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PROFESSOR: Hi.
Today's lesson, well, I settled for the title,
"Circular Functions." But I guess it could have been
called a lot of different things.
It could've been called 'Trigonometry without
Triangles'.
It could have been called 'Trigonometry Revisited'.
And the whole point is that much of what today's lecture
hinges on is a hang-up that bothered me, and which I think
may bother you and is worthwhile discussing.
I remember, when I was in high school, I asked my
trigonometry teacher, why would I have to know
trigonometry?
And his answer was, surveyors use it.
And at that particular time, I didn't know what I was going
to be, but I knew what I wasn't going to be.
I wasn't going to be a surveyor.
And I kind of took the course kind of lightly, and really
got clobbered a year or two later when I got into calculus
and physics courses.
So what I would like to do today is to introduce the
notion of what we call circular functions, and point
out what the connection is between these and the
trigonometric functions that we learned when we studied the
subject that we call trigonometry, and which might
better have been called numerical geometry.
Let me get to the point right away.
Let's imagine that I say circular functions to you.
I think it's rather natural that, as soon as I say that,
you think of a circle.
And because you think of a circle, let me draw a circle
here, and let me assume that the radius of the circle is 1.
In other words, I have the circle here, 'x squared' plus
'y squared' equals 1.
Now, the thing is this.
When I talk about--
And I'm assuming now that you are familiar with the
trigonometric functions in the traditional sense.
And in fact, the first section of our supplementary notes in
the reading material that goes with the present lecture takes
care of the fact that, if you don't recall some of these
things too well, there's ample opportunity for refreshing
your minds and getting some review in here.
But the idea is something like this.
When we're talking about calculus, we talk about
functions of a real variable.
We are assuming that our functions have the property
that the domain is a set of suitably chosen real numbers,
and the image is a suitably chosen set of real numbers.
We do not think of inputs as being angles and
things of this type.
And so the question is, how can we define, for example--
let's call it the 'sine machine'.
Let me come down here.
I'll call it the 'sine machine'.
If the input is the number 't', I want the output, say,
to be 'sine t'.
But you see, now I'm talking about a number, not an angle.
Well, one way of doing this thing visually is the old idea
of the number line.
Let us think of a number as being a length, the same as we
do in coordinate geometry.
We knock off lengths along the x-axis and the y-axis.
Let me think of 't' as being a length.
As such, I can take 't' and lay it off along my circle in
such a way that the length originates at 'S' and
terminates, shall we say, at some point 'P' whose
coordinates are 'x' and 'y'.
Now, notice what I'm saying here.
I lay the length off along the circumference.
I'll talk more about that a little bit later.
Now, so far, so good.
No mention of the word "angle" here or anything like this.
Now, wherever t terminates-- and again, conventions here,
if 't' is positive, I lay if off along the circle in the
so-called positive direction, namely, what?
Counter-clockwise.
If 't' is negative, I'll lay it off in the clockwise
direction, et cetera.
The usual trigonometric conventions.
Now what I do is is, at the point 'P', I drop a
perpendicular.
And I define the sine of 't' to be the length, 'PR', and
the cosine of 'P' to be the length, 'OR'.
In other words, I could write that like this.
I could write down that I'm defining 'sine t' to be the
length of 'RP' in that direction, meaning, of course,
that this is just a fancy way of saying that the sine of 't'
will just be the y-coordinate of the point at which the
length 't' terminates on the circle.
And in a similar way, 'cosine t' will be the directed length
from 'O' to 'R', or more conventionally, the
x-coordinate.
Now, notice I can do this with any length.
Whatever length I'm given, I just mark this length off.
It's a finite length.
Eventually, it has to terminate some
place on the circle.
Wherever it terminates, the x-coordinate of the point of
termination is called the cosine of 't', and the
y-coordinate is called the sine of 't'.
And notice that, in this way, both the sine and the cosine
are functions which map real numbers into real numbers.
So that part, I hope, is clear.
Notice again, I can mimic the usual traditional
trigonometry.
I can define the tangent of t to be the number 'sine t',
divided by the number 'cosine t', et cetera.
And I'll leave those details to the reading material.
I can ascertain rather interesting results the same
way as I could in regular traditional trigonometry.
In fact, I can get some certain results very nicely.
I remember, for example--
Well, I won't even go into these.
But how did you talk about the sine of 0 when one talked
about traditional trigonometry?
How did you embed a 0-degree angle into a triangle, and
things of this type.
