
because they want to be the
new equity holders.

00:10:33.820 --> 00:10:36.075
When this company emerges from
bankruptcy-- let's say that

00:10:36.075 --> 00:10:38.740
this is how it emerges from
bankruptcy-- they want to be

00:10:38.740 --> 00:10:40.850
these guys, the new
equity holders.

00:10:40.850 --> 00:10:43.100
Because usually when you value
it, you want to undervalue it

00:10:43.100 --> 00:10:44.540
a little bit.

00:10:44.540 --> 00:10:47.490
I know I've overdrawn this
picture a little bit too much.

00:10:47.490 --> 00:10:50.700
But the debt guys, especially
the senior debt guys, they

00:10:50.700 --> 00:10:51.510
want to be safe.

00:10:51.510 --> 00:10:52.810
They want to say,
you know what?

00:10:52.810 --> 00:10:54.590
We've already been hurt
by this company.

00:10:54.590 --> 00:10:56.300
They're already not
paying our debt.

00:10:56.300 --> 00:10:58.845
We want to assign as low a
possible value to the company

00:10:58.845 --> 00:11:01.870
as possible-- in this
case $5 million-- so

00:11:01.870 --> 00:11:02.940
that we make sure.

00:11:02.940 --> 00:11:06.645
Hopefully the company ends up
being worth $10 million again,

00:11:06.645 --> 00:11:11.780
in which case these guys
right here make out

00:11:11.780 --> 00:11:14.030
like bandits, right?

00:11:14.030 --> 00:11:16.080
If the company was really worth
$10 million but the

00:11:16.080 --> 00:11:20.450
bankruptcy court values it at $5
million, these guys get all

00:11:20.450 --> 00:11:21.280
of the shares of the company.

00:11:21.280 --> 00:11:23.880
These guys get wiped out, even
though the company really was

00:11:23.880 --> 00:11:24.810
worth something.

00:11:24.810 --> 00:11:27.030
So let's say the company emerges
from bankruptcy like

00:11:27.030 --> 00:11:30.360
this, but it actually turns out
there were $10 million.

00:11:30.360 --> 00:11:31.800
Then let's say a year
later the company

00:11:31.800 --> 00:11:33.170
starts doing well again.

00:11:33.170 --> 00:11:35.570
And let's say that someone could
value the company again

00:11:35.570 --> 00:11:36.060
at $10 million.

00:11:36.060 --> 00:11:38.970
Now it only has $2
million of debt.

00:11:38.970 --> 00:11:42.050
And now you have $8 million
worth of equity.

00:11:42.050 --> 00:11:45.330
So these guys-- maybe they were
owed $2 or $3 million

00:11:45.330 --> 00:11:48.470
before, and they got $3 million
of the new equity,

00:11:48.470 --> 00:11:49.780
they might have made
out like bandits.

00:11:49.780 --> 00:11:51.790
Because now all of a
sudden, that equity

00:11:51.790 --> 00:11:52.550
could be worth a lot.

00:11:52.550 --> 00:11:53.610
That's not always the case.

00:11:53.610 --> 00:11:56.670
But that's the view from the
debt holders' point of view.

00:11:56.670 --> 00:11:58.510
The equity holders, you can
imagine, they don't want to be

00:11:58.510 --> 00:11:59.980
left with nothing.

00:11:59.980 --> 00:12:01.640
They'll hire their
own bankers.

00:12:01.640 --> 00:12:03.315
And their bankers, they'll
probably submit a plan that

00:12:03.315 --> 00:12:03.930
says, no, no, no, no.

00:12:03.930 --> 00:12:06.890
This company is worth
at least $8 million.

00:12:09.680 --> 00:12:11.910
So up here $8 million.

00:12:11.910 --> 00:12:16.470
And we think it can handle
$4 million of debt.

00:12:16.470 --> 00:12:18.550
So they'd want a scenario like
this, where they think the

00:12:18.550 --> 00:12:20.270
company's worth $8 million.

00:12:22.850 --> 00:12:25.060
It can handle $4 million
worth of debt.

00:12:25.060 --> 00:12:28.310
And so it has $4 million
worth of equity.

00:12:28.310 --> 00:12:31.520
And of course, the first $6
million of the value-- so the

00:12:31.520 --> 00:12:35.000
$4 million of debt, and then $2
million of the equity will

00:12:35.000 --> 00:12:36.940
go to the debt holders, right?

00:12:36.940 --> 00:12:40.090
Because they were owed $6
million to begin with.

00:12:40.090 --> 00:12:43.620
And then what's left over, which
is essentially-- so this

00:12:43.620 --> 00:12:45.780
is $2 million of equity, and
then you'd have $2 million of

00:12:45.780 --> 00:12:50.050
equity here-- this $2 million
of new equity, right?

00:12:50.050 --> 00:12:52.700
This is the new shares of the
company will be given to the

00:12:52.700 --> 00:12:54.470
old shareholders
of the company.

00:12:54.470 --> 00:12:56.320
So that's what the shareholders
want.

00:12:56.320 --> 00:12:59.560
I know this gets a little
confusing, but it all ends up

00:12:59.560 --> 00:13:03.140
being valuing the assets as you
emerge from bankruptcy.

00:13:03.140 --> 00:13:04.020
You say, you know,
it's generating

00:13:04.020 --> 00:13:05.500
cash, it's worth something.

00:13:05.500 --> 00:13:08.240
And then you pay people off
according to seniority.

00:13:08.240 --> 00:13:09.500
And first you pay them off.

00:13:09.500 --> 00:13:11.250
You say, OK, I still
owe you some money.

00:13:11.250 --> 00:13:13.960
But this company can't support
$6 million of debt.

00:13:13.960 --> 00:13:16.090
It can now support $2 million.

00:13:16.090 --> 00:13:18.680
And whatever's left, people
are paid with actually

00:13:18.680 --> 00:13:21.150
shares-- new shares--
of the company.

00:13:21.150 --> 00:13:22.080
Not the old shares.

00:13:22.080 --> 00:13:23.820
So the old shares
will go to 0.

00:13:23.820 --> 00:13:26.560
So you can imagine a world
where GM goes bankrupt.

00:13:26.560 --> 00:13:29.520
Right now, the shares
of GM go to 0.

00:13:29.520 --> 00:13:32.330
GM old goes to 0.

00:13:32.330 --> 00:13:34.750
But the assets keep operating,
and that's why some people are

00:13:34.750 --> 00:13:36.970
a little bit misleading in
this whole automotive

00:13:36.970 --> 00:13:37.810
bankruptcy debate.

00:13:37.810 --> 00:13:40.440
They're kind of using scare
tactics to say, oh, if GM goes

00:13:40.440 --> 00:13:42.790
bankrupt, then these assets are
just going to disappear.

00:13:42.790 --> 00:13:44.070
No, they'll just
keep operating.

00:13:44.070 --> 00:13:46.750
If it makes sense to operate
them, they'll keep operating.

00:13:46.750 --> 00:13:49.170
The only people who
will lose big are

00:13:49.170 --> 00:13:50.620
the old equity holders.

00:13:50.620 --> 00:13:52.900
And then some of the unsecured,
the more junior

00:13:52.900 --> 00:13:55.700
levels of debt, will probably
lose some money.

00:13:55.700 --> 00:13:57.290
But if the assets are
worth operating,

00:13:57.290 --> 00:13:58.175
they'll continue to operate.

00:13:58.175 --> 00:14:00.690
And if the people, if it makes
sense to have them employed,

00:14:00.690 --> 00:14:02.410
they'll keep working.

00:14:02.410 --> 00:14:04.230
See you in the next video.


Chapter 7: Bankruptcy liquidation

WEBVTT
Kind: captions
Language: en

00:00:00.740 --> 00:00:03.470
In the last set of videos, we've
hopefully familiarized

00:00:03.470 --> 00:00:05.250
ourselves with the different
ways that a

00:00:05.250 --> 00:00:06.630
company can raise capital.

00:00:06.630 --> 00:00:09.140
It can do it through
debt or equity.

00:00:09.140 --> 00:00:11.030
And we learned that debt
securities are

00:00:11.030 --> 00:00:12.200
often called bonds.

00:00:12.200 --> 00:00:14.470
And equity securities you're
probably familiar with.

00:00:14.470 --> 00:00:16.610
Those are stocks.

00:00:16.610 --> 00:00:18.620
And then I left you with
a cliffhanger.

00:00:18.620 --> 00:00:20.470
Let draw it so I don't
get ahead of myself.

00:00:20.470 --> 00:00:24.570
So these are the assets of a
company and it was able to

00:00:24.570 --> 00:00:25.430
generate these assets.

00:00:25.430 --> 00:00:27.020
So there's a couple of ways
you can generate assets.

00:00:27.020 --> 00:00:30.270
You can get investors through
equity, and we've done several

00:00:30.270 --> 00:00:31.040
videos on that.

00:00:31.040 --> 00:00:33.810
You start with the angel
investors, or maybe your rich

00:00:33.810 --> 00:00:35.750
uncle, and then eventually get
venture capitalists, and you

00:00:35.750 --> 00:00:37.320
do an initial public offering.

00:00:37.320 --> 00:00:38.760
And then you can do follow-on
offerings.

00:00:38.760 --> 00:00:40.380
And so on and so forth.

00:00:40.380 --> 00:00:44.380
Now we see governments will buy
equity in you if you are a

00:00:44.380 --> 00:00:45.880
bank that's too big to fail.

00:00:45.880 --> 00:00:48.420
But we'll do a whole
playlist on that.

00:00:48.420 --> 00:00:49.350
So equity.

00:00:49.350 --> 00:00:52.270
That's one way that you can
get cash or get capital so

00:00:52.270 --> 00:00:54.560
that you can buy assets
to run your business.

00:00:54.560 --> 00:00:57.030
The other way is you can borrow
money from people.

00:00:57.030 --> 00:00:59.820
So the equity holders
are actually the

00:00:59.820 --> 00:01:00.790
owners of the company.

00:01:00.790 --> 00:01:02.230
So you might have been part of
the equity holder, and you

00:01:02.230 --> 00:01:04.260
have to sell some of the equity,
or sell some shares in

00:01:04.260 --> 00:01:06.090
your company for someone
else to give money.

00:01:06.090 --> 00:01:08.820
Then they become kind of
like your partner.

00:01:08.820 --> 00:01:10.910
And the other way is you
could borrow money.

00:01:10.910 --> 00:01:12.290
Let me draw that.

