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JOEL LEWIS: Hi.

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Welcome back to recitation.

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Today I wanted to talk about
something that's mentioned in

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the notes but wasn't
covered in lecture

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because of the exam review.

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So this is, the subject is
hyperbolic trig functions.

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So first I just wanted to define
them for you and graph

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them so we can get a little
bit of a feeling for what

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these functions are like, and
then I'm going to explain to

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you why they have the
words hyperbolic and

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trig in their names.

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So these are some interesting
functions.

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They're not, they don't, aren't
quite as important as

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your usual, sort of circular
trig functions.

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But yeah, so let me introduce
them and let me jump in just

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with their definition.

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So there are two most
important ones.

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Just like a regular
trigonometric functions

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there's the sine and the cosine
and then you can write

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the other four trigonometric
functions in terms of them.

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So for hyperbolic trig
functions we have the

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hyperbolic scoine and
the hyperbolic sine.

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So the notation here,
we write c o s h.

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So the h for hyperbolic.

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So hyperbolic cosine.

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And usually we pronounce this
"cosh." And similarly, for the

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hyperbolic sine we write s i n
h, for hyperbolic sine, except

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in the reverse order.

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And we usually pronounce this
"sinch," so in American

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English as if there were
an extra c in there.

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Sinch.

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OK.

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So these functions have fairly
simple definitions in terms of

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the exponential function--

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e to the x.

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So cosh of x is defined to be e
to the x plus e to the minus

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x divided by 2.

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And sinh of x is defined to be
e to x minus e to the minus x

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divided by 2.

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So if you remember what your
graph of e to the x looks

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like, and your graph of e to the
minus x, it's not hard to

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see that the graphs of cosh x
and sinh x, should look sort

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of like this.

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So for cosh x, so we see as x
gets big, so e to the minus x

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is going to 0.

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It's not very important.

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So it mostly is driven by
this e to the x part.

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And as x gets negative and big,
then this is going to 0

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and this is getting
larger and larger.

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So we got something that
looks like this.

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So it looks a little bit, in
this picture it looks a little

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bit like a parabola, but the
growth here is exponential at

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both sides.

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So in fact, this is growing
much, much, much faster than,

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say 1 plus x squared.

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So it's a much steeper curve.

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OK.

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And it reaches its minimum here
at x equals 0-- it has

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the value 1 plus 1 over 2.

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So its minimum there is
x equals 0 as its

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minimum value 1.

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For sinh, OK, so we're taking
the difference of them.

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So it's similar when x is
positive and large, e to the x

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is big, and e to the minus
x is pretty small, almost

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negligible.

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So we got exponential growth
off that side.

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When x becomes negative and
large, e to the x is going to

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0, e to the minus x is becoming
large, but it's,

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we've got a minus sign here.

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So as x goes to minus infinity,
this curve goes also

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to minus infinity.

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And again, the growth here is
exponential in both cases.

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And if you were curious, say
about what the slope there at

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the origin is, you could quickly
take a derivative and

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check that that's passing
through the origin

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with slope 1 there.

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OK.

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So this is a sort of basic
picture of what

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these curves look like.

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They have some nice properties,
and let me talk

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about them.

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So for example, one nice thing
you might notice about these

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functions is that it's easy to
compute their derivatives.

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Right?

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So if we look at d dx of cosh
x in order to compute that,

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well, just look at the
definition of cosh.

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So it's really just a sum of
two exponential functions.

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Exponential functions are easy
to take the derivatives.

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Take the derivative of e to
the x, you get e to the x.

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Take the derivative of e to the
minus x, well, OK, so it's

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a little chain rule so you
get a minus 1 in front.

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So the derivative of cosh x is
e to the x minus e to the

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minus x over 2.

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But we have a name for this.

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This is actually just sinh x.

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So the derivative of cosh is
sinh, and the derivative of

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sinh, well, OK.

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You look at the same thing, take
this formula, take its

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derivative, well, e to the x.

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Take its derivative,
you get e to the x.

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e to the minus x.

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Take its derivative, you get
minus e to the minus x, so

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those two minus signs cancel
out and become a plus.

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So this is e to the x plus
e to the minus x over 2,

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which is cosh x.

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So here you have some behavior
that's a little bit

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reminiscent of the behavior
of trig functions.

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Right?

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For trig functions, if you take
the derivative of sine

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you get cosine.

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And if you take the derivative
of cosine you almost get back

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sine, but you get minus sine.

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So here you don't have
that extra negative

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sign floating around.

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Right?

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So you, when you take the
derivative of cosh you get

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sinh on the nose.

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No minus sign needed.

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So that's interesting.

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But the real reason that these
have the words trig in their

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name is actually a little
bit deeper.

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So let me come over here and
draw a couple pictures.

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So the normal trig functions--

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what sometimes we call the
circular trig functions if we

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want to distinguish them from
the hyperbolic trig

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functions--

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they're closely, so circular
trig functions, they're

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closely related to
the unit circle.

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So the unit circle has equation
x squared plus y

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squared equals 1.

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It's a circle.

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Well, close enough, right?

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And what is the nice
relationship between this

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circle and the trig functions?

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Well, if you choose any point
on this circle, then there

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exists some value of t such
that this point has

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coordinates cosine
t comma sine t.

