JOEL LEWIS: Hi.
Welcome back to recitation.
In lecture you discussed the sum of the inverse
trigonometric functions as part of your discussion of
inverse functions in general, and implicit differentiation.
And I just wanted to talk about one, briefly, that you
didn't mention in lecture, as far as I recall, which is the
inverse cosine.
So what I'm going to call the arc cosine function.
So I just wanted to go briefly through its graph and its
derivative.
So here I have the graph of the curve, y equals cosine x.
So this is a, you know--
you should have seen this before, I hope.
So it has at x equals 0 it has its maximum value 1.
And then to the right it goes down.
Its first 0 is at pi over 2.
And then it has its trough at x equals pie.
And then it goes back up again.
And, OK, and it's an even function that looks the same
to the right and the left of the y-axis.
And it's periodic with period 2 pi.
And it's also, you know, what you get by shifting the sine
function, the graph of the sine function to the
left by pi over 2.
So, OK, so this is y equals cosine x.
So in order to graph y equals arc cosine of x, we do what we
do for every inverse function, which is we just take the
graph and we reflect it across the line, y equals x.
So I've done that over here.
So this is what we get when we reflect this curve--
the y equals cosine x curve-- when we reflect it through
that diagonal line, y equals x.
So one thing you'll notice about this is
that it's not a function.
Right?
This curve is not the graph of a function because every, all
these humps on cosine x--
there are more humps out here--
those horizontal lines cut the humps in many points.
And when you reflect you get vertical lines that cut this
curve in many points.
So it doesn't pass the vertical line test.
So in order to get a function out of this, what we have to
do is we just have to take a chunk of this curve that does
pass the vertical line test. And so there are many, many
ways we could do this.
And we choose one basically arbitrarily, meaning we could
make a different choice, and we could do all of our
trigonometry around some other choice, but it's convenient to
just choose one and if everyone agrees that that's
what that one is then we can use it and it's nice.
We have a function and we can, the other ones are all closely
related to this one choice that we can make.
So in particular here, I think there's an easiest choice,
which is we take the curve, y equals arc cosine x to be just
this one piece of the arc here.
So this has maximum--
so it goes from x equals minus 1 to x equals 1.
And when x is minus 1 we have y is pi, and then when x
equals 1 y is 0.
So this is the, this curve is the graph of the function, y
equals arc cosine of x.
And if you want-- so there's a notation that mathematicians
use sometimes to show that we're talking about the
particular arc cosine function that has this as its domain
and this as its range.
So we sometimes write arc cosine and
it takes this domain--
the values between 1 and 1--
and it spits out values between 0 and pi.
So this is a sort of fancy notation that mathematicians
use to say the arc cosine function takes values in the
interval minus 1, 1-- so it takes values between
negative 1 and 1--
and it spits out values in the intervals 0, pi.
So every value that it spits out is between 0 and pi.
OK, so if you graph the function, so now this is a
proper function.
Right?
It's single-valued, it passes the vertical
line test. So, OK.
And so that's the graph of y equals arc cosine of x.
So the other thing that we did in lecture, I think we talked
about arc sine and we graphed it.
And we talked about arc tan and we graphed it.
And we also computed their derivatives.
So let's do that for the arc cosine, as well.
So what have we got?
Well so, in order to compute the derivative--
this function is defined as an universe function--
so we do the same thing that we did in lecture, which is we
use this trick from implicit differentiation.
So in particular, we have that if y is equal to arc cosine of
x then we can take the cosine of both sides.
And cosine of arc cosine, since we've chosen it as an
inverse function, that just gives us back x.
So we get cosine of y is equal to x.
And now we can differentiate.
So what we're after is the derivative of arc cosine of x,
so we're after dy dx.
So we differentiate this through with respect to x.
So on the right hand side we just get 1.
And on the left hand side, well, we have a chain rule
here, right?
Because we have cosine of y, and y is a function of x.
So this is, so the derivative of cosine is minus sine y, and
then we have to multiply by the derivative of
y, which is dy dx.
Now, dy dx is the thing we're after, so we solve this
equation for dy dx and we get dy dx is equal to minus 1
divided by sine y.
OK, which is fine.
This is a nice formula, but what we'd really like,
ideally, is to express this back in terms of x.
And so we can, well we can substitute, right?
We have an expression for y in terms of x.
So that's y is equal to arc cosine of x.
So this is equal to minus 1 divided by sine of
arc cosine of x.
Now, this looks really ugly.
And here this is another place where we could stop, but
actually it turns out that because trigonometric
functions are nicely behaved we can make this nicer.
So I'm going to appeal here to the case where x is
between 0 and 1.
So then x, so then we have a right
triangle that we can draw.
And the other case you can do a similar argument with a unit
circle, but I'll just do this one case.
So if, OK so arc cosine of x.
What does that mean?
That is the angle whose cosine is x.
Right?
So if you draw a right triangle and you make this
angle arc--
two c's--
arc cosine of x.
Well, that angle has cosine equal to x so--
and this is a right triangle-- so it's adjacent side over the
hypotenuse is equal to x, and one easy way to get that
arrangement of things is say this side is x
and the side is 1.
So OK.
So what?
Why do I care?
Because I need sine of that angle.
So this is the angle arc cosine of x, so sine of that
angle is the opposite side over the hypotenuse.
And what's the opposite side?
Well I can use the Pythagorean theorem here, and the opposite
side is square root of 1 minus x squared.
That's the length of the opposite side.
So the sine of arc cosine of x is square root of 1 minus x
squared divided by 1.
So sine of arc cosine of x is just square root
of 1 minus x squared.
So we can write this in the somewhat nicer form, minus 1
over the square root of one minus x squared.
So if you remember what the derivative of arc sine of x
was, you'll notice that this is a very
similar looking function.
And this is just because cosine and sine are very
similar looking functions.
So in fact, the graph of arc cosine is just a reflection of
the graph of arc sine, and that's why the derivatives are
so closely related to each other.
So OK, so there you go.
You've got the graph of arc cosine up there and you've got
the formula for its derivative, so that sort of
completes the tour of the most important inverse
trigonometric functions.
So I think I'll end there.