00:01:12.290 --> 00:01:14.070
That we'll just put generally
as liability.

00:01:14.070 --> 00:01:16.990
Debt isn't the only kind of
liability, but that's a pretty

00:01:16.990 --> 00:01:19.430
reasonable simplification
for now.

00:01:19.430 --> 00:01:20.220
There's other things.

00:01:20.220 --> 00:01:22.930
In general, liability means you
owe something to somebody

00:01:22.930 --> 00:01:24.180
in the future.

00:01:30.390 --> 00:01:31.760
So these are liabilities.

00:01:31.760 --> 00:01:33.970
And we'll assume right
now that your debt

00:01:33.970 --> 00:01:35.070
is your main liability.

00:01:35.070 --> 00:01:36.550
You might have other
liabilities.

00:01:36.550 --> 00:01:38.760
You might have some type of
legal liability, where someone

00:01:38.760 --> 00:01:43.190
is suing you or you had sprayed
asbestos on a bunch of

00:01:43.190 --> 00:01:45.920
playgrounds, thinking that it
was actually good for the

00:01:45.920 --> 00:01:48.770
playground equipment and now
there's all of this liability

00:01:48.770 --> 00:01:50.670
because, well, you
get the idea.

00:01:50.670 --> 00:01:54.610
But from now on we'll have the
simplification that debt is

00:01:54.610 --> 00:01:55.270
your liability.

00:01:55.270 --> 00:01:58.150
And we said there's different
kinds of debt.

00:01:58.150 --> 00:02:02.410
If you securitize it,
it's often a bond.

00:02:02.410 --> 00:02:02.690
Right?

00:02:02.690 --> 00:02:05.860
That would be a certificate
that's an IOU from a company.

00:02:05.860 --> 00:02:07.940
It'll pay you coupons or
interest and so forth.

00:02:07.940 --> 00:02:09.789
Or you can also just get regular
bank debt, where you

00:02:09.789 --> 00:02:11.320
owe the bank money.

00:02:11.320 --> 00:02:14.510
And I left you with a question
the last time around.

00:02:14.510 --> 00:02:18.460
I said, let's say this company
goes into bankruptcy.

00:02:18.460 --> 00:02:21.440
And let's say that these assets
aren't worth what we

00:02:21.440 --> 00:02:22.170
think they are, right?

00:02:22.170 --> 00:02:24.250
In this world, if we just
have to sell off

00:02:24.250 --> 00:02:26.730
these assets, fine.

00:02:26.730 --> 00:02:28.440
The debt guys would
get paid off.

00:02:28.440 --> 00:02:31.400
And the equity guys would get
left over with whatever else.

00:02:31.400 --> 00:02:34.410
So let's say if this
was on our books.

00:02:34.410 --> 00:02:36.140
Whenever you hear things like
book value, and I've done a

00:02:36.140 --> 00:02:38.670
couple of videos on book value
versus market value, but the

00:02:38.670 --> 00:02:40.930
book value is essentially
what you have on

00:02:40.930 --> 00:02:41.680
your accounting books.

00:02:41.680 --> 00:02:44.570
You say that this is
worth $10 million.

00:02:44.570 --> 00:02:44.800
Right?

00:02:44.800 --> 00:02:48.010
Let's say we've bought land
and factories and whatever

00:02:48.010 --> 00:02:50.160
else worth $10 million.

00:02:50.160 --> 00:02:54.160
Let's say your debt
is $6 million.

00:02:54.160 --> 00:02:56.360
Then your equity would
be worth $4 million.

00:02:56.360 --> 00:02:58.780
And let's say, for whatever
reason, the economy turns

00:02:58.780 --> 00:03:04.580
south or maybe this was some
type of business that's now

00:03:04.580 --> 00:03:05.410
not viable.

00:03:05.410 --> 00:03:07.040
So it's going to go
into bankruptcy.

00:03:07.040 --> 00:03:09.770
And I'll get a little bit more
specific on the different

00:03:09.770 --> 00:03:10.650
types of bankruptcy.

00:03:10.650 --> 00:03:12.220
But we're assuming
liquidation.

00:03:12.220 --> 00:03:15.760
Actually I'll just get
specific right now.

00:03:15.760 --> 00:03:21.080
So when we say bankruptcy,
bankruptcy is

00:03:21.080 --> 00:03:22.130
a very common word.

00:03:22.130 --> 00:03:25.050
I think most people have a
general sense what it means.

00:03:25.050 --> 00:03:26.970
They know it's bad and it means
to some degree that a

00:03:26.970 --> 00:03:29.510
company can't operate
as it was before.

00:03:29.510 --> 00:03:31.420
But there's a lot of confusion
over what it means.

00:03:31.420 --> 00:03:33.940
There's actually two types
of bankruptcy.

00:03:33.940 --> 00:03:35.190
There's liquidation.

00:03:37.740 --> 00:03:39.780
And that's essentially saying
that, you know what, this

00:03:39.780 --> 00:03:41.580
business doesn't
make any sense.

00:03:41.580 --> 00:03:43.290
It doesn't make sense
to have the

00:03:43.290 --> 00:03:45.350
employees and run the factories.

00:03:45.350 --> 00:03:47.840
You're never going to make any
money, so you might as well

00:03:47.840 --> 00:03:50.550
just sell everything you have.
You liquidate it all.

00:03:50.550 --> 00:03:51.310
That's one type.

00:03:51.310 --> 00:03:53.810
And that falls under the
category of Chapter 7.

00:03:53.810 --> 00:03:55.620
And we're just talking
about corporate

00:03:55.620 --> 00:03:56.730
bankruptcy right now.

00:03:56.730 --> 00:03:58.250
There's also personal
bankruptcy.

00:03:58.250 --> 00:04:00.825
And maybe we'll do a couple
of videos on that.

00:04:00.825 --> 00:04:04.200
It might be especially relevant
in this economy.

00:04:04.200 --> 00:04:07.090
Well, the other type is
reorganization or

00:04:07.090 --> 00:04:08.340
restructuring.

00:04:14.510 --> 00:04:16.990
And restructuring says,
you know what?

00:04:16.990 --> 00:04:18.459
This factory here,
it's actually

00:04:18.459 --> 00:04:19.610
making something useful.

00:04:19.610 --> 00:04:21.430
It's actually generating
money.

00:04:21.430 --> 00:04:24.430
And actually we can get more
value for what we have here if

00:04:24.430 --> 00:04:25.600
we keep it running.

00:04:25.600 --> 00:04:27.890
And we will just keep it
running, and we'll restructure

00:04:27.890 --> 00:04:28.610
the company.

00:04:28.610 --> 00:04:31.190
And usually that means changing
this side of it.

00:04:31.190 --> 00:04:33.130
So maybe we'll cancel some
debt and all of that.

00:04:33.130 --> 00:04:36.080
And I'll show you how that's
done in a reasonably fair way.

00:04:36.080 --> 00:04:40.400
But just to understand kind of
a simplified scenario, let's

00:04:40.400 --> 00:04:42.120
take liquidation into
consideration.

00:04:42.120 --> 00:04:48.480
So let's say that this was my
website selling shoes online,

00:04:48.480 --> 00:04:50.710
and that all of a sudden
people have

00:04:50.710 --> 00:04:52.880
stopped wearing shoes.

00:04:52.880 --> 00:04:56.060
It's just gone out of fashion,
so it makes no sense anymore

00:04:56.060 --> 00:04:57.110
to sell shoes online.

00:04:57.110 --> 00:05:00.320
So I'm just going to liquidate
my assets, my real estate that

00:05:00.320 --> 00:05:02.270
I might have, my warehouses,
et cetera, et cetera.

00:05:02.270 --> 00:05:04.160
My question that I left you with
in the last video was,

00:05:04.160 --> 00:05:05.570
who gets it?

00:05:05.570 --> 00:05:08.470
So let's say when we liquidate
it-- so we go into bankruptcy

00:05:08.470 --> 00:05:11.690
and essentially all of the
assets are taken into

00:05:11.690 --> 00:05:14.260
possession by the bankruptcy
court-- they're going to sell

00:05:14.260 --> 00:05:14.820
these assets.

00:05:14.820 --> 00:05:16.460
And let's say when they sell
them, they don't get $10

00:05:16.460 --> 00:05:17.790
million for these assets.

00:05:21.020 --> 00:05:23.590
They only get $5 million
for them.

00:05:23.590 --> 00:05:24.250
Right?

00:05:24.250 --> 00:05:26.390
I paid for them thinking that
they were useful in some way,

00:05:26.390 --> 00:05:33.540
but they end up not to
be, so my assets--

00:05:33.540 --> 00:05:35.430
You know what, I just realized
when I talked earlier about

00:05:35.430 --> 00:05:37.170
there's two ways to raise
capital, there's a third way

00:05:37.170 --> 00:05:37.700
to raise capial.

00:05:37.700 --> 00:05:37.930
Right?

00:05:37.930 --> 00:05:39.580
You can sell shares.

00:05:39.580 --> 00:05:41.350
You can issue debt.

00:05:41.350 --> 00:05:42.380
You can borrow money.

00:05:42.380 --> 00:05:44.480
Obviously the third way is
actually just make money.

00:05:44.480 --> 00:05:44.900
Right?

00:05:44.900 --> 00:05:46.810
Once you start a company,
hopefully you generate

00:05:46.810 --> 00:05:49.590
earnings, and that'll also
generate cash or capital that

00:05:49.590 --> 00:05:50.790
you can reinvest in
the business.

00:05:50.790 --> 00:05:51.640
And we'll talk about that.

00:05:51.640 --> 00:05:53.640
But I just wanted to make it
clear that that's obviously

00:05:53.640 --> 00:05:56.250
the best way to generate capital
for your business is

00:05:56.250 --> 00:05:58.520
when the business itself
generates capital.

00:05:58.520 --> 00:06:01.180
So let's say that these assets,
when you actually sell

00:06:01.180 --> 00:06:05.760
them off, aren't worth
$10 million anymore.

00:06:05.760 --> 00:06:09.340
Let me make the pointer
smaller.

00:06:09.340 --> 00:06:11.730
They're worth $5 million.

00:06:15.320 --> 00:06:17.320
So my question in the
last video is, who

00:06:17.320 --> 00:06:18.750
gets this $5 million?

00:06:18.750 --> 00:06:21.670
Do you somehow split it evenly
between all of these people?