Notice that in terms of my tradition here--
and we'll summarize these results in a minute-- but
notice, for example, that the sine of 0 comes out to be 0
very nicely, because when 't' is 0, the length 0
terminates at 'S'.
'S' is on the x-axis.
That makes, what? 'y' equal to 0.
Notice also that, if the radius of my circle is 1, the
circumference is 2 pi.
So for example, what I usually think of a 90-degree angle
would be the length pi/2.
And without making any fuss over this, again, leaving most
of the details to the reading and to the simplicity of just
plugging these things in, we arrive at these rather
familiar results.
We also get, very quickly, in addition to these results,
things like the fundamental result that we always like
with trigonometric functions.
That's 'sine squared t' plus 'cosine squared t' is 1.
And how do we know that?
Remember that 'cosine t' was just another name for the
x-coordinate at which the point
terminated on the circle.
In other words, notice that 'cosine
squared t' is 'x squared'.
'Sine squared t' is 'y squared'.
The x-coordinate and the y-coordinate are related by
the fact that, what?
The sum of the squares to be on the circle is equal to 1.
We could even graph 'sine t' without any problem at all.
Namely, we observe that when 't' is 0, 'sine t' is 0.
Notice that as we go along the circle, the sine increases up
until we get to pi/2, at which it peaks at 1, then decreases
at pi, back down to 0.
And if that's giving you trouble to follow, let's
simply come back to our diagram to make sure that we
understand this.
In other words, all we're saying is, as 't' gets longer,
its y-coordinate increases from 0 to a maximum of 1, when
the particle was over here.
Then, as 't' goes from to pi/2 to pi, the length of the
y-coordinate decreases until it again becomes 0.
And again, without making much more ado over this, we get the
usual curve that we associate with the sine function even
when we thought of it as a traditional
trigonometric problem.
But the major point that I want you to see right now--
and we won't worry about why I want to do this--
I can define the trigonometric functions in such a way that
their domains are real numbers rather than angles.
And in fact, this is the main reason why people invented the
notion of radian measure.
Let me see if I can't make that a little bit clearer,
once and for all.
You see, the question is this.
Let's suppose I'm talking--
Oh, let me give you some letters over here.
We'll put a 'Q' over here.
Let's talk about angle, 'QOS'.
That's a right angle.
It's 1/4 of a rotation of the circle.
Now, the question that I have in mind is, if something is
1/4 of a rotation, why do you need two different ways of
saying that?
Why do we have to say it's 90 degrees or pi/2 radians, and
bring in a new measure when we already have another way of
measuring circles, angles of circles?
The idea is something like this.
Let's again mimic the idea of taking the length 't' and
laying it off along the circle like this.
Now, here's the idea.
Remember, the radius of this circle is 1.
So notice that 'PR', in other words, this y-coordinate, is
what we call, by definition, the sine of 't'.
In other words, just above we said that 'sine t' was the
length 'R' to 'P' in that direction.
Now, the point is, I said disregard traditional
trigonometry, but we can't really disregard it.
It exists.
For the person who's had traditional trigonometry, how
would he tend to look at this length divided by this length?
He would think of that as being what?
It's side opposite over hypotenuse.
That also suggests sine.
And the sine of what?
Well, the sine of what angle this is.
Now, the thing is this.
Somehow or other, to avoid ambiguity, if we could have
called whatever measure this angle was measured in terms
of, if we could have called that unit 't', then notice
that the sine of the angle 't' would have been numerically
the same as the sine of the number 't'.
And again, if this seems like a hard point to understand, we
explore this in great detail in our notes.
But the idea is this.
You see, somehow or other, if 'sine t' is going to have two
different meanings, we would like to make sure that we pick
the kind of a unit where it makes no difference whether
you're thinking of 't' as being a number or thinking of
't' as being a length.
For example, suppose I now invent the word "radian" to
mean the following.
An angle is said to have 't' radians.
If, when made the central angle of a unit circle, a
circle whose radius is 1, it subtends an arc whose length
is 't' units of length.
See, in other words, I would define the measure called
radians so that an angle of 't' radians intercepts the
length 't' over here.
In that way, 'sine t' is unambiguous whether you're
talking about an angle or a length.
For example, when I say the sine of pi/1
radians, what do I mean?
I mean the angle which is the sine of the angle which
intercepts a length, an arc, pi/2 units long.
Well, see, pi/2 is this length.
I'm now talking about this angle here.
And the sine, therefore, of pi/2 radians, in terms of
classical trigonometry, is 1.
But that's also what the sine of a number pi/2 was.
