WEBVTT
00:00:00.040 --> 00:00:02.400
The following content is
provided under a Creative
00:00:02.400 --> 00:00:03.690
Commons License.
00:00:03.690 --> 00:00:06.630
Your support will help MIT
OpenCourseWare continue to
00:00:06.630 --> 00:00:09.990
offer high quality educational
resources for free.
00:00:09.990 --> 00:00:12.830
To make a donation, or to view
additional materials from
00:00:12.830 --> 00:00:16.760
hundreds of MIT courses, visit
MIT OpenCourseWare at
00:00:16.760 --> 00:00:18.010
ocw.mit.edu.
00:00:28.873 --> 00:00:30.870
PROFESSOR: Hi.
00:00:30.870 --> 00:00:33.840
Today's lesson, well, I
settled for the title,
00:00:33.840 --> 00:00:36.440
"Circular Functions." But I
guess it could have been
00:00:36.440 --> 00:00:38.280
called a lot of different
things.
00:00:38.280 --> 00:00:40.530
It could've been called
'Trigonometry without
00:00:40.530 --> 00:00:41.490
Triangles'.
00:00:41.490 --> 00:00:44.740
It could have been called
'Trigonometry Revisited'.
00:00:44.740 --> 00:00:48.450
And the whole point is that much
of what today's lecture
00:00:48.450 --> 00:00:52.280
hinges on is a hang-up that
bothered me, and which I think
00:00:52.280 --> 00:00:55.220
may bother you and is worthwhile
discussing.
00:00:55.220 --> 00:00:57.570
I remember, when I was in
high school, I asked my
00:00:57.570 --> 00:00:59.860
trigonometry teacher, why
would I have to know
00:00:59.860 --> 00:01:00.720
trigonometry?
00:01:00.720 --> 00:01:03.280
And his answer was,
surveyors use it.
00:01:03.280 --> 00:01:06.030
And at that particular time, I
didn't know what I was going
00:01:06.030 --> 00:01:08.940
to be, but I knew what
I wasn't going to be.
00:01:08.940 --> 00:01:10.440
I wasn't going to
be a surveyor.
00:01:10.440 --> 00:01:13.690
And I kind of took the course
kind of lightly, and really
00:01:13.690 --> 00:01:17.300
got clobbered a year or two
later when I got into calculus
00:01:17.300 --> 00:01:18.680
and physics courses.
00:01:18.680 --> 00:01:22.230
So what I would like to do
today is to introduce the
00:01:22.230 --> 00:01:26.450
notion of what we call circular
functions, and point
00:01:26.450 --> 00:01:30.630
out what the connection is
between these and the
00:01:30.630 --> 00:01:33.680
trigonometric functions that we
learned when we studied the
00:01:33.680 --> 00:01:36.890
subject that we call
trigonometry, and which might
00:01:36.890 --> 00:01:39.890
better have been called
numerical geometry.
00:01:39.890 --> 00:01:41.890
Let me get to the point
right away.
00:01:41.890 --> 00:01:44.970
Let's imagine that I say
circular functions to you.
00:01:44.970 --> 00:01:47.280
I think it's rather natural
that, as soon as I say that,
00:01:47.280 --> 00:01:48.270
you think of a circle.
00:01:48.270 --> 00:01:51.480
And because you think of a
circle, let me draw a circle
00:01:51.480 --> 00:01:55.660
here, and let me assume that the
radius of the circle is 1.
00:01:55.660 --> 00:01:59.660
In other words, I have the
circle here, 'x squared' plus
00:01:59.660 --> 00:02:02.750
'y squared' equals 1.
00:02:02.750 --> 00:02:04.250
Now, the thing is this.
00:02:04.250 --> 00:02:05.620
When I talk about--
00:02:05.620 --> 00:02:08.430
And I'm assuming now that you
are familiar with the
00:02:08.430 --> 00:02:11.550
trigonometric functions in
the traditional sense.