00:06:21.670 --> 00:06:24.000
Or does one of them get more
of it, or one of them gets

00:06:24.000 --> 00:06:25.000
less of it?

00:06:25.000 --> 00:06:28.890
And I think you'll get a sense
based on where I took the $5

00:06:28.890 --> 00:06:31.680
million out of, who
gets the money.

00:06:31.680 --> 00:06:33.630
It's the debt holders.

00:06:33.630 --> 00:06:36.100
And the way I drew it right
here, you can kind of view it

00:06:36.100 --> 00:06:38.760
as you go up in this
direction, you're

00:06:38.760 --> 00:06:40.010
getting more senior.

00:06:42.580 --> 00:06:43.850
Or if you're going down
in this way, you're

00:06:43.850 --> 00:06:46.110
getting more junior.

00:06:46.110 --> 00:06:48.750
And seniority, when you talk
about a company's capital

00:06:48.750 --> 00:06:50.760
structure, is just, you know
what, if there's anything

00:06:50.760 --> 00:06:52.860
left, who gets their
money first?

00:06:52.860 --> 00:06:55.160
And even within the
debt, you'll have

00:06:55.160 --> 00:06:57.280
different layers of debt.

00:06:57.280 --> 00:06:59.520
There might be different debt
holders who have different

00:06:59.520 --> 00:07:01.030
levels of seniority.

00:07:01.030 --> 00:07:04.950
So this one might be called
senior secured debt.

00:07:08.490 --> 00:07:10.370
Senior means they're high
up on the stack.

00:07:10.370 --> 00:07:13.080
They are one of the first people
to get their money.

00:07:13.080 --> 00:07:15.590
And secured means there's
actually some collateral on

00:07:15.590 --> 00:07:19.170
the asset side that they get
if the company can't pay.

00:07:19.170 --> 00:07:21.060
So maybe this is like
a piece of land.

00:07:21.060 --> 00:07:22.490
Right?

00:07:22.490 --> 00:07:25.240
So just in kind of our everyday
personal finance

00:07:25.240 --> 00:07:28.150
world, your mortgage is
actually secured debt.

00:07:28.150 --> 00:07:29.990
It's secured by the collateral
of your home.

00:07:29.990 --> 00:07:32.830
If you can't pay the debt,
the bank comes

00:07:32.830 --> 00:07:33.470
and takes your home.

00:07:33.470 --> 00:07:35.040
It forecloses on the property.

00:07:35.040 --> 00:07:36.320
So that's what secured means.

00:07:36.320 --> 00:07:38.540
It means that there's some
collateral, and in the event

00:07:38.540 --> 00:07:41.750
of a bankruptcy this guy can
immediately go and get the

00:07:41.750 --> 00:07:43.620
collateral that his debt
is secured by.

00:07:43.620 --> 00:07:47.310
So this is considered a very,
very senior form of debt.

00:07:47.310 --> 00:07:48.620
Senior secured.

00:07:48.620 --> 00:07:52.960
Then you might have here, you
might have senior unsecured.

00:07:56.620 --> 00:07:59.070
And there's a lot of words
around, senior, junior,

00:07:59.070 --> 00:08:00.110
subordinate, and all of that.

00:08:00.110 --> 00:08:02.470
But just to get a sense that
there's just a hierarchy here.

00:08:02.470 --> 00:08:04.620
Some people are the first people
to get the money, and

00:08:04.620 --> 00:08:06.440
then whatever money is left goes
to this person, then if

00:08:06.440 --> 00:08:08.460
there's any money left, it goes
to this person, and then

00:08:08.460 --> 00:08:10.630
if there's anything left
it goes to this person.

00:08:10.630 --> 00:08:12.460
And once you're in bankruptcy
court it does tend to be a

00:08:12.460 --> 00:08:15.163
negotiation between the
different, you can almost view

00:08:15.163 --> 00:08:16.120
it as buckets, of debt.

00:08:16.120 --> 00:08:19.010
And we'll do a more complicated
example in the

00:08:19.010 --> 00:08:19.590
future on that.

00:08:19.590 --> 00:08:21.760
We'll actually delve into the
details of bankruptcy.

00:08:21.760 --> 00:08:23.170
But this is the general
notion.

00:08:23.170 --> 00:08:26.760
That the senior guys get made
whole first, then the more

00:08:26.760 --> 00:08:29.420
junior guys get whatever's left,
and so on and so forth.

00:08:29.420 --> 00:08:31.990
And if there's no money for the
equity, there's no money

00:08:31.990 --> 00:08:32.679
for the equity.

00:08:32.679 --> 00:08:33.610
And that makes sense, right?

00:08:33.610 --> 00:08:36.400
Because the debt holders, all
they were getting-- their

00:08:36.400 --> 00:08:38.070
upside was just interest,
right?

00:08:38.070 --> 00:08:40.640
So they also should get limited
downside in the event

00:08:40.640 --> 00:08:41.980
things should turn bad.

00:08:41.980 --> 00:08:43.500
Equity holders, they kind
of took a gamble.

00:08:43.500 --> 00:08:46.000
If things were great, they would
get all of the upside.

00:08:46.000 --> 00:08:47.770
And now that things turn
bad, they take

00:08:47.770 --> 00:08:48.500
a lot of the downside.

00:08:48.500 --> 00:08:50.350
And they're actually lucky that
they don't owe money.

00:08:50.350 --> 00:08:52.730
That's actually the-- I guess
you could call it-- the beauty

00:08:52.730 --> 00:08:55.060
of a corporate structure, that
you have limited liability.

00:08:55.060 --> 00:08:57.640
In some times in history, these
people would actually

00:08:57.640 --> 00:08:58.770
owe the difference.

00:08:58.770 --> 00:09:01.730
They would actually owe
this extra $1 million.

00:09:01.730 --> 00:09:03.150
They would all go to debtor's
prison and all that.

00:09:03.150 --> 00:09:04.670
But we'll talk more about
it in the future.

00:09:04.670 --> 00:09:07.190
So anyway, just going back on
the different tranches of

00:09:07.190 --> 00:09:08.730
debt, or buckets of debt.

00:09:08.730 --> 00:09:09.840
So we could call this
senior unsecured.

00:09:09.840 --> 00:09:12.380
And that means that they're
still senior.

00:09:12.380 --> 00:09:15.780
They're still fairly high
up the seniority ladder.

00:09:15.780 --> 00:09:16.440
But they're unsecured.

00:09:16.440 --> 00:09:18.680
There's no particular assets
that they can go run.

00:09:18.680 --> 00:09:21.190
But as long as there's enough
for them, they'll get it.

00:09:21.190 --> 00:09:23.190
So let me put some
numbers here.

00:09:23.190 --> 00:09:26.390
So let's say there was, I don't
know, $1 million of

00:09:26.390 --> 00:09:27.750
senior secured.

00:09:27.750 --> 00:09:32.570
Let's say there's $2 million
of senior unsecured.

00:09:32.570 --> 00:09:40.930
And let's say that this is $2
million of subordinated--

00:09:40.930 --> 00:09:41.870
subordinated just means they're

00:09:41.870 --> 00:09:43.330
not senior-- unsecured.

00:09:46.280 --> 00:09:48.780
So in this reality, what would
happen is the bankruptcy court

00:09:48.780 --> 00:09:50.780
would liquidate all this stuff
and then they'll hand it out

00:09:50.780 --> 00:09:51.620
in order of seniority.

00:09:51.620 --> 00:09:53.620
These guys get their
$1 million back.

00:09:53.620 --> 00:09:54.980
So they're made whole.

00:09:54.980 --> 00:09:57.000
And they probably charged a
lower interest rate, because

00:09:57.000 --> 00:09:59.680
they didn't perceive their risk
that high to begin with.

00:09:59.680 --> 00:10:02.290
These guys, right here, the
senior unsecured, they'll get

00:10:02.290 --> 00:10:03.300
the next $2 million.

00:10:03.300 --> 00:10:05.980
And then there's $1
million left.

00:10:05.980 --> 00:10:06.490
Right?

00:10:06.490 --> 00:10:09.700
And that $1 million will go
to the subordinated debt.

00:10:09.700 --> 00:10:11.870
So they'll get 50% of
their money back.

00:10:11.870 --> 00:10:15.120
So they took a little bit of a
hit, but that's OK because

00:10:15.120 --> 00:10:18.300
when things were good, they
probably got higher interest

00:10:18.300 --> 00:10:19.650
to compensate them
for their risk.

00:10:19.650 --> 00:10:23.810
Usually as you get more and more
junior and you take on

00:10:23.810 --> 00:10:26.630
more risk, you get more upside,
or more interest. And

00:10:26.630 --> 00:10:30.930
in this case the equity
holders get nothing.

00:10:30.930 --> 00:10:32.790
They get wiped out.

00:10:32.790 --> 00:10:33.860
So it just goes to 0.

00:10:33.860 --> 00:10:36.190
So that's the answer
to the question.

00:10:36.190 --> 00:10:37.360
I said, who gets the money?

00:10:37.360 --> 00:10:39.430
Well, it's the debt holders
get first dibs.

00:10:39.430 --> 00:10:45.720
And if there was actually $7
million here instead of $5

00:10:45.720 --> 00:10:50.820
million, then you would have
paid the six off completely,

00:10:50.820 --> 00:10:53.810
and then the equity holders
would've gotten $1 million.

00:10:53.810 --> 00:10:55.580
And so they would have gotten
something if there was enough

00:10:55.580 --> 00:10:57.840
money to hand it to them.

00:10:57.840 --> 00:11:01.950
Anyway, in the next video I'll
cover-- this was liquidation,

00:11:01.950 --> 00:11:03.420
where we just say this
isn't worth running.

00:11:03.420 --> 00:11:06.440
Let's just give it all away, or
let's sell it, and give it

00:11:06.440 --> 00:11:09.675
back to our creditors.

00:11:09.675 --> 00:11:12.430
In the next video I'll talk
about reorganization, where we

00:11:12.430 --> 00:11:13.190
say, hey, you know what?

00:11:13.190 --> 00:11:14.480
This business is a
good business.

00:11:14.480 --> 00:11:17.110
It just has too many
liabilities.

00:11:17.110 --> 00:11:18.870
See you in the next video.


Polar coordinates 3

WEBVTT
Kind: captions
Language: en

00:00:00.750 --> 00:00:04.250
All right, let's keep
converting Cartesian functions

00:00:04.250 --> 00:00:05.940
to polar coordinates.