This explains the convention that one says when one uses
radians, you can leave the label off.
All we're saying is that, if we had used degrees, there
would have been an ambiguity.
Certainly, the sine of 3 degrees is not the same
as the sine of 3.
You see, 3 degrees is a rather small angle.
But 3 is a rather great length when you're talking about the
arc of the unit circle here.
Remember, 1/2 circle is pi units long, so 3 would be just
about this long.
In other words, notice that 3 radians and 3 degrees are
entirely different things.
But the beauty is what?
That if we agreed to use radian measure, then we have
no ambiguity when we talk about the sine.
The sine of the number 't' will equal the sine of the
angle 't' radians.
The cosine of a number 't' will equal the cosine of the
angle 't' radians.
In a certain sense, it was analogous to when we talked
about the derivative 'dy/dx', then wanted to define
differentials 'dy' and 'dx' separately, so that 'dy'
divided by 'dx' would be the same as 'dy/dx', that we
wanted to avoid any ambiguity where the same symbol could be
interpreted in two different ways to give
two different answers.
By the way, again, there is nothing sacred about our
choice of why we pick circular functions.
We could have picked hyperbolic functions.
Namely, why couldn't we have started, say, with one branch
of the hyperbola, 'x squared' minus 'y squared' equals 1.
Given a length 't', why couldn't we have measured 't'
off along the hyperbola?
Say this way if 't' is positive, the other way if 't'
is negative.
And then what we could have done is drop the
perpendicular again.
And we could have defined what?
The y-coordinate to be the hyperbolic.
Well, we couldn't call it cosine anymore because it
would be confused with the circular functions.
We could have invented a name, as we later will, called the
'hyperbolic cosine'.
I won't go into any more detail on this.
See, this is an abbreviation for hyperbolic
cosine, meaning this--
I'm sorry, I got this backwards.
Call the x-coordinate the hyperbolic cosine, the
y-coordinate the hyperbolic sine.
You don't have to know anything about advanced
mathematics to see this.
All I'm saying is, I could just as easily have taken any
geometric figure, marked off lengths along it, taken the
x-coordinates and the y-coordinates, and seen what
relationships they obey.
You see, as such, there's nothing sacred about working
on a circle.
Not only that, but even after you agree to work on the
circle, there are many other ways that one
could have done this.
For example, someone might have said, look it, when you
take this length called 't', why did you elect to mark it
off along the circle?
Why couldn't you have taken a radius equal to 't', taken 'S'
as a center, and swung an arc that met the circle, and call
this length 't'?
You see, instead of measuring along the circle, measure
along the straight line.
Again, you could have done this if you wanted to.
Why you would've wanted to do this?
Well, you have the same right to do this
as I had to do mine.
Of course, you have to be a little bit careful.
For example, in this particular configuration,
notice that, if this is how you're going to define your
trigonometric function, your input, your domain, has to be
somewhere between 0 and 2.
In other words, you cannot have a length longer than 2,
because notice that the diameter of the
circle is only 2.
And therefore, if 't' were greater than 2, when you swung
an arc from the point 'S', it wouldn't meet
the circle at all.
Well, that's no great handicap.
It's no great disaster.
You still have the right to make up whatever
functions you want.
I will try to make it clearer why we chose these circular
functions from a physical point of view as we go along.
What I thought I'd like to do now is, having motivated, that
we can invent the trigonometric functions in
terms of numbers definitions along this circle.
And coupled with the fact that, in radian measure, you
can have a very nice identification between what's
happening pictorially and what's happening analytically,
to show, for example, that in terms of our subject called
calculus, that we're pretty much home free once we learn
these basic ideas.
You see, the important point is that, in a manner of
speaking, we have finished differential calculus.
We know what all the recipes are We know what properties
things have.
So all of the rules that we learned will apply to any
particular type of function that we're talking about.
For example, let's suppose we define 'f of
x' to be 'sine x'.
And we want to find the derivative of 'sine x'.
Notice that 'f prime of x' evaluated at any number 'x1'
has already been defined for us.
It's the limit as 'delta x' approaches 0, 'f of 'x1 plus
delta x'', minus 'f of x1' over 'delta x'.
This is true for any function 'f'.
In particular, if 'f of x' is 'sine x', all we get is what?
That the derivative is the limit as 'delta x' approaches
0, sine of 'x1 plus delta x' minus sine of
'x1' over 'delta x'.
Now you see, on this particular score,
nobody can fault us.
This is still the basic definition.