00:02:11.550 --> 00:02:15.030
And in fact, the first section
of our supplementary notes in
00:02:15.030 --> 00:02:19.190
the reading material that goes
with the present lecture takes
00:02:19.190 --> 00:02:21.050
care of the fact that, if you
don't recall some of these
00:02:21.050 --> 00:02:24.210
things too well, there's ample
opportunity for refreshing
00:02:24.210 --> 00:02:26.580
your minds and getting
some review in here.
00:02:26.580 --> 00:02:28.280
But the idea is something
like this.
00:02:28.280 --> 00:02:31.610
When we're talking about
calculus, we talk about
00:02:31.610 --> 00:02:33.710
functions of a real variable.
00:02:33.710 --> 00:02:37.240
We are assuming that our
functions have the property
00:02:37.240 --> 00:02:41.890
that the domain is a set of
suitably chosen real numbers,
00:02:41.890 --> 00:02:45.120
and the image is a suitably
chosen set of real numbers.
00:02:45.120 --> 00:02:48.290
We do not think of inputs
as being angles and
00:02:48.290 --> 00:02:49.530
things of this type.
00:02:49.530 --> 00:02:52.730
And so the question is, how can
we define, for example--
00:02:52.730 --> 00:02:54.100
let's call it the
'sine machine'.
00:02:54.100 --> 00:02:55.510
Let me come down here.
00:02:55.510 --> 00:02:56.700
I'll call it the
'sine machine'.
00:02:56.700 --> 00:03:00.590
If the input is the number 't',
I want the output, say,
00:03:00.590 --> 00:03:01.810
to be 'sine t'.
00:03:01.810 --> 00:03:07.300
But you see, now I'm talking
about a number, not an angle.
00:03:07.300 --> 00:03:12.020
Well, one way of doing this
thing visually is the old idea
00:03:12.020 --> 00:03:13.250
of the number line.
00:03:13.250 --> 00:03:17.500
Let us think of a number as
being a length, the same as we
00:03:17.500 --> 00:03:19.700
do in coordinate geometry.
00:03:19.700 --> 00:03:22.680
We knock off lengths along the
x-axis and the y-axis.
00:03:22.680 --> 00:03:25.850
Let me think of 't'
as being a length.
00:03:25.850 --> 00:03:32.540
As such, I can take 't' and lay
it off along my circle in
00:03:32.540 --> 00:03:36.250
such a way that the length
originates at 'S' and
00:03:36.250 --> 00:03:39.430
terminates, shall we say,
at some point 'P' whose
00:03:39.430 --> 00:03:41.710
coordinates are 'x' and 'y'.
00:03:41.710 --> 00:03:43.050
Now, notice what I'm
saying here.
00:03:43.050 --> 00:03:47.190
I lay the length off along
the circumference.
00:03:47.190 --> 00:03:50.360
I'll talk more about that
a little bit later.
00:03:50.360 --> 00:03:52.090
Now, so far, so good.
00:03:52.090 --> 00:03:54.970
No mention of the word "angle"
here or anything like this.
00:03:54.970 --> 00:03:58.340
Now, wherever t terminates-- and
again, conventions here,
00:03:58.340 --> 00:04:01.830
if 't' is positive, I lay if
off along the circle in the
00:04:01.830 --> 00:04:04.580
so-called positive direction,
namely, what?
00:04:04.580 --> 00:04:05.880
Counter-clockwise.
00:04:05.880 --> 00:04:09.710
If 't' is negative, I'll lay
it off in the clockwise
00:04:09.710 --> 00:04:11.030
direction, et cetera.
00:04:11.030 --> 00:04:13.220
The usual trigonometric
conventions.
00:04:13.220 --> 00:04:18.579
Now what I do is is, at the
point 'P', I drop a
00:04:18.579 --> 00:04:19.829
perpendicular.