00:00:05.940 --> 00:00:14.120
The next one I have here is
3y minus 7x is equal to 10.

00:00:14.120 --> 00:00:16.620
I cut and pasted our tool
kit here, and let's

00:00:16.620 --> 00:00:17.110
see what we can do.

00:00:17.110 --> 00:00:20.160
We want to convert this to
a function of r and theta.

00:00:20.160 --> 00:00:22.350
So the simplest thing,
we have a y and an x,

00:00:22.350 --> 00:00:23.320
we just substitute.

00:00:23.320 --> 00:00:25.930
We know that y is equal to r
sine of theta and x is equal

00:00:25.930 --> 00:00:28.110
to r cosine of theta, so
let's just substitute.

00:00:28.110 --> 00:00:33.180
So 3 times y, well, y is the
same thing as r sine of

00:00:33.180 --> 00:00:37.740
theta minus 7 times x.

00:00:37.740 --> 00:00:43.550
Well, that's 7 times r cosine
theta is equal to-- now,

00:00:43.550 --> 00:00:45.860
remember, these just came from
SOHCAHTOA, nothing

00:00:45.860 --> 00:00:47.080
fancier than that.

00:00:47.080 --> 00:00:49.320
Let's see if we can simplify
this or write it explicitly

00:00:49.320 --> 00:00:51.590
in terms of theta.

00:00:51.590 --> 00:00:59.460
So we could factor out an r, so
you get r times 3 sine theta

00:00:59.460 --> 00:01:01.865
minus 7 cosine theta, right?

00:01:01.865 --> 00:01:04.230
We just factored out an r,
and that's equal to 10.

00:01:04.230 --> 00:01:06.960
And now divide both sides by
this big expression, and you're

00:01:06.960 --> 00:01:14.860
left with r is equal to 10, 3
sine of theta-- 10 divided by 3

00:01:14.860 --> 00:01:19.370
sine of theta minus
7 cosine of theta.

00:01:19.370 --> 00:01:20.870
And we could also write this,
you know, we could say, well,

00:01:20.870 --> 00:01:22.400
this r is a function of theta.

00:01:22.400 --> 00:01:25.800
We could write r of
theta, right? r is a

00:01:25.800 --> 00:01:26.840
function of theta.

00:01:26.840 --> 00:01:28.480
I just felt like doing that so
that you could show that the

00:01:28.480 --> 00:01:32.130
function notation even works
in polar coordinates.

00:01:32.130 --> 00:01:33.980
Let's do another one.

00:01:33.980 --> 00:01:36.080
Let me cut and paste
our tool kit.

00:01:36.080 --> 00:01:37.740
Look at that.

00:01:37.740 --> 00:01:41.480
All right, next one.

00:01:41.480 --> 00:01:46.760
So it says y is equal
to 2x minus 3.

00:01:46.760 --> 00:01:47.880
Well, we can do the same thing.

00:01:47.880 --> 00:01:51.390
We know what y and x are in
terms of r and theta. y is

00:01:51.390 --> 00:01:53.780
equal to r sine of theta,
so let's write that.

00:01:53.780 --> 00:01:59.130
r sine of theta is
equal to 2 times x.

00:01:59.130 --> 00:02:04.420
Well, x is r cosine
of theta minus 3.

00:02:04.420 --> 00:02:06.440
Let's see if we can separate
the r's and thetas, so let's

00:02:06.440 --> 00:02:09.170
subtract this from both sides.

00:02:09.170 --> 00:02:17.490
So we get r sine of theta
minus 2r cosine of theta

00:02:17.490 --> 00:02:19.560
is equal to minus 3.

00:02:19.560 --> 00:02:23.050
Just like the last problem, we
can factor an r out, so we get

00:02:23.050 --> 00:02:31.010
r times sine of theta minus 2
cosine of theta is

00:02:31.010 --> 00:02:31.990
equal to minus 3.

00:02:31.990 --> 00:02:35.190
Now, divide both sides by this
expression, so you're left with

00:02:35.190 --> 00:02:41.930
r is equal to minus 3 divided
by sine of theta minus

00:02:41.930 --> 00:02:44.450
2 cosine of theta.

00:02:44.450 --> 00:02:46.240
There you go.

00:02:46.240 --> 00:02:49.680
Now they want us to
do some problems.

00:02:49.680 --> 00:02:54.110
Let me clear everything.

00:02:54.110 --> 00:02:55.220
I want to do some problems.

00:02:55.220 --> 00:02:56.130
We're going to convert
the other way.

00:02:56.130 --> 00:02:58.890
Let's write our tool kit down.

00:02:58.890 --> 00:03:01.380
It's proved to be useful.

00:03:01.380 --> 00:03:04.170
All right, now they want to
convert from polar to Cartesian

00:03:04.170 --> 00:03:08.560
coordinates, and so they give
us the polar function r is

00:03:08.560 --> 00:03:11.800
equal to 4 sine of theta.

00:03:11.800 --> 00:03:15.290
So how do we convert this
into a function of x and y?

00:03:15.290 --> 00:03:15.950
So let's see here.

00:03:15.950 --> 00:03:21.190
There's no obvious-- let's
think about it a little bit.

00:03:21.190 --> 00:03:23.240
So we know we have this
equation y is equal

00:03:23.240 --> 00:03:24.510
to r sine of that.

00:03:24.510 --> 00:03:26.290
So can we write sine of theta?

00:03:26.290 --> 00:03:28.190
Well, let me do a
little side here.

00:03:28.190 --> 00:03:32.690
So y is equal to
r sine of theta.

00:03:32.690 --> 00:03:34.900
If we divide both sides
by r, what do we get?

00:03:34.900 --> 00:03:40.500
We get y/r is equal to
sine of theta, right?

00:03:40.500 --> 00:03:41.460
Well, that seems to be helpful.

00:03:41.460 --> 00:03:43.350
We have a sine of theta there,
and then we have something that

00:03:43.350 --> 00:03:45.550
at least it gets sort of the
theta out of the equation.

00:03:45.550 --> 00:03:47.460
We're still left with an
r, but it makes it a

00:03:47.460 --> 00:03:48.610
lot easier to look at.

00:03:48.610 --> 00:03:49.250
So let's do that.

00:03:49.250 --> 00:03:51.970
Sine of theta is equal to y/r,
so let's substitute that there.

00:03:51.970 --> 00:03:55.470
So we get r is equal to
4 times sine of theta,

00:03:55.470 --> 00:03:57.055
which is y/r, so 4y/r.

00:03:59.640 --> 00:04:02.140
We can multiply both sides of
the equation by r, and you're

00:04:02.140 --> 00:04:05.930
left with r squared
is equal to 4y.

00:04:05.930 --> 00:04:07.570
And we know what r
squared is equal to.

00:04:07.570 --> 00:04:09.260
It tells us right there.
r squared is equal to x

00:04:09.260 --> 00:04:10.250
squared plus y squared.

00:04:10.250 --> 00:04:16.820
So you get x squared plus
y squared is equal to 4y.

00:04:16.820 --> 00:04:17.490
And there we have it.

00:04:17.490 --> 00:04:18.950
We have it at least
in an implicit form.

00:04:18.950 --> 00:04:21.410
We don't have an explicit
equation, but I think that's

00:04:21.410 --> 00:04:23.100
good enough for now.

00:04:23.100 --> 00:04:25.300
So you see, it's really just a
lot of algebra and I guess a

00:04:25.300 --> 00:04:27.190
little bit of trigonometry, but
really a lot of algebra and

00:04:27.190 --> 00:04:28.880
just kind of figuring out
how to use this tool kit.

00:04:28.880 --> 00:04:30.610
Let's do a couple more.

00:04:30.610 --> 00:04:36.780
So let's say-- let me
paste the tool kit.

00:04:36.780 --> 00:04:41.020
Let's say that we have the
polar coordinates r is equal

00:04:41.020 --> 00:04:45.360
to sine of theta plus
cosine of theta.

00:04:45.360 --> 00:04:47.320
And just so you don't lose the
big picture of what we're

00:04:47.320 --> 00:04:50.280
doing, we could graph this.

00:04:50.280 --> 00:04:51.950
You could put your graphing
calculator on the polar

00:04:51.950 --> 00:04:54.030
coordinates, and I'll do future
videos where we do graph it,

00:04:54.030 --> 00:04:55.310
and it'll produce some graph.

00:04:55.310 --> 00:04:58.200
As theta changes, as we go
around the circle, the radius

00:04:58.200 --> 00:04:59.480
will change, and it'll
produce something.

00:04:59.480 --> 00:05:00.810
I don't know what it might
look like, like flower

00:05:00.810 --> 00:05:01.580
petals or something.

00:05:01.580 --> 00:05:03.650
I don't know I'll have to-- I
don't have the intuition of

00:05:03.650 --> 00:05:05.940
exactly what this'll look like,
but what we're saying is when

00:05:05.940 --> 00:05:08.100
we convert to x and y, if you
actually graphed in the

00:05:08.100 --> 00:05:10.210
Cartesian coordinates, it would
look the exact same way.

00:05:10.210 --> 00:05:11.100
That's all we're doing.

00:05:11.100 --> 00:05:13.210
How do you express the
same relationship in

00:05:13.210 --> 00:05:15.010
terms of x and y's?

00:05:15.010 --> 00:05:15.955
So how do we do this?

00:05:15.955 --> 00:05:20.820
Well, in the last one, we can--
these two, we can rewrite.

00:05:20.820 --> 00:05:24.540
We can divide both sides
of these equations by r.

00:05:24.540 --> 00:05:28.770
This is the same thing
as sine of theta is

00:05:28.770 --> 00:05:32.590
equal to y/r, right?

00:05:32.590 --> 00:05:34.320
I just divide both sides by r.

00:05:34.320 --> 00:05:36.120
And this is the same
thing as cosine of

00:05:36.120 --> 00:05:40.140
theta is equal to x/r.

00:05:40.140 --> 00:05:42.060
Just divided both sides
of this equation by

00:05:42.060 --> 00:05:43.500
r, and you get this.

00:05:43.500 --> 00:05:47.480
So now we can use this to
substitute back here, so r

00:05:47.480 --> 00:05:48.740
is equal to sine of theta.

00:05:48.740 --> 00:05:53.860
Well, we know that's equal to
y/r plus cosine of theta.

00:05:53.860 --> 00:05:55.480
Well, cosine of theta
is equal to x/r.

