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JOEL LEWIS: Hi.

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Welcome back to recitation.

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In lecture you discussed
the sum of the inverse

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trigonometric functions as part
of your discussion of

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inverse functions in general,
and implicit differentiation.

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And I just wanted to talk about
one, briefly, that you

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didn't mention in lecture, as
far as I recall, which is the

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inverse cosine.

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So what I'm going to call
the arc cosine function.

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So I just wanted to go briefly
through its graph and its

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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--

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you should 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 0 is at pi over 2.

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And then it has its trough
at x equals pie.

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And then it goes
back up again.

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And, OK, and it's an even
function that looks the same

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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 get by shifting the sine

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function, the graph of the
sine function to the

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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
arc cosine of x, we do what we

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do for every inverse function,
which is we just take the

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graph and we reflect it 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 this curve--

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

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that diagonal line,
y equals x.

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So one thing you'll notice
about this is

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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, all

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these humps on cosine x--

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there are more humps
out here--

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those horizontal lines cut
the humps in many points.

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And when you reflect you get
vertical lines that cut this

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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

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

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pass the vertical line test.
And so there are many, many

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ways we could do this.

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And we choose one basically
arbitrarily, meaning we could

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make a different choice, and
we could do all of our

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trigonometry around some other
choice, but it's convenient to

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just choose one and if everyone
agrees that that's

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what that one is then we can
use it and it's nice.

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We have a function and we can,
the other ones are all closely

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related to this one choice
that we can make.

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So in particular here, I think
there's an easiest choice,

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which is we take the curve, y
equals arc cosine x to be just

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this one piece of
the arc here.

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So this has maximum--

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so it goes from x equals
minus 1 to x equals 1.

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

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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 arc cosine 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 about the

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particular arc cosine function
that has this as its domain

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and this as its range.

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So we sometimes write
arc cosine and

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it takes this domain--

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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 that mathematicians

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use to say the arc cosine
function takes values in the

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interval minus 1, 1-- so
it takes values between

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negative 1 and 1--

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and it spits out values in
the intervals 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, so now this is a

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proper function.

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Right?

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It's single-valued, it
passes the vertical

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line test. So, OK.

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

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So the other thing that we did
in lecture, I think we talked

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about arc sine and
we graphed it.

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And we talked about arc
tan 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
arc cosine, 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 universe function--

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so we do the same thing that we
did in lecture, which is we

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use this trick from implicit
differentiation.

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So in particular, we have that
if y is equal to arc cosine of

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x then we can take the
cosine of both sides.

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And cosine of arc cosine, since
we've chosen it as an

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inverse function, that
just gives us 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 arc cosine of x,

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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 have a chain rule

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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, and

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then we have to multiply
by the derivative of

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y, which is dy dx.

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Now, dy dx is the thing we're
after, so we solve this

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equation for dy dx and we get
dy dx is equal to minus 1

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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 arc cosine of x.

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So this is equal to minus
1 divided by sine of

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arc cosine 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, but

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actually it turns out that
because trigonometric

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functions are nicely behaved
we can make this nicer.

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So I'm going to appeal here
to the case where x is

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between 0 and 1.

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So then x, so then
we have a right

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triangle that we can draw.

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And the other case you can do a
similar argument with a unit

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

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So if, OK so arc cosine 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

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angle arc--

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two c's--

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arc cosine 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 over the

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hypotenuse is equal to x, and
one easy way to get that

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arrangement of things is
say this side is x

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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 arc cosine
of x, so sine of that

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angle 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, and the opposite

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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 arc cosine of x
is square root of 1 minus x

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squared divided by 1.

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So sine of arc cosine of
x is just square root

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of 1 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 arc sine of x

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was, you'll notice that
this is a very

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similar looking function.

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And this is just because cosine
and sine are very

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similar looking functions.

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So in fact, the graph of arc
cosine is just a reflection of

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the graph of arc sine, and
that's why the derivatives are

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so closely related
to each other.

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So OK, so there you go.

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You've got the graph of arc
cosine up there and you've got

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the formula for its derivative,
so that sort of

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completes the tour of the
most important inverse

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trigonometric functions.

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So I think I'll end there.

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