00:04:23.050 --> 00:04:30.020
And I define the sine of 't'
to be the length, 'PR', and
00:04:30.020 --> 00:04:34.150
the cosine of 'P' to be
the length, 'OR'.
00:04:34.150 --> 00:04:36.500
In other words, I could
write that like this.
00:04:36.500 --> 00:04:40.540
I could write down that I'm
defining 'sine t' to be the
00:04:40.540 --> 00:04:45.010
length of 'RP' in that
direction, meaning, of course,
00:04:45.010 --> 00:04:48.600
that this is just a fancy way of
saying that the sine of 't'
00:04:48.600 --> 00:04:52.200
will just be the y-coordinate
of the point at which the
00:04:52.200 --> 00:04:55.260
length 't' terminates
on the circle.
00:04:55.260 --> 00:05:00.760
And in a similar way, 'cosine t'
will be the directed length
00:05:00.760 --> 00:05:03.950
from 'O' to 'R', or more
conventionally, the
00:05:03.950 --> 00:05:05.400
x-coordinate.
00:05:05.400 --> 00:05:07.730
Now, notice I can do this
with any length.
00:05:07.730 --> 00:05:11.940
Whatever length I'm given, I
just mark this length off.
00:05:11.940 --> 00:05:13.120
It's a finite length.
00:05:13.120 --> 00:05:14.760
Eventually, it has
to terminate some
00:05:14.760 --> 00:05:16.000
place on the circle.
00:05:16.000 --> 00:05:19.590
Wherever it terminates, the
x-coordinate of the point of
00:05:19.590 --> 00:05:23.280
termination is called the
cosine of 't', and the
00:05:23.280 --> 00:05:25.960
y-coordinate is called
the sine of 't'.
00:05:25.960 --> 00:05:29.500
And notice that, in this way,
both the sine and the cosine
00:05:29.500 --> 00:05:35.246
are functions which map real
numbers into real numbers.
00:05:35.246 --> 00:05:37.070
So that part, I hope,
is clear.
00:05:39.910 --> 00:05:43.970
Notice again, I can mimic
the usual traditional
00:05:43.970 --> 00:05:45.130
trigonometry.
00:05:45.130 --> 00:05:48.880
I can define the tangent of t
to be the number 'sine t',
00:05:48.880 --> 00:05:51.770
divided by the number 'cosine
t', et cetera.
00:05:51.770 --> 00:05:55.450
And I'll leave those details
to the reading material.
00:05:55.450 --> 00:05:58.870
I can ascertain rather
interesting results the same
00:05:58.870 --> 00:06:03.520
way as I could in regular
traditional trigonometry.
00:06:03.520 --> 00:06:06.650
In fact, I can get some certain
results very nicely.
00:06:06.650 --> 00:06:09.230
I remember, for example--
00:06:09.230 --> 00:06:10.550
Well, I won't even
go into these.
00:06:10.550 --> 00:06:14.460
But how did you talk about the
sine of 0 when one talked
00:06:14.460 --> 00:06:16.230
about traditional
trigonometry?
00:06:16.230 --> 00:06:20.220
How did you embed a 0-degree
angle into a triangle, and
00:06:20.220 --> 00:06:21.320
things of this type.
00:06:21.320 --> 00:06:24.130
Notice that in terms of
my tradition here--
00:06:24.130 --> 00:06:26.500
and we'll summarize these
results in a minute-- but
00:06:26.500 --> 00:06:30.510
notice, for example, that the
sine of 0 comes out to be 0
00:06:30.510 --> 00:06:34.790
very nicely, because when
't' is 0, the length 0
00:06:34.790 --> 00:06:36.630
terminates at 'S'.
00:06:36.630 --> 00:06:38.690
'S' is on the x-axis.
00:06:38.690 --> 00:06:41.990
That makes, what?
'y' equal to 0.
00:06:41.990 --> 00:06:46.900
Notice also that, if the radius
of my circle is 1, the
00:06:46.900 --> 00:06:49.200
circumference is 2 pi.