00:05:58.200 --> 00:06:00.762
And now we can multiply both
sides of this equation by r,

00:06:00.762 --> 00:06:06.120
and we're left with r squared
is equal to y plus x.

00:06:06.120 --> 00:06:07.980
And we know what r
squared is now.

00:06:07.980 --> 00:06:10.160
That should be second
nature, hopefully.

00:06:10.160 --> 00:06:14.240
So it's x squared plus
y squared, right?

00:06:14.240 --> 00:06:17.760
That's r squared is
equal to x-- sorry,

00:06:17.760 --> 00:06:19.990
is equal to y plus x.

00:06:19.990 --> 00:06:21.570
There you go.

00:06:21.570 --> 00:06:24.920
Let's do one more.

00:06:24.920 --> 00:06:26.270
Let's do it without our
tool kit, and I'll do it

00:06:26.270 --> 00:06:28.360
in a different color.

00:06:28.360 --> 00:06:33.830
Let's say we had r is equal
to a squared, so they're

00:06:33.830 --> 00:06:35.030
leaving it abstract.

00:06:35.030 --> 00:06:37.990
But a is some constant, so we
hopefully know our intuition

00:06:37.990 --> 00:06:39.410
from a couple of videos ago
that this should be

00:06:39.410 --> 00:06:40.390
a circle, right?

00:06:40.390 --> 00:06:44.630
If a was 3, then we'd say,
oh, r is equal to 9, right?

00:06:44.630 --> 00:06:46.310
3 squared, and then there'd
just be a circle with a

00:06:46.310 --> 00:06:47.400
constant radius of 9.

00:06:47.400 --> 00:06:52.560
So how would you write this
in Cartesian coordinates?

00:06:52.560 --> 00:06:55.940
Well, we know that x
squared plus y squared

00:06:55.940 --> 00:06:57.490
is equal to r squared.

00:06:57.490 --> 00:06:58.870
So one thing we could do,
maybe just square both

00:06:58.870 --> 00:06:59.750
sides of this equation.

00:06:59.750 --> 00:07:02.610
So you get r squared is equal
to-- what's a squared squared?

00:07:02.610 --> 00:07:04.540
Well, it's a to the
fourth, right?

00:07:04.540 --> 00:07:08.260
It's a to the 2 times
2, so a to the fourth.

00:07:08.260 --> 00:07:10.220
And we know what r squared is.

00:07:10.220 --> 00:07:13.750
It's x squared plus y squared,
so you get x squared plus y

00:07:13.750 --> 00:07:16.920
squared is equal to
a to the fourth.

00:07:16.920 --> 00:07:18.580
Not too bad.

00:07:18.580 --> 00:07:21.050
Not too bad.

00:07:21.050 --> 00:07:23.660
And that's really-- I mean,
that'll get you pretty far in

00:07:23.660 --> 00:07:26.100
converting between Cartesian
and polar coordinates.

00:07:26.100 --> 00:07:28.600
In the next video, we'll maybe
explore a couple of these

00:07:28.600 --> 00:07:32.080
graphs to hopefully give you a
little intuition of really how

00:07:32.080 --> 00:07:34.120
the relationships between
r and theta really works.

00:07:34.120 --> 00:07:36.100
See you in the next video.


Polar coordinates 2

WEBVTT
Kind: captions
Language: en

00:00:00.700 --> 00:00:03.540
We're now equipped, I think,
with all of the tools to

00:00:03.540 --> 00:00:05.780
convert back and forth between
polar and rectangular

00:00:05.780 --> 00:00:06.280
coordinates.

00:00:06.280 --> 00:00:08.470
And hopefully you have a
reasonable intuition.

00:00:08.470 --> 00:00:11.380
I mean, they're both just
two ways of specifying a

00:00:11.380 --> 00:00:12.950
point in two dimensions.

00:00:12.950 --> 00:00:15.500
Cartesian, you kind of say how
far left and right you go--

00:00:15.500 --> 00:00:16.300
that's the x-coordinate.

00:00:16.300 --> 00:00:19.000
And then the y-coordinate tells
you how far up and down you

00:00:19.000 --> 00:00:20.120
go when you get to a point.

00:00:20.120 --> 00:00:23.070
When you do polar coordinates,
the angle tells you what

00:00:23.070 --> 00:00:26.030
direction to point in, and then
the r, or you could view it as

00:00:26.030 --> 00:00:29.330
the radius, tells you how far
to walk in that direction.

00:00:29.330 --> 00:00:29.580
Right?

00:00:29.580 --> 00:00:31.765
So they're just two ways to
specify a point in space.

00:00:31.765 --> 00:00:35.720
And as we'll see some functions
are easier to specify, or

00:00:35.720 --> 00:00:38.410
they're better described, in
polar coordinates, while other

00:00:38.410 --> 00:00:40.710
functions are better described
in Cartesian coordinates.

00:00:40.710 --> 00:00:42.570
But before we actually do
functions, let's do a couple

00:00:42.570 --> 00:00:48.410
of quick problems using the
tools that we've built.

00:00:48.410 --> 00:00:52.570
Let's say that we're given
the polar coordinate

00:00:52.570 --> 00:00:58.090
4, 150 degrees, and we
wanted to convert this to

00:00:58.090 --> 00:00:58.990
Cartesian coordinates.

00:00:58.990 --> 00:01:01.420
So we want to
convert it to x, y.

00:01:03.940 --> 00:01:07.510
Our first thing that we could
do is just to get an intuition

00:01:07.510 --> 00:01:08.750
of where this even is.

00:01:08.750 --> 00:01:10.300
Right now I think you're
already at the point where if I

00:01:10.300 --> 00:01:12.020
give you an x and y coordinate
you already have a

00:01:12.020 --> 00:01:14.130
visualization of
where that shows up.

00:01:14.130 --> 00:01:18.660
But polar might be relatively
new, so let's try to draw it.

00:01:21.400 --> 00:01:24.170
So for me, when I do polar
I like to think about

00:01:24.170 --> 00:01:24.970
the angle first.

00:01:24.970 --> 00:01:26.300
So 150 degrees.

00:01:26.300 --> 00:01:27.770
So if we start at the x-axis.

00:01:27.770 --> 00:01:28.000
Let's see.

00:01:28.000 --> 00:01:29.430
That's 90 degrees.

00:01:29.430 --> 00:01:31.910
All the way around
is 180 degrees.

00:01:31.910 --> 00:01:35.750
So this will be 30 degrees
away from 180, so it's going

00:01:35.750 --> 00:01:37.260
to be in this direction.

00:01:37.260 --> 00:01:37.550
Right?

00:01:37.550 --> 00:01:42.830
This angle right here
is 150 degrees.

00:01:42.830 --> 00:01:46.370
And I do find degrees easier
to visualize than radians.

00:01:46.370 --> 00:01:49.040
That's, I think, just because
we use that in everyday life.

00:01:49.040 --> 00:01:54.090
And then we're saying, OK I've
pointed you in this direction.

00:01:54.090 --> 00:01:55.330
How far should you go?

00:01:55.330 --> 00:01:58.680
It says, well you go 4 units
in the 150 degree direction.

00:01:58.680 --> 00:02:00.740
So you go 4 units.

00:02:00.740 --> 00:02:01.880
I don't know what that'll be.

00:02:01.880 --> 00:02:04.160
I don't want to do that.

00:02:04.160 --> 00:02:05.990
So I want to go 4 units
in that direction.

00:02:10.005 --> 00:02:12.220
Let me do it in a
different color.

00:02:12.220 --> 00:02:19.030
I'm just assuming this is
one, two, three, four units.

00:02:19.030 --> 00:02:21.250
So that's 4, 150 degrees.

00:02:21.250 --> 00:02:24.050
So we want to convert this
to Cartesian coordinates.

00:02:24.050 --> 00:02:26.070
And so we already have an
intuition that our x-coordinate

00:02:26.070 --> 00:02:28.630
should be some negative number,
and our y-coordinate should

00:02:28.630 --> 00:02:29.720
be some positive number.

00:02:29.720 --> 00:02:30.030
Right?

00:02:30.030 --> 00:02:31.340
And let's see if
it all bears out.

00:02:31.340 --> 00:02:35.690
So what are the tools that we
came up with in the last video?

00:02:35.690 --> 00:02:40.540
Well, we came up with x is
equal to r cosine theta.

00:02:40.540 --> 00:02:42.650
And that just fell
out of SOH-CAH-TOA.

00:02:42.650 --> 00:02:43.990
Nothing fancy there.

00:02:43.990 --> 00:02:49.610
And y is equal to r sine theta.

00:02:49.610 --> 00:02:52.010
So this coordinate, x is
equal to r cosine theta.

00:02:52.010 --> 00:02:55.390
So this is going to be-- let me
write it in a different color.

00:02:55.390 --> 00:02:59.180
x is r-- so now we're dealing
with Cartesian coordinates--

00:02:59.180 --> 00:03:00.300
r cosine theta.

00:03:00.300 --> 00:03:01.280
So r is 4.

00:03:01.280 --> 00:03:07.370
So it's 4 cosine
of 150 degrees.

00:03:07.370 --> 00:03:13.970
And then the y is r, which is
4, times sine of 150 degrees.

00:03:17.660 --> 00:03:21.240
And now we can get
the calculator out.

00:03:21.240 --> 00:03:23.070
OK.

00:03:23.070 --> 00:03:23.530
So let's see.

00:03:23.530 --> 00:03:25.890
What's the cosine of-- well,
I'll just write it out.

00:03:25.890 --> 00:03:31.160
So 4 times the cosine of-- and
make sure your calculator

00:03:31.160 --> 00:03:32.600
is in degree mode.

00:03:32.600 --> 00:03:36.450
You don't want to make it think
you're saying 150 radians.

00:03:36.450 --> 00:03:39.320
So it's minus 3.46.

00:03:39.320 --> 00:03:43.930
So the x-coordinate
is minus 3.46.

00:03:43.930 --> 00:03:46.286
And what is the y-coordinate?

00:03:46.286 --> 00:03:55.305
It's 4 times sine of 150
degrees, which is equal to 2.

00:03:57.870 --> 00:03:59.090
And that makes sense, right?

00:03:59.090 --> 00:04:01.940
I mean if we eyeball it,
that looks about right.

00:04:01.940 --> 00:04:05.870
If we go one, two, three, a
little bit more than three,

00:04:05.870 --> 00:04:09.420
that looks like minus 3.46.

00:04:09.420 --> 00:04:11.120
And then we go up one, two.

