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JOEL LEWIS: Hi.

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Welcome back to recitation.

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In lecture you discussed some
of the inverse trigonometric

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functions as part
of your discussion

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of inverse functions in general
and implicit differentiation.

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And I just wanted to
talk about one, briefly,

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that you didn't mention in
lecture, as far as I recall,

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which is the inverse cosine.

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So what I'm going to call
the arccosine function.

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So I just wanted to go
briefly through its graph

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and its derivative.

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So here I have the graph of
the curve y equals cosine x.

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So this is a-- you
know, you should

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have seen this before, I hope.

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So it has-- at x equals 0
it has its maximum value 1.

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And then to the
right it goes down.

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Its first zero is at pi over 2.

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And then it has its
trough at x equals pi.

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And then it goes back up again.

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And, OK, and it's
an even function

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that looks the same to the right
and the left of the y-axis.

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And it's periodic
with period 2 pi.

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And it's also,
you know, what you

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get by shifting
the sine function,

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the graph of the sine function,
to the left by pi over 2.

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So, OK, so this is
y equals cosine x.

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So in order to graph
y equals arccosine

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of x, we do what we do for
every inverse function,

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which is we just take the
graph and we reflect it

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across the line y equals x.

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So I've done that over here.

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So this is what we
get when we reflect

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this curve-- the y equals cosine
x curve-- when we reflect it

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through that diagonal
line, y equals x.

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So one thing you'll
notice about this

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is that it's not a function.

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Right?

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This curve is not the graph
of a function because every,

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all these humps on
cosine x-- there

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are more humps out here--
those horizontal lines cut

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the humps in many points.

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And when you reflect
you get vertical lines

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that cut this curve
in many points.

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So it doesn't pass the
vertical line test.

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So in order to get a function
out of this, what we have to do

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is we just have to take a
chunk of this curve that does

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pass the vertical line test.

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And so there are many,
many ways we could do this.

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And we choose one
basically arbitrarily,

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meaning we could make
a different choice,

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and we could do all
of our trigonometry

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around some other choice, but
it's convenient to just choose

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one and if everyone agrees that
that's what that one is then we

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can use it and it's nice.

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We have a function and
we can-- the other ones

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are all closely related to this
one choice that we can make.

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So in particular
here, I think there's

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an easiest choice,
which is we take

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the curve y equals arccosine
x to be just this one

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piece of the arc here.

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So this has maximum-- so it
goes from x equals minus 1 to x

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equals 1.

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And when x is minus
1 we have y is pi,

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and then when x equals 1 y is 0.

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So this is the-- this curve
is the graph of the function y

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equals arccosine of x.

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And if you want-- so there's
a notation that mathematicians

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use sometimes to show
that we're talking

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about the particular
arccosine function that

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has this as its domain
and this as its range.

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So we sometimes write
arccosine and it

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takes this domain-- the
values between 1 and 1--

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and it spits out values
between 0 and pi.

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So this is a sort
of fancy notation

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that mathematicians use to say
the arc cosine function takes

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values in the
interval minus 1, 1--

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so it takes values
between negative 1 and 1--

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and it spits out values
in the interval 0, pi.

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So every value that it spits
out is between 0 and pi.

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OK, so if you
graph the function,

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so now this is a
proper function, right?

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It's single-valued, it passes
the vertical line test.

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So, OK.

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And so that's the graph of
y equals arccosine of x.

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So the other thing
that we did in lecture,

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I think we talked about
arcsine and we graphed it.

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And we talked about
arctan and we graphed it.

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And we also computed
their derivatives.

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So let's do that for
the arccosine, as well.

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So, what have we got?

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Well so, in order to
compute the derivative--

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this function is defined
as an inverse function--

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so we do the same
thing that we did

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in lecture, which
is we use this trick

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from implicit differentiation.

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So in particular,
we have that if y

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is equal to arccosine
of x then we can

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take the cosine of both sides.

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And cosine of
arccosine, since we've

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chosen it as an inverse
function, that just gives us

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back x.

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So we get cosine
of y is equal to x.

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And now we can differentiate.

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So what we're after is the
derivative of arccosine

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of x, so we're after dy/dx.

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So we differentiate this
through with respect to x.

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So on the right-hand
side we just get 1.

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And on the left-hand
side, well, we

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have a chain rule here, right?

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Because we have cosine of
y, and y is a function of x.

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So this is, so the derivative
of cosine is minus sine y,

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and then we have to multiply
by the derivative of y,

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which is dy/dx.

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Now, dy/dx is the
thing we're after,

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so we solve this
equation for dy/dx

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and we get dy/dx is equal to
minus 1 divided by sine y.

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OK, which is fine.

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This is a nice formula,
but what we'd really like,

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ideally, is to express
this back in terms of x.

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And so we can, well we
can substitute, right?

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We have an expression
for y in terms of x.

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So that's y is equal
to arccosine of x.

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So this is equal
to minus 1 divided

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by sine of arccosine of x.

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Now, this looks really ugly.

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And here this is another
place where we could stop,

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but actually it turns out that
because trigonometric functions

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are nicely behaved we
can make this nicer.

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So I'm going to appeal
here to the case

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where x is between 0 and 1.

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So then x, so then we
have a right triangle

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that we can draw.

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And the other case you
can do a similar argument

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with a unit circle, but
I'll just do this one case.

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So, if--

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OK, so arccosine of x.

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What does that mean?

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That is the angle
whose cosine is x.

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Right?

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So if you draw a right triangle
and you make this angle arc--

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two c's-- arccosine of x.

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Well, that angle has
cosine equal to x so--

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and this is a right triangle--
so it's adjacent side

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over the hypotenuse
is equal to x,

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and one easy way to get that
arrangement of things is

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say this side is x
and the side is 1.

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So OK.

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So what?

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Why do I care?

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Because I need
sine of that angle.

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So this is the angle arccosine
of x, so sine of that angle

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is the opposite side
over the hypotenuse.

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And what's the opposite side?

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Well I can use the
Pythagorean theorem here,

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and the opposite side is square
root of 1 minus x squared.

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That's the length of
the opposite side.

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So the sine of arccosine
of x is square root of 1

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minus x squared divided by 1.

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So sine of arccosine of x
is just square root of 1

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minus x squared.

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So we can write this in the
somewhat nicer form, minus 1

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over the square root
of one minus x squared.

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So if you remember what the
derivative of arcsine of x was,

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you'll notice that this is a
very similar looking function.

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And this is just because
cosine and sine are

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very similar looking functions.

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So in fact, the
graph of arccosine

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is just a reflection of
the graph of arcsine,

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and that's why the derivatives
are so closely related

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to each other.

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So OK, so there you go.

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You've got the graph
of arccosine up there

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and you've got the formula
for its derivative,

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so that sort of
completes the tour

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of the most important inverse
trigonometric functions.

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So I think I'll end there.