00:06:49.200 --> 00:06:52.530
So for example, what I usually
think of a 90-degree angle
00:06:52.530 --> 00:06:55.250
would be the length pi/2.
00:06:55.250 --> 00:06:58.370
And without making any fuss over
this, again, leaving most
00:06:58.370 --> 00:07:01.450
of the details to the reading
and to the simplicity of just
00:07:01.450 --> 00:07:05.420
plugging these things in, we
arrive at these rather
00:07:05.420 --> 00:07:07.430
familiar results.
00:07:07.430 --> 00:07:12.460
We also get, very quickly, in
addition to these results,
00:07:12.460 --> 00:07:15.950
things like the fundamental
result that we always like
00:07:15.950 --> 00:07:17.520
with trigonometric functions.
00:07:17.520 --> 00:07:21.890
That's 'sine squared t' plus
'cosine squared t' is 1.
00:07:21.890 --> 00:07:23.310
And how do we know that?
00:07:23.310 --> 00:07:27.620
Remember that 'cosine t' was
just another name for the
00:07:27.620 --> 00:07:29.280
x-coordinate at which the point
00:07:29.280 --> 00:07:30.670
terminated on the circle.
00:07:30.670 --> 00:07:32.610
In other words, notice
that 'cosine
00:07:32.610 --> 00:07:34.970
squared t' is 'x squared'.
00:07:34.970 --> 00:07:37.510
'Sine squared t'
is 'y squared'.
00:07:37.510 --> 00:07:41.310
The x-coordinate and the
y-coordinate are related by
00:07:41.310 --> 00:07:43.350
the fact that, what?
00:07:43.350 --> 00:07:47.930
The sum of the squares to be on
the circle is equal to 1.
00:07:47.930 --> 00:07:53.380
We could even graph 'sine t'
without any problem at all.
00:07:53.380 --> 00:07:58.280
Namely, we observe that when
't' is 0, 'sine t' is 0.
00:07:58.280 --> 00:08:04.250
Notice that as we go along the
circle, the sine increases up
00:08:04.250 --> 00:08:10.510
until we get to pi/2, at which
it peaks at 1, then decreases
00:08:10.510 --> 00:08:13.330
at pi, back down to 0.
00:08:13.330 --> 00:08:15.910
And if that's giving you trouble
to follow, let's
00:08:15.910 --> 00:08:18.480
simply come back to our diagram
to make sure that we
00:08:18.480 --> 00:08:20.580
understand this.
00:08:20.580 --> 00:08:25.200
In other words, all we're saying
is, as 't' gets longer,
00:08:25.200 --> 00:08:31.240
its y-coordinate increases from
0 to a maximum of 1, when
00:08:31.240 --> 00:08:33.000
the particle was over here.
00:08:33.000 --> 00:08:37.340
Then, as 't' goes from to pi/2
to pi, the length of the
00:08:37.340 --> 00:08:42.770
y-coordinate decreases until
it again becomes 0.
00:08:42.770 --> 00:08:46.740
And again, without making much
more ado over this, we get the
00:08:46.740 --> 00:08:51.490
usual curve that we associate
with the sine function even
00:08:51.490 --> 00:08:54.860
when we thought of it
as a traditional
00:08:54.860 --> 00:08:56.240
trigonometric problem.
00:08:56.240 --> 00:08:59.470
But the major point that I want
you to see right now--
00:08:59.470 --> 00:09:01.800
and we won't worry about
why I want to do this--
00:09:01.800 --> 00:09:05.650
I can define the trigonometric
functions in such a way that
00:09:05.650 --> 00:09:09.150
their domains are real numbers
rather than angles.
00:09:09.150 --> 00:09:13.700
And in fact, this is the main
reason why people invented the
00:09:13.700 --> 00:09:15.780
notion of radian measure.
00:09:15.780 --> 00:09:18.940
Let me see if I can't make that
a little bit clearer,
00:09:18.940 --> 00:09:20.780
once and for all.