00:04:11.120 --> 00:04:13.310
That looks like
about positive 2.

00:04:13.310 --> 00:04:14.670
So our intuition was right.

00:04:14.670 --> 00:04:18.050
You could either kind of orient
yourself into the 150 degree

00:04:18.050 --> 00:04:20.950
direction and then march
forward in that direction, or

00:04:20.950 --> 00:04:23.520
you could say I'm going to go
minus 3.46 to the

00:04:23.520 --> 00:04:25.140
left and 2 up.

00:04:25.140 --> 00:04:28.440
So we've just done a
transition from polar to

00:04:28.440 --> 00:04:29.400
Cartesian coordinates.

00:04:29.400 --> 00:04:30.390
Not too bad.

00:04:30.390 --> 00:04:32.880
So now let's step it up a
little bit and actually convert

00:04:32.880 --> 00:04:36.140
functions from Cartesian to
polar and polar to Cartesian,

00:04:36.140 --> 00:04:38.970
and back and forth.

00:04:38.970 --> 00:04:41.640
So I just did a random
web search for problems

00:04:41.640 --> 00:04:43.440
and I found these.

00:04:43.440 --> 00:04:48.770
The first one, they want us to
convert this, x squared plus y

00:04:48.770 --> 00:04:53.260
squared is equal to 4,
to polar coordinates.

00:04:53.260 --> 00:04:53.770
Right?

00:04:53.770 --> 00:04:55.670
And hopefully you have some
intuition already that

00:04:55.670 --> 00:04:56.640
this is a circle.

00:04:56.640 --> 00:04:56.940
Right?

00:04:56.940 --> 00:04:58.850
And it's actually even more
obvious when you do it

00:04:58.850 --> 00:04:59.600
in polar coordinates.

00:04:59.600 --> 00:05:01.230
So what were the
toolkits we had built?

00:05:01.230 --> 00:05:05.630
We had figured out that x
squared plus y squared

00:05:05.630 --> 00:05:06.560
is equal to r squared.

00:05:06.560 --> 00:05:08.490
That was just the
Pythagorean theorem.

00:05:08.490 --> 00:05:16.430
We knew that the tangent of
theta was equal to the opposite

00:05:16.430 --> 00:05:20.230
over the adjacent-- y over x.

00:05:20.230 --> 00:05:26.380
And we also learned that y is
equal to r sine of theta, and

00:05:26.380 --> 00:05:30.440
x is equal to r cosine theta.

00:05:30.440 --> 00:05:32.780
And you can prove all these
just from SOH-CAH-TOA and

00:05:32.780 --> 00:05:33.440
the Pythagorean theorem.

00:05:33.440 --> 00:05:36.500
But if you had to memorize one
thing going into an exam on

00:05:36.500 --> 00:05:38.490
polar coordinates it would
be these four equations.

00:05:38.490 --> 00:05:39.790
Not that I'm a big fan.

00:05:39.790 --> 00:05:41.590
I actually don't have
these memorized.

00:05:41.590 --> 00:05:44.290
I always spent, literally,
the first 10 seconds of the

00:05:44.290 --> 00:05:47.050
exam writing these down or
reproving them, and that

00:05:47.050 --> 00:05:48.090
way you'll never forget it.

00:05:48.090 --> 00:05:50.920
But anyway, let's convert
this to polar coordinates.

00:05:50.920 --> 00:05:52.840
Well, you should already
see we have an x

00:05:52.840 --> 00:05:54.600
squared plus y squared.

00:05:54.600 --> 00:05:56.170
And we already say, well
x squared plus y squared

00:05:56.170 --> 00:05:57.410
is equal to r squared.

00:05:57.410 --> 00:05:58.410
So we could just rewrite this.

00:05:58.410 --> 00:06:01.770
This r squared is equal to 4.

00:06:01.770 --> 00:06:06.400
And then you say, oh maybe r
is equal to plus or minus 2.

00:06:06.400 --> 00:06:06.930
Right?

00:06:06.930 --> 00:06:08.110
And it actually doesn't matter.

00:06:08.110 --> 00:06:10.160
You could just say
r is equal to 2.

00:06:10.160 --> 00:06:10.890
And what does that mean?

00:06:10.890 --> 00:06:15.100
Actually, I think it'll be
instructive to graph this one.

00:06:15.100 --> 00:06:17.620
I wasn't planning on it,
but I think it'll be

00:06:17.620 --> 00:06:18.310
neat to show you.

00:06:18.310 --> 00:06:20.130
I mean r equals 2, that's
such a simple equation.

00:06:20.130 --> 00:06:22.890
How can that be a circle?

00:06:22.890 --> 00:06:24.080
Well what is a circle?

00:06:24.080 --> 00:06:26.380
A circle is something
of constant radius

00:06:26.380 --> 00:06:28.180
around some point.

00:06:28.180 --> 00:06:28.540
Right?

00:06:28.540 --> 00:06:30.110
So this is staying
a constant radius.

00:06:30.110 --> 00:06:31.670
Radius is equal to 2.

00:06:31.670 --> 00:06:36.470
So if we take a distance
of-- so this is one, two.

00:06:36.470 --> 00:06:38.700
So it says no matter what
the angle is, right?

00:06:38.700 --> 00:06:41.740
This function doesn't
even involve the angle.

00:06:41.740 --> 00:06:42.840
It says the radius is constant.

00:06:42.840 --> 00:06:44.280
It doesn't matter
what angle I have.

00:06:44.280 --> 00:06:46.170
So if the angle is
0 the radius is 2.

00:06:46.170 --> 00:06:48.130
If the angle is, I don't
know, 30 degrees the

00:06:48.130 --> 00:06:49.280
radius is still 2.

00:06:49.280 --> 00:06:51.490
If the angle is 60
degrees it's still 2.

00:06:51.490 --> 00:06:53.870
So the radius is 2
no matter what.

00:06:53.870 --> 00:06:55.310
And I should probably
use the circle tool.

00:06:55.310 --> 00:06:56.550
So as we go around
it doesn't matter.

00:06:56.550 --> 00:06:59.380
The radius is always
going to be 2.

00:06:59.380 --> 00:07:01.410
This distance is always going
to be 2, no matter what

00:07:01.410 --> 00:07:03.150
direction we're pointed in.

00:07:03.150 --> 00:07:06.220
And to some degree, that's
the definition of a circle.

00:07:06.220 --> 00:07:08.650
A circle, in polar coordinates,
is actually a lot

00:07:08.650 --> 00:07:10.800
easier to write down.

00:07:10.800 --> 00:07:12.380
Radius is equal to 2.

00:07:12.380 --> 00:07:13.780
That's a circle.

00:07:13.780 --> 00:07:14.880
Let's do the next one.

00:07:14.880 --> 00:07:18.170
Actually, I'm going to erase
this because I want to

00:07:18.170 --> 00:07:20.130
reuse this over here.

00:07:20.130 --> 00:07:24.880
So let me just black
that stuff out.

00:07:24.880 --> 00:07:25.245
OK.

00:07:27.810 --> 00:07:30.390
And then here I go.

00:07:30.390 --> 00:07:31.180
All right.

00:07:31.180 --> 00:07:37.890
The next one they give us is
x squared plus y squared is

00:07:37.890 --> 00:07:43.790
equal to 9 times y/x squared.

00:07:43.790 --> 00:07:44.200
All righty.

00:07:44.200 --> 00:07:46.660
If we do some pattern matching,
well x squared plus y squared

00:07:46.660 --> 00:07:48.200
we know that's equal
to r squared.

00:07:48.200 --> 00:07:53.700
So this is equal to r squared
is equal to 9 times--

00:07:53.700 --> 00:07:54.710
let's see, y over x.

00:07:54.710 --> 00:07:55.860
That looks suspicious.

00:07:55.860 --> 00:07:58.810
Tangent of theta
is equal to y/x.

00:07:58.810 --> 00:08:00.500
So this is tangent of theta.

00:08:00.500 --> 00:08:05.540
So 9 times tangent
of theta squared.

00:08:05.540 --> 00:08:08.540
And maybe we want to take the
square root of both sides.

00:08:08.540 --> 00:08:08.810
Right?

00:08:08.810 --> 00:08:11.030
So r is equal to-- remember, we
have to take the square root of

00:08:11.030 --> 00:08:12.395
both of these, because we're
taking the square root of

00:08:12.395 --> 00:08:13.270
the whole both sides.

00:08:13.270 --> 00:08:18.500
So it's r is equal to, we could
say plus or minus 2 tangent

00:08:18.500 --> 00:08:21.000
of theta, but it actually
doesn't matter once again.

00:08:21.000 --> 00:08:22.410
And that happens a lot
in polar coordinates.

00:08:25.230 --> 00:08:29.290
Actually, let me show you what
a negative radius means, just

00:08:29.290 --> 00:08:31.950
so that you get the intuition
why in that previous example it

00:08:31.950 --> 00:08:33.930
didn't matter whether I took
the radius as minus

00:08:33.930 --> 00:08:35.830
2 or positive 2.

00:08:35.830 --> 00:08:38.160
So for any given angle-- I just
want to give you an intuition

00:08:38.160 --> 00:08:42.680
of what a negative radius is--
if I give you a positive radius

00:08:42.680 --> 00:08:46.180
we're going to go in that
direction for this angle.

00:08:46.180 --> 00:08:48.260
But if I give you a negative
radius we go in the

00:08:48.260 --> 00:08:49.310
other direction.

00:08:49.310 --> 00:08:51.620
So this would be a positive
radius, and this would

00:08:51.620 --> 00:08:52.380
be a minus radius.

00:08:52.380 --> 00:08:53.390
You just go backwards.

00:08:53.390 --> 00:08:55.580
You walk backwards
in that direction.

00:08:55.580 --> 00:08:58.720
So in that previous example we
could say, OK the radius is

00:08:58.720 --> 00:09:01.170
equal to 2 no matter what.

00:09:01.170 --> 00:09:01.490
Right?

00:09:01.490 --> 00:09:02.910
So when you're angle is
in this direction your

00:09:02.910 --> 00:09:04.090
radius is equal to 2.

00:09:04.090 --> 00:09:06.710
Or you could say the radius is
always equal to negative 2.

00:09:06.710 --> 00:09:06.990
Right?

00:09:06.990 --> 00:09:08.610
When you're pointed in
this direction the radius

00:09:08.610 --> 00:09:10.800
is equal to minus 2.

