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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.