00:09:20.780 --> 00:09:22.270
You see, the question is this.
00:09:22.270 --> 00:09:23.990
Let's suppose I'm talking--
00:09:23.990 --> 00:09:26.060
Oh, let me give you some
letters over here.
00:09:26.060 --> 00:09:28.580
We'll put a 'Q' over here.
00:09:28.580 --> 00:09:32.030
Let's talk about angle, 'QOS'.
00:09:35.330 --> 00:09:36.560
That's a right angle.
00:09:36.560 --> 00:09:39.790
It's 1/4 of a rotation
of the circle.
00:09:39.790 --> 00:09:43.570
Now, the question that I have
in mind is, if something is
00:09:43.570 --> 00:09:46.250
1/4 of a rotation, why do you
need two different ways of
00:09:46.250 --> 00:09:47.310
saying that?
00:09:47.310 --> 00:09:52.260
Why do we have to say it's 90
degrees or pi/2 radians, and
00:09:52.260 --> 00:09:56.270
bring in a new measure when we
already have another way of
00:09:56.270 --> 00:09:59.250
measuring circles, angles
of circles?
00:09:59.250 --> 00:10:01.020
The idea is something
like this.
00:10:01.020 --> 00:10:05.430
Let's again mimic the idea of
taking the length 't' and
00:10:05.430 --> 00:10:07.820
laying it off along the
circle like this.
00:10:07.820 --> 00:10:11.190
Now, here's the idea.
00:10:11.190 --> 00:10:15.040
Remember, the radius of
this circle is 1.
00:10:15.040 --> 00:10:20.660
So notice that 'PR', in other
words, this y-coordinate, is
00:10:20.660 --> 00:10:23.750
what we call, by definition,
the sine of 't'.
00:10:23.750 --> 00:10:28.030
In other words, just above we
said that 'sine t' was the
00:10:28.030 --> 00:10:31.250
length 'R' to 'P' in
that direction.
00:10:31.250 --> 00:10:34.160
Now, the point is, I said
disregard traditional
00:10:34.160 --> 00:10:36.860
trigonometry, but we can't
really disregard it.
00:10:36.860 --> 00:10:38.410
It exists.
00:10:38.410 --> 00:10:41.480
For the person who's had
traditional trigonometry, how
00:10:41.480 --> 00:10:47.580
would he tend to look at this
length divided by this length?
00:10:47.580 --> 00:10:49.930
He would think of that
as being what?
00:10:49.930 --> 00:10:52.610
It's side opposite
over hypotenuse.
00:10:52.610 --> 00:10:54.620
That also suggests sine.
00:10:54.620 --> 00:10:55.890
And the sine of what?
00:10:55.890 --> 00:10:59.570
Well, the sine of what
angle this is.
00:10:59.570 --> 00:11:00.940
Now, the thing is this.
00:11:00.940 --> 00:11:04.530
Somehow or other, to avoid
ambiguity, if we could have
00:11:04.530 --> 00:11:07.520
called whatever measure this
angle was measured in terms
00:11:07.520 --> 00:11:11.530
of, if we could have called
that unit 't', then notice
00:11:11.530 --> 00:11:15.550
that the sine of the angle 't'
would have been numerically
00:11:15.550 --> 00:11:18.760
the same as the sine
of the number 't'.
00:11:18.760 --> 00:11:22.140
And again, if this seems like a
hard point to understand, we
00:11:22.140 --> 00:11:24.950
explore this in great
detail in our notes.
00:11:24.950 --> 00:11:26.670
But the idea is this.
00:11:26.670 --> 00:11:30.760
You see, somehow or other, if
'sine t' is going to have two
00:11:30.760 --> 00:11:33.620
different meanings, we would
like to make sure that we pick
00:11:33.620 --> 00:11:37.470
the kind of a unit where it
makes no difference whether
00:11:37.470 --> 00:11:40.130
you're thinking of 't' as being
a number or thinking of
00:11:40.130 --> 00:11:41.960
't' as being a length.