00:09:10.800 --> 00:09:12.090
So it's always going
to be minus 2.

00:09:12.090 --> 00:09:14.270
But it ends up being the
same thing either way.

00:09:14.270 --> 00:09:16.600
And actually it turns out the
same thing for this equation,

00:09:16.600 --> 00:09:19.240
so you could just say r is
equal to 3 tangent of theta.

00:09:19.240 --> 00:09:20.910
If you're not sure, I guess
it doesn't hurt to put

00:09:20.910 --> 00:09:23.020
the plus or minus there.

00:09:23.020 --> 00:09:23.990
Next one.

00:09:23.990 --> 00:09:26.220
Let me erase this one again.

00:09:26.220 --> 00:09:27.570
Actually, I'm almost
out of time.

00:09:27.570 --> 00:09:30.210
Let me continue this
in the next video.


Polar coordinates 1

WEBVTT
Kind: captions
Language: en

00:00:00.820 --> 00:00:04.540
Everything we've done up to now
has dealt with Cartesian

00:00:04.540 --> 00:00:06.310
coordinates, even though you
might have not realized

00:00:06.310 --> 00:00:07.930
that they were Cartesian
coordinates, because I never

00:00:07.930 --> 00:00:10.290
called it that before just now.

00:00:10.290 --> 00:00:11.420
So what is a Cartesian
coordinate?

00:00:11.420 --> 00:00:13.620
Let me draw the axis
that you're hopefully

00:00:13.620 --> 00:00:14.490
familiar with by now.

00:00:14.490 --> 00:00:16.420
If you're not, review
those videos.

00:00:16.420 --> 00:00:20.550
Let me just draw the
y-axis and the x-axis.

00:00:20.550 --> 00:00:22.720
And so, Cartesian coordinates--
and actually, they can apply to

00:00:22.720 --> 00:00:23.830
more than just 2 dimensions.

00:00:23.830 --> 00:00:27.190
But 2 dimensions is what
we tend to deal with.

00:00:27.190 --> 00:00:30.880
Then if I want to specify any
point in 2 dimensional space, I

00:00:30.880 --> 00:00:34.170
just tell you how far in the x
direction I have to go, and

00:00:34.170 --> 00:00:35.400
how far in the y direction.

00:00:35.400 --> 00:00:37.160
And I can specify that point.

00:00:37.160 --> 00:00:40.150
Like, let's see, if I said,
well, let me give the Cartesian

00:00:40.150 --> 00:00:42.590
coordinates for that
point right there.

00:00:42.590 --> 00:00:46.550
And I might say, well, to get
to that point right there, I

00:00:46.550 --> 00:00:50.570
have to go to the
right 3 steps.

00:00:50.570 --> 00:00:52.610
So this is 3, right here.

00:00:52.610 --> 00:00:54.650
And then I have to
go up 4 steps.

00:00:54.650 --> 00:00:57.510
1, 2, 3, 4.

00:00:57.510 --> 00:00:58.640
And you end up there.

00:00:58.640 --> 00:01:01.110
And by convention, we call this
the x-coordinate, and we call

00:01:01.110 --> 00:01:05.630
this the y-coordinate, and
we call this 3 comma 4.

00:01:05.630 --> 00:01:07.530
And this is just a convention
where we say the first

00:01:07.530 --> 00:01:11.730
coordinate is how far in the x
direction we go, and the second

00:01:11.730 --> 00:01:14.250
coordinate is how far in the up
and down, or the y

00:01:14.250 --> 00:01:14.940
direction, we go.

00:01:14.940 --> 00:01:16.270
So this is 3 comma 4.

00:01:16.270 --> 00:01:19.020
And we could have gone 3 to the
right and 4 up, or we could

00:01:19.020 --> 00:01:20.880
have gone 4 up and
3 to the right.

00:01:20.880 --> 00:01:23.740
And we'd have gotten to
the exact same point.

00:01:23.740 --> 00:01:26.440
This is just one way to
really specify a point

00:01:26.440 --> 00:01:27.440
in 2 dimensions.

00:01:27.440 --> 00:01:29.370
Another option, and you might
have done this in everyday

00:01:29.370 --> 00:01:32.490
life, is to say, hey, let me
just point you in a direction,

00:01:32.490 --> 00:01:34.680
and then you go a certain
distance in that direction.

00:01:34.680 --> 00:01:36.470
So I could have said,
you know what?

00:01:36.470 --> 00:01:39.530
Let me just point you
in this direction.

00:01:39.530 --> 00:01:40.940
Let me just point you in
that direction, and you

00:01:40.940 --> 00:01:42.970
need to go this far.

00:01:42.970 --> 00:01:46.120
You need to go this far
in that direction.

00:01:46.120 --> 00:01:48.000
That would also get you
to the same point.

00:01:48.000 --> 00:01:50.920
So how do I specify
the direction?

00:01:50.920 --> 00:01:52.840
Well, how about I
call this 0 degrees?

00:01:52.840 --> 00:01:54.420
How about I call
this the x-axis?

00:01:54.420 --> 00:01:55.790
How about I call
that 0 degrees?

00:01:55.790 --> 00:01:57.070
And we'll deal with degrees.

00:01:57.070 --> 00:01:59.400
But hopefully, you know
how to convert between

00:01:59.400 --> 00:02:00.260
degrees and radians.

00:02:00.260 --> 00:02:02.810
So you could just as easily
convert to radians.

00:02:02.810 --> 00:02:06.270
But you could specify the
direction if you call this 0,

00:02:06.270 --> 00:02:09.720
saying, ok, this is theta
degrees, and I want you-- I'm

00:02:09.720 --> 00:02:11.490
going to point you in a degree
in-- you know, you could

00:02:11.490 --> 00:02:13.570
always imagine someone
who's blindfolded.

00:02:13.570 --> 00:02:15.200
You point them in this
direction and you

00:02:15.200 --> 00:02:17.700
say, walk r units.

00:02:17.700 --> 00:02:20.280
And then they'll end up
at that same point.

00:02:20.280 --> 00:02:22.740
And so you could also specify
this point-- instead of

00:02:22.740 --> 00:02:27.160
specifying, this is the
Cartesian coordinates, this is

00:02:27.160 --> 00:02:30.570
x comma y-- you could also
specify it-- maybe, if we can

00:02:30.570 --> 00:02:31.710
figure out a way to do it.

00:02:31.710 --> 00:02:34.250
And I'll do it in magenta.

00:02:34.250 --> 00:02:37.580
You can also specify
it as r comma theta.

00:02:37.580 --> 00:02:40.990
Which essentially says,
you know, walk r units

00:02:40.990 --> 00:02:42.650
in the theta direction.

00:02:42.650 --> 00:02:44.660
Well, let's figure
out what that is.

00:02:44.660 --> 00:02:46.360
Because I think it's kind of
abstract now, and might be a

00:02:46.360 --> 00:02:48.240
little difficult to understand.

00:02:48.240 --> 00:02:50.050
So we're going to break out
a little trigonometry and

00:02:50.050 --> 00:02:52.490
actually just a little
Pythagorean theorem.

00:02:52.490 --> 00:02:55.090
Can we figure out r and theta?

00:02:55.090 --> 00:02:58.010
Well, r, hopefully, is
the easier one here.

00:02:58.010 --> 00:03:00.950
Because if you think about it,
this is a right triangle.

00:03:00.950 --> 00:03:02.030
This is a right triangle.

00:03:02.030 --> 00:03:04.380
That has distance 3.

00:03:04.380 --> 00:03:07.410
This is distance 4.

00:03:07.410 --> 00:03:08.840
This is a right triangle.

00:03:08.840 --> 00:03:10.090
So what is r here?

00:03:10.090 --> 00:03:11.630
Pythagorean Theorem.

00:03:11.630 --> 00:03:14.260
3 squared plus 4 squared is
equal to the hypotenuse

00:03:14.260 --> 00:03:15.580
squared, or r squared.

00:03:15.580 --> 00:03:20.890
So we could say, 3 squared
plus 4 squared is

00:03:20.890 --> 00:03:23.420
equal to r squared.

00:03:23.420 --> 00:03:28.860
That's 9 plus 16 is
equal to r squared.

00:03:28.860 --> 00:03:31.880
25 is equal to r squared.

00:03:31.880 --> 00:03:34.990
We don't want to deal with
negative distances, so 5 is

00:03:34.990 --> 00:03:36.900
equal to r, r is equal to 5.

00:03:36.900 --> 00:03:37.380
Fair enough.

00:03:37.380 --> 00:03:39.480
So we already figured out
that r is equal to 5.

00:03:39.480 --> 00:03:41.900
How do we figure out theta?

00:03:41.900 --> 00:03:43.040
So what do we know here?

00:03:43.040 --> 00:03:44.950
We know-- well, we're trying
to figure out theta.

00:03:44.950 --> 00:03:47.480
And we know it's
opposite side, right?

00:03:47.480 --> 00:03:49.370
This is just going
back to SOHCAHTOA.

00:03:49.370 --> 00:03:52.140
Let me write down SOHCAHTOA.

00:03:52.140 --> 00:03:54.660
If this is completely
unfamiliar to you, you might

00:03:54.660 --> 00:03:59.170
want to watch the basic
trigonometry video.

00:03:59.170 --> 00:04:02.610
So we know the opposite
side of theta, right?

00:04:02.610 --> 00:04:03.570
That's 4.

00:04:03.570 --> 00:04:04.950
And we also know
the adjacent side.

00:04:04.950 --> 00:04:06.060
That's 3.

00:04:06.060 --> 00:04:09.090
So which of the trig
functions uses the

00:04:09.090 --> 00:04:11.470
opposite and the adjacent?

00:04:11.470 --> 00:04:12.090
Well, the TOA.

00:04:12.090 --> 00:04:13.620
Opposite and adjacent.

00:04:13.620 --> 00:04:14.750
So tangent.

00:04:14.750 --> 00:04:21.100
So tangent of theta is equal to
the opposite-- which is really

00:04:21.100 --> 00:04:26.680
the y-coordinate, which is
equal to 4-- over the adjacent.

00:04:26.680 --> 00:04:29.180
Which is the x-coordinate,
which is 3.

00:04:29.180 --> 00:04:31.720
So tangent of theta
is equal to 4/3.

00:04:31.720 --> 00:04:33.810
And to solve this, you
essentially just take

00:04:33.810 --> 00:04:36.600
the inverse tangent of
both sides of this.

