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.