All that happens computationally is that, if
we're not familiar with our new functions called the
trigonometric functions, we might not know how to express
sine of 'x1 plus delta x' in a more convenient form.
What do we mean by a more convenient form?
Well, notice again, as is always the case when we take a
derivative, as delta x approaches 0, our numerator
becomes 'sine x1' minus 'sine x1', which is 0/0.
And we're back to our familiar taboo form of 0/0.
Somehow or other, we're going to have to make a refinement
on our numerator that will allow us to get
rid of a 0/0 form.
Well, to make a long story short, if we happen to know
the addition formula for the sine--
in other words, 'sine 'x1 plus delta x'' is ''sine x1'
'cosine delta x'', plus ''sine delta x' 'cosine x1''--
then we subtract off 'sine x1' and divide by 'delta x', and
then we factor and collect terms.
We see what?
Without any knowledge of calculus at
all, but just what?
By our definition of derivative, just by our
definition, coupled with properties of the
trigonometric functions, we wind up with the fact that 'f
prime of x1' is this particular limit.
Now certainly, our limit theorems don't change.
The limit of a sum is still going to be
the sum of the limits.
The limit of a product will still be the
product of the limits.
So all in all, what we have to sort of do is figure out what
these limits will be.
Certainly, as 'delta x' approaches 0, this will stay
'cosine x1'.
Certainly this will stay 'sine x1', because 'x1' is a fixed
number that doesn't depend on 'delta x'.
But notice, rather interestingly, that both of my
expressions in parentheses happen to take on that 0/0
form if we're not careful.
Namely, if you replace 'delta x' by 0, sine 0 is 0, 0/0 is
0, and we run into trouble here if we replace 'delta x'
by 0, which of course we can't do.
This is the same definition of limit as we had before.
'Delta x' gets arbitrarily close to 0, but never is
allowed to get there.
Well, you see, if nothing else, this motivates why we
would like to learn this particular type of limit.
In other words, what we would like to know is, how do you--
the 'delta x' symbol here isn't that important.
'Delta x' just stands for any number.
Notice that what we would like to know is, if you take the
sine of something over that same something, and take the
limit as that same something goes to 0, we would like to
know what that becomes.
In a similar way, we would like to know how to handle
this quotient here, because notice that when 'delta x' is
0, cosine 0 is 1.
This is 1 minus 1 over 0.
It's another 0/0 form.
So the problem that we're confronted with is that, what
we would like to do is to figure out how to handle the
limit of 'sine t' over 't' as 't' approaches 0.
Now, what's 't' here? 't' is a number.
Remember that.
This is the big pitch I've been making.
We're thinking of 't' as a number.
If, on the other hand, you feel more comfortable thinking
in terms of traditional trigonometry--
and let's face it, the more background you've had in
traditional trigonometry, the more comfortable you're going
to feel using it.
Let's simply agree to do this, that if it bothers you to
think of this as a length divided by a length, et
cetera, and that this is a length or a number, let's
agree that we will go back to angles but use radian measure.
Why?
Because if the angle is measured in radians, the sine
of the angle 't' radians is the same as the number, the
sine, of the number 't'.
Well again, here's how this problem is tackled.
What we do is we mark off the angle of 't' radians.
Remember that we have the unit circle.
And what we very cleverly do is we catch our wedge, our
circular wedge, between two right triangles.
Again, without making a big issue over this, notice that
this length is 'sine t', this length is 'cosine t', so the
area of the small triangle is 'sine t' times
'cosine t' over 2.
See, 'sine t' times 'cosine t' over 2.
Now, on the other hand, since that's caught in our wedge,
what is the area of our wedge?
Well, since the area of the entire circle is pi-- see, pi
'R squared' and 'R' is 1--
since the area of the entire circle is pi--
and we're taking what?
't' of the 2pi.
So there are two pi radians in a circle.
So the sector of the circle that we have, it's 't/2pi' of
the entire circle.
And by the way, this is done more rigorously and carried
out in detail in the notes.
Let me point out that, if we insisted on working with
degrees, instead of 't/2pi', we just
would have had 't/360'.
Because, you see, if we're dealing with degrees, the
entire angle and measure of the circle is 360 degrees, and
we would have had 't/360'.
But here we've used the fact that we're
dealing with radians.
And finally, the bigger triangle, which includes the
wedge, has, as its base, 1, so that's the radius.
And since the tangent is side opposite over side adjacent,
this length is 'tangent t'.
And so what we have is what?
