www.mathcentre.ac.uk/.../Hyperbolic%20functions.mp4

In this video, we'll define
the hyperbolic functions

hyperbolic sign, hyperbolic
cosine and Hyperbolic Tangent.

We'll define hyperbolic cosine

first. We write hyperbolic
cosine like this.

We pronounce
it Cosh

X. And it's defined
using the exponential functions

like this. Cos X equals.

Each the X Plus E to the

minus X. All divided by
two.

Hyperbolic sign is
written like this.

There are two
ways of pronouncing

hyperbolic sign. In this video
I'll call it shine, but you will

come across some people who call

it's inch. And Shine X is
defined with the exponential

function, again like this.

We have each the X minus this
time, each the minus X all

divided by two.

Now we can use our knowledge of
the graphs of E to the X

to workout the graphs of Shine X
and Koch X.

Here I've got some graphs of E
to the X over 2 and E to the

minus X over 2.

First, we'll workout what
cautious 0 is.

A quickly remind you that

quarterbacks. Is each the X plus
E to the minus X over 2?

So Koch of

0. Is eat 0,
which is one.

Plus E to the zero again, which
is one all divided by two.

Which is 2 over 2, which is one.

So we can Mark Koch
0 onto a graph here.

Now we can workout. What happens
is X gets big.

We can split up how we write
quash X into this form. We have

each the X over 2 plus E
to the minus X over 2.

Now we already have the graphs
of E to the X over 2 and E to

the minus X over 2 here, and we
can see that as X gets big.

East the X over 2 gets very very

big. But each of the minus X
over 2 gets very very small.

So as X gets bigger, the second
part of the sum.

Gets very very close to 0.

So the whole thing.

Gets very very close to
just eat the X.

Over 2. So
now we can rule

that side of the
graph.

I'll just turn around the graph
so I can actually draw it.

Now remember that be'cause each
the minus X over 2.

Is always bigger than zero even
though it gets very very small.

Call shacks will always stay

above. The graph of each of
the axe over 2.

The grass get closer
together as X gets big, but

Cosh X must always stay
above E to the X over 2.

Now we'll see what happens when
ex gets large and negative.

Now when
ex gets

larger negative.
We can see that each the X

over 2 gets very very small,
but E to the minus X over 2

gets very very big.

So writing Cosh X again.

As each the X over 2 plus
E to the minus X over 2.

We can see that the first part
of the sum, this time gets very

very small. And the second part
of the sum gets very very big.

So the whole thing.

As X gets larger, negative will
get closer and closer to eat.

The minus X over 2.

So we can draw in that side
of the graph now.

And remember again.

That E to the X over 2 even
though it's getting very, very

small, will always be bigger

than 0. So the graph will always

stay above. East the minus X

over 2. As X gets more negative,
the graphs get closer together.

But the graph will always stay
above each the minus X over 2.

So here is a graph.

Of Cos X.

Now you can see that this graph
is symmetric about the Y axis.

And that tells us that cause

checks. Is always equal
to Koch of minus

X?

Now let's deal with the
graph was trying X.

This time to help me. I've got a
graph of each of the X over 2.

And minus E to the minus
X over 2.

Let's see what happens
when X equals 0.

I'll just remind
you that Shine X.

Is each the X minus E to
the minus X over 2?

So shine. Of

0. Is each zero?

Which is one.

Minus Y to 0, which is one.

All over 2 so that zero over
2 which is 0.

Right, so we can plot that

point. Onto a
graph at 00.

Now we'll see what happens
as X gets large.

Like before, we can split up the
way we write Shine X.

So we write Shine X.

As each the X over
2 minus E to the

minus X over 2.

Now, using the same logic as we

did before. And remembering what
the graphs of E to the X&E to

the minus X over 2 looked like.

We can see that.

The first part of the some here.

Gets very, very big as X
gets large.

But the second part of the sum.

Gets very, very small as X gets

large. So again.

Floor Jacks.
The whole of Shine

X becomes closer and

closer. To eat the
X over 2.

So now we
can draw in

that half of
the graph.

No,
just

this

time.
The graph will stay below each

the X over 2.

And that's because we're taking

away. E to the minus X over 2.

Now let's see what happens when
ex gets larger negative.

