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MIT OpenCourseWare at ocw.mit.edu. 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. Help OCW continue to provide
free and open access to MIT courses by making a donation
at ocw.mit.edu/donate.