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Now it happens that the value
of t is actually the angle

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that that radius makes with
the positive axis.

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But not going to worry
about that right now.

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It's not the key
idea of import.

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So as t varies through the real
numbers, the point cosine

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t, sine t, that varies and it
just goes around this curve.

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So it traces out this
circle exactly.

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So the hyperbolic trig functions
show up in a very

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similar situation.

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But instead of looking at the
unit circle, what we want to

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look at is the unit rectangular
hyperbola.

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So what do I mean by that?

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Well, so instead of taking the
equation x squared plus y

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squared equals 1, which gives a
circle, I'm going to look at

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a very similar equation that
gives a hyperbola.

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So this is the equation,
x squared minus y

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squared equals 1.

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So if you if you graph this
equation, what you'll see is

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that, well, it passes through
the point 1, 0.

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And then we've got one branch
here, we've got a little out

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asymptote there.

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So it's got a right branch
like that, and also it's

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symmetric across the y-axis.

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So there's a symmetric
left branch here.

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So this is the graph of the
equation, x squared minus y

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squared equals 1.

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So it's this hyperbola.

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Now what I claim is that cosh
and sinh have the same

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relationship to this hyperbola
as cosine and

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sine have to the circle.

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Well, so I'm fudging
a little bit.

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So it turns out it's only the
right half of the hyperbola.

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So what do I mean by that?

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Well, here's what
I'd like to do.

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Set x equals--

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so we're going to introduce
a new variable, u--

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I'm going to set x equal cosh
u and y equals sinh u.

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And I'm going to look
at the quantity x

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squared minus y squared.

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So x squared minus y squared.

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So this is, so we use most
of the same notations for

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hyperbolic trig functions
that we do for

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regular trig functions.

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So this is cosh squared u
minus sinh squared u.

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And now we can plug in the
formulas for cosh and sinh

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that we have. So this is equal
to e to the u plus e to the

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minus u over 2 quantity squared
minus e to the u minus

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e to the minus u over
2 quantity squared--

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and now we can expand out both
of these factors and, both of

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these squares, rather, and
put them together.

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So over 2 squared is over 4 and
we square this and we get

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e to the 2u.

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OK, so then we get 2 times e to
the u times e to the minus

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u-- but e to the u times e to
the minus u is just 1--

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so plus 2 plus e to the minus
2u minus e to the 2u minus 2

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plus e to the minus 2u-- so
same thing over here--

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over 4.

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OK, so the e to the 2u's cancel
and the e to the minus

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2u's cancel and we're left
with 2 minus minus 2.

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That's 4.

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So this is 4 over 4, so
this is equal to 1.

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OK.

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So if x is equal to cosh u and
y is equal to sinh u, then x

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squared minus y squared
is equal to 1.

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So if we choose a point cosh
u, sinh u for sum u, that

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point lies on this hyperbola.

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That's what this says.

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That this point, OK so the
point cosh u, sinh u is

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somewhere on this hyperbola.

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And what's also true is the
sort of reverse statement.

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If you look at all such points,
if you let u vary and

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look through the real numbers
and you ask what happens to

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this point cosh u, sinh u, the
answers is that it traces out

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the right half of
this hyperbola.

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If you go back to the graph of
y equals cosh x, you'll see

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that the hyperbolic cosine
function is always positive.

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So we can't--

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over here-- we can't
trace out this left

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branch where x is negative.

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Although it's easy enough to say
what does trace out this

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left branch.

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Since it's just the mirror
image, this is traced out by

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minus cosh u comma sinh u.

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So there's a, so the hyperbolic
trig functions have

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the same relationship to this
branch of this hyperbola that

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the regular trig functions
have to the circle.

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So there's where the words
hyperbolic and trig

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functions come from.

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So let me say one more thing
about them, which is that we

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saw that they have
this analogy with

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regular trig functions.

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Right?

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So instead of satisfying cosine
squared plus sine

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squared equals 1, they satisfy
cosh squared minus sinh

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squared equals 1.

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And instead of satisfying the
derivative of sine equals

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cosine and the derivative of
cosine equals minus sine, they

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satisfy derivative of cosh
equals sinh and derivative of

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sinh equals cosh.

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So similar relationships.

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Not exactly the same,
but similar.

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So this is true of a lot of
trig relationships that

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there's a corresponding formula
for the hyperbolic

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trig functions.

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So one example of such a formula
is your, for example,

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your angle addition formulas.

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So I'm going to just leave this
is an exercise for you.

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So let me, I guess I'll just
stick it in this funny little

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00:12:32,390 --> 00:12:33,640
piece of board right here.

251
00:12:33,640 --> 00:12:36,480

252
00:12:36,480 --> 00:12:37,730
So, exercise.

253
00:12:37,730 --> 00:12:41,530

254
00:12:41,530 --> 00:13:03,580
Find sinh of x plus y and cosh
of x plus y in terms of sinh

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00:13:03,580 --> 00:13:14,380
x, cosh x, sinh y, and cosh y.

256
00:13:14,380 --> 00:13:18,150
So in other words, find the
corresponding formula to the

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00:13:18,150 --> 00:13:20,160
angle addition formula
in that case of the

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00:13:20,160 --> 00:13:21,980
hyperbolic trig functions.

259
00:13:21,980 --> 00:13:23,960
So I'll leave you with that.

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