Again, just so you can
see what's happening, I'll write

up are split up form

shine X. I
remember as

ex got

larger negative. Each of

the X. Got very very small.

For the first part of the son
gets very very small.

Eat the minus X over 2.

Got very very big.

So this whole thing as X gets
larger, negative gets closer and

closer to. Just the second bit
of the sum, because the first

bit is getting closer to 0.

So for large negative.

X.

Sean X. Gets
very, very close to.

Minus each the minus X.

Over 2.

So we can draw the second bit of

a graph and now. Now again,
the graph must

stay above minus
E to the

minus X over

2. Be'cause each the X over
2 is never 0 even though it gets

very very small.

So that adds a bit on to minus E
to the minus X over 2. It keeps

the graph above it.

Now you know tist that shine X
for LG X is close to eat the X

over 2. And Koch X also remember
was close to E to the X over 2

for LG X. So that must mean.

Floor Jacks.
Khashab

X. And Shine
X are very close together.

And we can see that
if I show you a

graph of Cynex and koshek
spotted together.

Got a slightly different scale
this time so it looks a bit

wider. But there is caution X.

And they're showing X, which
gets very, very close to quash X

as X gets large.

Now we'll define
hyperbolic tangent.

We can define hyperbolic tangent
in terms of shine and Koch.

We write Hyperbolic Tangent.

Like this? And again, there are
two possible pronunciations for

this. Some people call it touch.

But I'm in this video,
we'll call it fan.

And we define it to be.

Sean X.
Over Cosh X.

And again, there's an
exponential function definition

which we can work out by looking
at this definition.

Because this is Sean X which is
E to the X minus E to the

minus X over 2.

Divided by Cosh X, which is E to
the X Plus E to the minus X over

2. And that's when you work it
out is if the X minus E to the

minus X all divided by.

Each the X Plus E
to the minus X.

Now this thing doesn't look very
useful for working out the

graph. It's a bit complicated,
so this time will use the graph

of shine and Koch to workout the
graphs of ban.

Now here I've got
a set of axes

which I'll use to
draw down onto.

I've got an asymptotes here at Y
equals 1 and one at White was

minus one and you'll see why in

a minute. I remember
that 4X large.

We saw that

Cosh X. Was
very, very close.

To Shine X.

So that must mean

that. Shine X.

Over Cosh
X.

Must be very close to one.

So as X

gets large. We should expect
Linix to get very close to one.

Now for large negative X.

We
saw

that.

Call shacks Was very close
to eat a minus X.

Over 2.

And Shine X.

Was very close to minus E to the
minus X over 2.

So you can see here that Shine

X. Is very very close to minus

Cosh X. So that must
mean that Shine X.

Over Cosh X.

Is very very close
to minus one.

So as X gets larger, negative.

We should expect the graph to
get very very close to minus

one. Now just to finish off,
will see what happens when X

equals 0. So
for X

equals 0.

Thanks.
Which is shine.

Over Cos X.

Will be well shine of X.

Was zero, so that would be 0.

Call Shabaks was one.

So fun of X when X equals
0 is just zero.

Now, putting these three bits of

information together. We can
plot 00 on that.

And his ex got large.

Thanks, got close to one so we
can see this thing getting

closer to one.

And his ex got large and

negative. Thanks, got close to
minus one so we can draw this.

Going down toward minus one.

But remember that.

These never actually reach
one or minus one. They just

get very very close to them
as X gets large reserves gets

more negative.

Also notice that.

Fun. Of

ex. Is equal to

minus fan? Of minus X.

Now hyperbolic functions
has my densities

which are very
similar to trig

identities, but not
quite the same.

In this video I'll show you and
prove for you, too, and then

I'll show you some more at the
end, which you control yourself.

The first one will prove is this

one. It's caution.

X squared

minus shine. X

squared Equals 1.

Not prove this will substitute
in the exponential definitions

for Koch and shine.

So we have the costs. X was each
the X Plus E to the minus X over

2, and we want that all squared.

We take away shine X squared,
which was E to the X minus E to

the minus X over 2 all squared.

Now notice that in both bits of

this. We have the same

denominator squared. So we can
take that out of the factor.

So we have that.

As a quarter.

Of the numerator squared.