That ''sine t' 'cosine t/2' must be less than this, which
in turn must be less than this, multiplying through by 2
and dividing through by 'sine t'.
And by the way, this hinges on the fact that 't' is positive.
Again, in our notes, we treat the case where 't' is negative
to arrive at the same result.
Remembering that 'tan t' is 'sine t' over 'cosine t', we
wind up with this result.
And now, observing that as 't' approaches 0,
this approaches 1.
This also approaches 1.
And 't' over 'sine t' is caught between these two.
We get that the limit of 't' over 'sine t' as 't'
approaches 0 is 1.
Now of course, since this limit is 1, the limit of the
reciprocal of this will be the reciprocal of this.
But what's very nice about the number 1 is that it's equal to
its own reciprocal.
In other words, what we've now shown is that the limit of
'sine t' over 't' as 't' approaches 0 is 1.
That, as I said before, is done in the text.
We do it in our notes.
But the thing that I hope this motivates is why we want to do
this in the first place.
Notice that this was a limit that we had to compute if we
wanted to compute the derivative of the sine.
Now, the next thing was, how do we handle '1 - cosine t'
over 't' as 't' approaches 0?
Again, leaving the details to you to sketch in as you see
fit, let me point out simply what the mathematics
involved here is.
You see, what we can handle is 'sine t' over 't'.
That means that what we would like to do is, whenever we're
given an alien form, we would somehow or other like to
figure some way of factoring a sine t over
t out of this thing.
When you look at '1 - cosine t', the identity, 'sine
squared' equals '1 - 'cosine squared t'',
should suggest itself.
Now, how do you get from '1 - cosine t' to '1 - 'cosine
squared t''?
You have to multiply by '1 + cosine t'.
And if you multiply by '1 + cosine t' upstairs, you must
multiply by '1 + cosine t' downstairs.
By the way, the only time you can't multiply by something is
when the thing is 0.
You can't put that into the denominator.
Notice that 'cosine t' is not 0 in a neighborhood
of 't' equals 0.
See, 'cosine t' behaves like 1 when 't' is near 0, so this is
a permissible step in this particular problem.
The point is, we now factor '1 - 'cosine squared t'' as 'sine
t' times 'sine t'.
See, that's 'sine squared t'.
We break up our 't' times '1 + cosine t' this way.
Now we know that the limit of a product is the product of
the limits.
This we already know goes to 1.
And as 't' approaches 0, from our previous limit work on the
like, notice here, the limit of a quotient is the quotient
of the limits, the numerator goes to 0, the denominator
goes to 2, because as t approaches 0, cosine 0 is 1.
At any rate, that's 0/2, which is 0.
And so this limit is 0.
Now, at the risk of giving you a slight headache as I take
the board down here, let me just review what
it was that we did.
You see, notice that, without any knowledge of these limits
at all, we were able to show that whatever the derivative
of 'sine x' was, it was this particular thing here.
Now what we've done is we've shown that this is 1, and
we've shown that this is 0.
And using our limit theorems, what we now see is what?
That if 'f of x' is 'sine x', 'f prime of x' is 'cosine x'.
Let me just write that down over here, that if 'y' equals
'sine x', 'dy/dx' is 'cosine x'.
And again, notice how much of the calculus involved here was
nothing new.
It goes back to the so-called baby chapter that nobody
likes, where we go back to epsilons, deltas, you see
derivatives by 'delta x', et cetera.
See, those recipes always remain the same.
What happens is, as you invent new functions, you need a
different degree of computational sophistication
to find the desired limits.
By the way, once you get over these hurdles, everything
again starts to go smoothly as before.
For example, our chain rule.
Suppose we have now that 'y' equals 'sine u', where 'u' is
some differentiable function of 'x'.
And we now want to find the 'dy/dx'.
Well, you see, the point is that we know that the
derivative of 'sine u' with respect to 'u' would be
'cosine u'.
What we want is the derivative of 'sine u'
with respect to 'x'.
And we motivate the chain rule the same way as we did before.
It happens to be that we're dealing with the specific
value called sine, but it could've
been any old function.
How would you differentiate 'f of u' with respect to 'x' if
you know how to differentiate 'f of u' with respect to 'u'?
And the answer is, you would just differentiate with
respect to 'u', and multiply that by a derivative of 'u'
with respect to 'x'.
In other words, we get the result what?
That since 'dy/du' is 'cosine u', we get that the derivative
of 'sine u' with respect to 'x' is
'cosine u' times 'du/dx'.
And by the way, one rather nice application of this is
that it gives us a very quick way of getting the derivative
of 'cosine x'.
