WEBVTT
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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
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mentioned in 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 them
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so we can get a little
bit of a feeling for what
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these functions are
like, and then I'm
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going to explain to you why
they have the words hyperbolic
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and 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
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as your usual, sort of
circular trig functions.
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But yeah, so let me introduce
them and let me jump
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in just 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
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you can write the other
four trigonometric functions
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in terms of them.
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So for hyperbolic
trig functions we
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have the hyperbolic cosine
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."
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And similarly, for
the hyperbolic sine
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we write s i n h,
for hyperbolic sine,
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except in the reverse order.
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And we usually pronounce this
"sinch," so in American English
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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
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in terms of the exponential
function, 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 x
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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
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looks like, and your
graph of e to the minus x,
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it's not hard to see that the
graphs of cosh x and sinh x
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should look sort of like this.
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So for cosh x, so we see as
x gets big, so e to the minus
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x 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
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it looks a little
bit like a parabola,
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but the growth here is
exponential at both sides.
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So in fact, this is growing
much, much, much faster
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than, 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
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equals 0-- it has the
value 1 plus 1 over 2.
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So its minimum there
is at x equals 0,
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it has its 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,
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e to the x is big,
and e to the minus
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x is pretty small,
almost negligible.
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So we got exponential
growth off that side.
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When x becomes
negative and large,
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e to the x is going to 0, e to
the minus x is becoming large,
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but it's-- we've got
a minus sign here.
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So as x goes to minus
infinity, this curve
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goes also 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
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at the origin is, you could
quickly take a derivative
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and check that that's passing
through the origin with slope 1
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there.
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OK.
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So this is a sort
of basic picture
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of what these curves look like.
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They have some nice properties,
and let me talk about them.
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So for example,
one nice thing you
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might notice about
these functions
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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
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it's 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 minus
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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,
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and the derivative
of sinh, well, OK.
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You look at the same
thing, take this formula,
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take its derivative.
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Well, e to the x, take its
derivative, you get e to the x.
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e to the minus x,
take its derivative,
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you get minus e to the minus
x, so those two minus signs
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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
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a little bit 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
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you almost get back sine,
but you get minus sine.
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So here you don't have
that extra negative sign
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floating around.
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Right?
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So you, when you take
the derivative of cosh
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you get 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
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the words trig in their 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-- what sometimes we
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call the circular trig functions
if we want to distinguish them
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from the hyperbolic trig
functions-- they're closely--
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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
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plus y 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
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between this circle
and the trig functions?
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Well, if you choose any
point on this circle,
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then there exists
some value of t
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such that this point
has coordinates
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cosine t comma sine t.
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Now it happens
that the value of t
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is actually the angle
that that radius makes
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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,
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the point cosine t,
sine t, that varies
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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
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in a very similar situation.
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But instead of
looking at the unit
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circle, what we
want to look at is
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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
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x squared plus y squared
equals 1, which gives a circle,
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I'm going to look at a
very similar equation that
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gives a hyperbola.
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So this is the equation
x squared minus y squared
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equals 1.
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So if you if you graph this
equation, what you'll see
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is that, well, it passes
through the point (1, 0).
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And then we've got
one branch here,
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we've got a little
asymptote there.
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So it's got a right
branch like that,
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and also it's 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
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x squared minus y
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
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the same relationship to this
hyperbola as cosine and sine
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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-- so we're going
to introduce a new variable,
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u-- 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
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x 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
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for hyperbolic trig
functions that we
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do for 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.
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So this is equal to e to the u
plus e to the minus u over 2,
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quantity squared, minus e to
the u minus e to the minus
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u over 2, quantity squared.
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And now we can expand
out both of these factors
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and-- both of these squares,
rather, and put them together.
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So over 2 squared is
over 4 and we square this
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and we get e to the 2u.
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OK, so then we get 2 times e to
the u times e to the minus u.
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But e to the u times e to the
minus u is just 1, so plus 2.
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Plus e to the minus 2u minus
e to the 2u minus 2 plus e
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to the minus 2u-- so same
thing over here-- over 4.
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OK, so the e to the 2u's cancel
and the e to the minus 2u's
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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,
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then x squared minus y
squared is equal to 1.