00:04:36.600 --> 00:04:38.530
So this is the same thing.

00:04:38.530 --> 00:04:41.050
And depending on your
calculator, or the convention

00:04:41.050 --> 00:04:42.060
you use, you might write this.

00:04:42.060 --> 00:04:44.560
When you take the inverse
tangent of both sides,

00:04:44.560 --> 00:04:46.020
you get-- well, I'll
just write it out.

00:04:46.020 --> 00:04:53.960
You could write it as the
arctan of the tangent of theta

00:04:53.960 --> 00:04:59.020
is equal to the arctan of 4/3.

00:04:59.020 --> 00:05:01.150
And this, of course-- the
arctan of the tangent.

00:05:01.150 --> 00:05:02.740
This is the same thing
as the inverse tan of

00:05:02.740 --> 00:05:03.620
the tangent of theta.

00:05:03.620 --> 00:05:05.590
This is just equal to theta.

00:05:05.590 --> 00:05:07.220
That just simplifies to theta.

00:05:07.220 --> 00:05:12.300
Is equal to arctan of 4/3.

00:05:12.300 --> 00:05:15.180
And then another way to write
arctan is often-- they'll often

00:05:15.180 --> 00:05:16.310
write it-- I'll write it.

00:05:16.310 --> 00:05:21.365
So the same exact statement
could be written as tan inverse

00:05:21.365 --> 00:05:22.780
to the negative 1 power.

00:05:22.780 --> 00:05:23.460
They sometimes say it.

00:05:23.460 --> 00:05:28.020
Although it's a little
misleading sometimes, when

00:05:28.020 --> 00:05:28.750
they write it like this.

00:05:28.750 --> 00:05:29.300
Because you don't know.

00:05:29.300 --> 00:05:30.800
Oh, am I taking the whole
tangent of theta to the

00:05:30.800 --> 00:05:31.900
negative power or something?

00:05:31.900 --> 00:05:35.480
But sometimes I'll write arctan
as 10 to the negative 1 power.

00:05:35.480 --> 00:05:36.280
But either way.

00:05:36.280 --> 00:05:39.610
So we can figure out theta by
taking the arctangent of 4/3.

00:05:39.610 --> 00:05:43.160
And most people do not have the
arctangent of 4/3 memorized.

00:05:43.160 --> 00:05:45.390
And I don't know an easy way
to figure it out, without

00:05:45.390 --> 00:05:47.050
getting my TI-85 out.

00:05:47.050 --> 00:05:49.600
So let's get it out.

00:05:49.600 --> 00:05:50.410
All right.

00:05:50.410 --> 00:05:53.190
So how do we know the arctan--
so we want to do this inverse--

00:05:53.190 --> 00:05:55.270
let me get the calculator up a
little higher, so you can see.

00:05:55.270 --> 00:05:57.550
We want to get this
inverse tan right here.

00:05:57.550 --> 00:06:04.440
So you press second inverse
tangent of 4 divided by 3.

00:06:04.440 --> 00:06:07.500
And I already set my
calculator to degree mode.

00:06:07.500 --> 00:06:10.055
It gives you 53.13 degrees.

00:06:13.320 --> 00:06:18.360
So theta is equal
to 53.13 degrees.

00:06:18.360 --> 00:06:19.010
So there you have it.

00:06:19.010 --> 00:06:21.610
We already knew that we could
specify this point in the 2

00:06:21.610 --> 00:06:24.020
dimensional plane by the point
x is equal to 3,

00:06:24.020 --> 00:06:25.550
y is equal to 4.

00:06:25.550 --> 00:06:30.320
We can also specify it by r
is equal to 5, and theta

00:06:30.320 --> 00:06:31.790
is equal to 53 degrees.

00:06:31.790 --> 00:06:33.590
So I'll write that.

00:06:33.590 --> 00:06:36.430
And polar coordinates, it can
be specified as r is equal to

00:06:36.430 --> 00:06:42.640
5, and theta is 53.13 degrees.

00:06:42.640 --> 00:06:50.930
So all that says is, OK, orient
yourself 53.13 degrees

00:06:50.930 --> 00:06:54.930
counterclockwise from the
x-axis, and then walk 5 units.

00:06:54.930 --> 00:06:57.000
And you'll get to the
exact same point.

00:06:57.000 --> 00:06:59.040
And that's all polar
coordinates are telling you.

00:06:59.040 --> 00:06:59.980
Let's do another one.

00:06:59.980 --> 00:07:01.640
Actually, before that, let's
see if we can come up with

00:07:01.640 --> 00:07:04.020
something general here.

00:07:04.020 --> 00:07:06.870
Because once you have the
general, then you can always

00:07:06.870 --> 00:07:08.850
solve for the particular.

00:07:08.850 --> 00:07:10.510
So let me just write
an arbitrary point.

00:07:10.510 --> 00:07:11.980
Last time I gave a
specific point.

00:07:11.980 --> 00:07:14.250
The point-- was it 3 comma 4?

00:07:14.250 --> 00:07:15.760
I think it was.

00:07:15.760 --> 00:07:20.090
So let's say I have
the point x comma y.

00:07:20.090 --> 00:07:22.120
It's right there.

00:07:22.120 --> 00:07:27.500
So its x-coordinate is there,
its y-coordinate is there.

00:07:27.500 --> 00:07:32.580
So how do we convert this to
r, to polar coordinates?

00:07:32.580 --> 00:07:33.720
r, theta.

00:07:33.720 --> 00:07:36.140
Well, let's do the same thing
we did in the last video.

00:07:36.140 --> 00:07:41.770
If this length is r, and
this angle is theta.

00:07:41.770 --> 00:07:43.640
So just going into what we
did in the last video.

00:07:43.640 --> 00:07:44.750
We use Pythagorean Theorem.

00:07:44.750 --> 00:07:47.340
We say that x squared plus--
because this distance

00:07:47.340 --> 00:07:48.580
here is also y, right?

00:07:48.580 --> 00:07:50.150
This distance here is x.

00:07:50.150 --> 00:07:54.446
Pythagorean Theorem, you get
x squared plus y squared

00:07:54.446 --> 00:07:56.130
is equal to r squared.

00:07:56.130 --> 00:07:58.090
That's just the
Pythagorean Theorem.

00:07:58.090 --> 00:08:01.460
And then you have the
tangent of theta.

00:08:01.460 --> 00:08:03.100
The tangent of this angle.

00:08:05.690 --> 00:08:06.250
SOHCAHTOA.

00:08:06.250 --> 00:08:07.220
TOA.

00:08:07.220 --> 00:08:08.670
Opposite over adjacent.

00:08:08.670 --> 00:08:12.560
Tangent of theta is equal to
the opposite-- which is y--

00:08:12.560 --> 00:08:15.320
over the adjacent-- which is x.

00:08:15.320 --> 00:08:16.340
And this is really
all you need.

00:08:16.340 --> 00:08:18.530
And if you wanted to go one
step further, and this is

00:08:18.530 --> 00:08:20.640
actually something you
should write at the

00:08:20.640 --> 00:08:22.490
side of your paper.

00:08:22.490 --> 00:08:24.800
How can we express y as
a function if we wanted

00:08:24.800 --> 00:08:25.420
to go the other way?

00:08:25.420 --> 00:08:28.490
If we were given r and
theta, how could we get y?

00:08:28.490 --> 00:08:29.790
Well, let's think about it.

00:08:29.790 --> 00:08:34.800
y-- if we want to know-- let's
see, if we are given r and y

00:08:34.800 --> 00:08:38.160
and we want to figure out
theta-- what deals with the r

00:08:38.160 --> 00:08:41.300
and the y, with respect to
theta? r is the hypotenuse

00:08:41.300 --> 00:08:42.610
and y is the opposite.

00:08:42.610 --> 00:08:43.840
Let me write SOHCAHTOA down.

00:08:49.830 --> 00:08:51.710
So what deals with the
opposite and the hypotenuse?

00:08:55.020 --> 00:08:55.950
Sine.

00:08:55.950 --> 00:09:03.410
So the sine of theta is equal
to the opposite-- which is the

00:09:03.410 --> 00:09:07.630
y side, which is equal to y--
over the hypotenuse,

00:09:07.630 --> 00:09:09.750
which is r.

00:09:09.750 --> 00:09:10.990
Equal to r.

00:09:10.990 --> 00:09:14.570
And so if you multiply both
sides by r, you get r

00:09:14.570 --> 00:09:19.410
sine theta is equal to y.

00:09:19.410 --> 00:09:20.680
And then let's do
it the other way.

00:09:20.680 --> 00:09:24.840
What if we wanted to have an
equation for theta, r, and x?

00:09:24.840 --> 00:09:26.860
x is adjacent to the angle.

00:09:26.860 --> 00:09:28.335
r is still the hypotenuse.

00:09:28.335 --> 00:09:30.680
So what deals with
adjacent and hypotenuse?

00:09:30.680 --> 00:09:31.820
Well, that's CAH.

00:09:31.820 --> 00:09:33.070
Cosine.

00:09:33.070 --> 00:09:38.530
So we have the cosine of theta
is equal to the adjacent,

00:09:38.530 --> 00:09:43.540
is equal to x, over the
hypotenuse, which is r.

00:09:43.540 --> 00:09:47.710
Multiply both sides by r
and you get x is equal

00:09:47.710 --> 00:09:50.980
to r cosine theta.

00:09:50.980 --> 00:09:53.560
And really, when you're
equipped with this formula--

00:09:53.560 --> 00:09:56.060
which really just comes from
our trig identities-- this

00:09:56.060 --> 00:09:57.770
formula, which you know, is
similar, just comes

00:09:57.770 --> 00:09:59.220
from SOHCAHTOA.

00:09:59.220 --> 00:10:01.090
This, which comes from
SOHCAHTOA-- and really

00:10:01.090 --> 00:10:04.180
is, you know-- if you're
given these first.

00:10:04.180 --> 00:10:07.350
If you're given these two, you
can easily derive that one.

00:10:07.350 --> 00:10:09.920
And this one, which is just
the Pythagorean Theorem,

00:10:09.920 --> 00:10:11.010
you're actually ready.

00:10:11.010 --> 00:10:14.010
You're fully equipped to
convert between polar and

00:10:14.010 --> 00:10:15.030
rectangular coordinates.

00:10:15.030 --> 00:10:16.480
And we'll do that in the
next video, because I just

00:10:16.480 --> 00:10:18.530
realized I'm out of time.