After all, our basic identity is that 'cosine x' is sine
pi/2 minus 'x'.
Again, a number or an angle, either way.
As long as the measurement is in radians, it makes no
difference whether you think of this as being an angle or
being a number.
The answer will be the same.
The idea is this.
To take the derivative of 'cosine x' with respect to
'x', all I have to differentiate is sine pi/2
minus 'x' with respect to 'x'.
But I know how to do that.
Namely, the derivative of sine pi/2 minus 'x' is cosine pi/2
minus 'x,' and by the chain rule, times the derivative of
this with respect to 'x'.
Well, pi/2 is a constant.
The derivative of 'minus x' is minus 1.
And then, remembering that the cosine of pi/2 minus 'x' is
'sine x', I now have the result that the derivative of
the cosine is minus the sine.
And again, I can do all sorts of things this way.
If I want the derivative of a tangent, I could write tangent
as sine over cosine.
Use the quotient rule.
You see, as soon as I make one breakthrough, all of the
previous body of calculus comes to my
rescue, so to speak.
By the way, what I'd like to do now is point out why, from
a physical point of view, we like circular functions to be
independent of angles and the like.
With the results that we've derived so far, it's rather
easy to derive one more result.
Namely, let's assume that a particle is
moving along the x-axis--
I'm going to start with the answer, sort of, and work
backwards--
according to the rule, 'x' equals 'sine kt', where 't' is
time and 'k' is a constant.
Then its speed, 'dx/dt', is what?
It's the derivative of 'sine kt', which is 'cosine kt',
times the derivative of what's inside with respect to 't'.
In other words, it's 'k cosine kt'.
The second derivative of 'x' with respect to 't', namely,
the acceleration is what?
How do you differentiate the cosine?
The derivative of the cosine is minus the sine.
By the chain rule, I must multiply by the derivative of
'kt' with respect to 't', which gives me another factor
of 't' over here.
Remembering that 'x' equals 'sine kt', I arrive at this
particular so-called differential equation.
And what does this say?
It says that 'd2x/ dt squared', the acceleration, is
proportional to the displacement, the distance
traveled, but in the opposite direction.
You see, 'k squared' can't be negative, so 'minus 'k
squared'' can't be positive.
This says what?
The acceleration is proportional to the
displacement, but in the opposite direction.
Does that problem require any knowledge of angles to solve?
Notice that this is a perfectly
good physical problem.
It's known as simple harmonic motion.
And all I'm trying to have you see is that, by inventing the
circular functions in the proper way, not only can we do
their calculus, but even more importantly, if we reverse
these steps, for example, we can show that, to solve the
physical problem of simple harmonic motion, we have to
know the so-called circular trigonometric functions.
And this is a far cry, you see, from using trigonometry
in the sense that the surveyor uses trigonometry.
You see, this ties up with my initial hang-up that I was
telling you about at the beginning of the program.
By the way, in closing, I should also make reference to
something that we pointed out in our last lecture, namely,
inverse differentiation.
Keep in mind, also, that as you read the calculus of the
trigonometric functions, that the fact that we know that the
derivative of sine u with respect to 'u' was 'cosine u'
gives us, with a switch in emphasis, the result that the
integral 'cosine u', 'du' is 'sine u' plus a constant.
And in a similar way, since the derivative of cosine is
minus the sine, the integral of 'sine u' with respect to
'u' is 'minus cosine u' plus a constant.
Be careful.
Notice how the sines can screw you up.
Namely, they're in the opposite sense when you're
integrating as when you were differentiating.
But again, these are the details which I expect you can
have come out in the wash rather nicely.
We can continue on this way, from knowing how to
differentiate 'sine x' to the nth power.
Namely, it's 'n - 1' 'x', times the derivative of 'sine
x', which is 'cosine x'.
We don't want this in here.
That's a differential form.
Without going into any detail here, notice that a
modification of this shows us that, if we differentiate
this, we wind up with this.
We could now take the time, if this were the proper place, to
develop all sorts of derivative formulas and
integral formulas.
As you study your study guide, you will notice that the
lesson after this is concerned with the calculus of the
circular functions.
My feeling is is that, with this as background, a very
good review of the previous part of the course will be to
see how much of this you can apply on your own to these new
functions called the circular functions.
Next time, we will talk, as you may be able to guess,
about the inverse circular functions
and why they're important.
But until next time, goodbye.
Funding for the publication of this video was provided by the
Gabriella and Paul Rosenbaum Foundation.
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