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So if we choose a point
(cosh u, sinh u) for some u,
00:10:26.270 --> 00:10:28.850
that 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)
00:10:35.540 --> 00:10:38.150
is somewhere on this hyperbola.
00:10:38.150 --> 00:10:41.040
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 look--
00:10:45.980 --> 00:10:47.640
through the real
numbers and you ask
00:10:47.640 --> 00:10:50.210
what happens to this
point (cosh u, sinh u),
00:10:50.210 --> 00:10:52.620
the answers is that it
traces out the right half
00:10:52.620 --> 00:10:53.700
of this hyperbola.
00:10:53.700 --> 00:10:56.805
If you go back to the
graph of y equals cosh x,
00:10:56.805 --> 00:10:59.750
you'll see that the hyperbolic
cosine function is always
00:10:59.750 --> 00:11:00.580
positive.
00:11:00.580 --> 00:11:04.070
So we can't-- over here, we
can't trace out this left
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branch where x is negative.
00:11:05.610 --> 00:11:07.700
Although it's easy
enough to say what does
00:11:07.700 --> 00:11:09.950
trace out this left branch.
00:11:09.950 --> 00:11:12.280
Since it's just
the mirror image,
00:11:12.280 --> 00:11:18.740
this is traced out by
minus cosh u comma sinh u.
00:11:22.220 --> 00:11:24.520
So there's a-- so the
hyperbolic trig functions have
00:11:24.520 --> 00:11:27.870
the same relationship to
this branch of this hyperbola
00:11:27.870 --> 00:11:31.120
that the regular trig
functions have to the circle.
00:11:31.120 --> 00:11:35.070
So there's where the words
hyperbolic and trig functions
00:11:35.070 --> 00:11:36.120
come from.
00:11:36.120 --> 00:11:39.230
So let me say one
more thing about them,
00:11:39.230 --> 00:11:41.830
which is that we
saw that they have
00:11:41.830 --> 00:11:45.060
this analogy with
regular trig functions.
00:11:45.060 --> 00:11:45.560
Right?
00:11:45.560 --> 00:11:47.820
So instead of satisfying
cosine squared
00:11:47.820 --> 00:11:52.350
plus sine squared equals 1, they
satisfy cosh squared minus sinh
00:11:52.350 --> 00:11:53.590
squared equals 1.
00:11:53.590 --> 00:11:56.770
And instead of satisfying
the derivative of sine
00:11:56.770 --> 00:12:00.250
equals cosine and the derivative
of cosine equals minus sine,
00:12:00.250 --> 00:12:03.150
they satisfy derivative
of cosh equals
00:12:03.150 --> 00:12:05.990
sinh and derivative
of sinh equals cosh.
00:12:05.990 --> 00:12:07.690
So similar relationships.
00:12:07.690 --> 00:12:09.680
Not exactly the
same, but similar.
00:12:09.680 --> 00:12:13.010
So this is true of a lot
of trig relationships,
00:12:13.010 --> 00:12:16.960
that there's a corresponding
formula for the hyperbolic trig
00:12:16.960 --> 00:12:17.560
functions.
00:12:17.560 --> 00:12:22.170
So one example of such
a formula is your--
00:12:22.170 --> 00:12:24.790
for example, your angle
addition formulas.
00:12:24.790 --> 00:12:28.160
So I'm going to just leave
this is an exercise for you.
00:12:28.160 --> 00:12:31.482
So let me, I guess
I'll just stick it
00:12:31.482 --> 00:12:33.440
in this funny little
piece of board right here.
00:12:36.284 --> 00:12:36.825
So, exercise.
00:12:41.530 --> 00:13:03.580
Find sinh of x plus y and cosh
of x plus y in terms of sinh
00:13:03.580 --> 00:13:14.380
x, cosh x, sinh y, and cosh y.
00:13:14.380 --> 00:13:17.890
So in other words, find
the corresponding formula
00:13:17.890 --> 00:13:20.070
to the angle addition
formula in that case
00:13:20.070 --> 00:13:21.980
of the hyperbolic
trig functions.
00:13:21.980 --> 00:13:23.637
So I'll leave you with that.