So we have each of the X Plus E
to the minus X squared.

Minus eat the AX minus E
to the minus X squared.

Now we want to expand out
these squared brackets.

So that is going to be.

Well, these brackets.

Are going to be each the X times
E to the X which is E to the 2X.

And because each the X Times E
to the minus X is one.

And we get that happening twice.

We get a plus two here.

And each the minus X Times E to
the minus X.

Is each the minus 2X?

And I will take away this
bracket squared which working

out before we can see is.

Each the two weeks again.

And this time we get a minus two

here. And we get.

Plus, if the minus 2X.

Now we can remove these

inner brackets. And we get a

quarter. Of.

Just take out the first brackets
first, E to the two X +2, plus E

to the minus 2X.

Put a minus here, so that's

minus. Each the 2X.

+2. Minus.
E to the minus 2X.

And of course, a lot of things
that are going to cancel here.

We have E to the two X minus
either 2X, so these go.

We have plus E to the minus
two X here and minus it again

there, so they go.

And then we just have 2 + 2, so
the whole thing turns out to be

1/4 of 2 + 2, which is 4.

Which is one.

So there we have the proof
that cause squared X minus

sign squared X, which is
all that is equal to 1.

And one more density or
proof you now is this

one. It's shine 2X.

Equal to two.

Sean X. Call
Josh X.

Now with this one, I think it's
easier to start with the right

hand side, so will do that
again. Will a substitute in the

exponential definitions to the
right hand side.

So 2.
Sean

X. Cosh

X. Is equal to.

Two lots of.

Each 3X minus each the minus X
over 2 times caution X which was

E to the X Plus E to
the minus X over 2.

Now you can see the two councils
with one of these denominators,

so will cancel those.

And take out this half as a

factor. So now
we have 1/2.

Each of the X minus E to the
minus X Times E to the X Plus

E to the minus X.

And if we just multiply out the

brackets. We get a half.

Get each the X Times E to the X,
which is each the two acts.

Each the X Times Plus E to the

minus X. Which is plus one.

Minus E to the minus X times Y
to the X, which is minus one.

And finally, minus each the

minus 2X. So obviously
these ones cancel.

And we get this bracket here,
which was E to the two

X minus. Eat the minus 2X.

All divided by two.

And that is the definition of

shine 2X. So that's
the second identity proved.

Now here is more identity's
which you can prove yourself.

The first one
is Koch 2X.

Is equal to.

Call shacks

squared.
Plus

Shine X squared.

We also have.

Shine of X

Plus Y.

Equals.

Shine X.
Gosh why?

Plus Shine

Y. Koch
X.

And also for Koch.

X plus Y.

We have cautious.

X. Call
Shuai Plus Shine

X. Shine

Y. And

two more.

Gosh. X over
2.

All squared. Is 1

plus. Cosh X all
divided by two.

And finish off shine.

X over 2.

All squared. Is
Cosh X minus one

all divided by two?

Now you can see how similar
these are to trig identities.

And in fact, the hyperbolic
functions and triggered entities

are very closely related.

And you can learn more about
that if you go and study complex

numbers. And before we
finish, we look at some

other functions which are
related to hyperbolic functions.

Firstly, some notation.

You've already seen me use

notation like. Sean X
squared, which means.

The whole of Shine X squared.

Just remember that if you see

something like. Sean X with a
minus one there that is not

equal to 1.

Over Shine X.

This refers to the inverse
function of shine.

If you remember that if we
have a function F.

F minus one the inverse function
works like this.

You have F minus one.
The inverse function of X

gives you this thing why
where F of Y is

equal to X.

And the same way.

Shine with the minus one there.

Of X gives You Y.

Will shine of why was
equal to X?

We can also define inverse
functions for caution than.

But for these we have to
restrict the domains of

the function.

For inverse Cosh, Cosh the minus
one of X, we need to have.

X greater than or equal to 1.

And for the inverse of
fan of X.

We need X to be greater than
minus one and less than one.

Some other functions you
might come across other

reciprocal functions.

These are such.

X. Which is defined
to be one over Cosh of X.

There are two more of these.

This Kosach
Of X, which

is one over
shine of X.

And finally, there's
cough X.

Which is one over fan.