00:11:41.960 --> 00:11:45.370
For example, suppose I now
invent the word "radian" to
00:11:45.370 --> 00:11:47.310
mean the following.
00:11:47.310 --> 00:11:49.910
An angle is said to
have 't' radians.
00:11:49.910 --> 00:11:53.920
If, when made the central angle
of a unit circle, a
00:11:53.920 --> 00:11:59.180
circle whose radius is 1, it
subtends an arc whose length
00:11:59.180 --> 00:12:02.270
is 't' units of length.
00:12:02.270 --> 00:12:05.570
See, in other words, I would
define the measure called
00:12:05.570 --> 00:12:09.690
radians so that an angle of
't' radians intercepts the
00:12:09.690 --> 00:12:11.490
length 't' over here.
00:12:11.490 --> 00:12:16.210
In that way, 'sine t' is
unambiguous whether you're
00:12:16.210 --> 00:12:19.005
talking about an angle
or a length.
00:12:19.005 --> 00:12:22.860
For example, when I say
the sine of pi/1
00:12:22.860 --> 00:12:26.150
radians, what do I mean?
00:12:26.150 --> 00:12:29.280
I mean the angle which is the
sine of the angle which
00:12:29.280 --> 00:12:34.350
intercepts a length, an
arc, pi/2 units long.
00:12:34.350 --> 00:12:36.570
Well, see, pi/2 is
this length.
00:12:36.570 --> 00:12:40.630
I'm now talking about
this angle here.
00:12:40.630 --> 00:12:43.640
And the sine, therefore, of
pi/2 radians, in terms of
00:12:43.640 --> 00:12:46.530
classical trigonometry, is 1.
00:12:46.530 --> 00:12:51.040
But that's also what the sine
of a number pi/2 was.
00:12:51.040 --> 00:12:53.890
This explains the convention
that one says when one uses
00:12:53.890 --> 00:12:56.240
radians, you can leave
the label off.
00:12:56.240 --> 00:12:59.870
All we're saying is that, if
we had used degrees, there
00:12:59.870 --> 00:13:01.900
would have been an ambiguity.
00:13:01.900 --> 00:13:08.120
Certainly, the sine of 3
degrees is not the same
00:13:08.120 --> 00:13:10.490
as the sine of 3.
00:13:10.490 --> 00:13:13.970
You see, 3 degrees is a
rather small angle.
00:13:13.970 --> 00:13:19.550
But 3 is a rather great length
when you're talking about the
00:13:19.550 --> 00:13:20.740
arc of the unit circle here.
00:13:20.740 --> 00:13:26.830
Remember, 1/2 circle is pi units
long, so 3 would be just
00:13:26.830 --> 00:13:28.000
about this long.
00:13:28.000 --> 00:13:32.180
In other words, notice that 3
radians and 3 degrees are
00:13:32.180 --> 00:13:33.240
entirely different things.
00:13:33.240 --> 00:13:34.280
But the beauty is what?
00:13:34.280 --> 00:13:38.130
That if we agreed to use radian
measure, then we have
00:13:38.130 --> 00:13:41.230
no ambiguity when we talk
about the sine.
00:13:41.230 --> 00:13:44.670
The sine of the number 't' will
equal the sine of the
00:13:44.670 --> 00:13:46.170
angle 't' radians.
00:13:46.170 --> 00:13:49.490
The cosine of a number 't' will
equal the cosine of the
00:13:49.490 --> 00:13:51.360
angle 't' radians.
00:13:51.360 --> 00:13:53.960
In a certain sense, it was
analogous to when we talked
00:13:53.960 --> 00:13:57.460
about the derivative 'dy/dx',
then wanted to define
00:13:57.460 --> 00:14:01.700
differentials 'dy' and 'dx'
separately, so that 'dy'
00:14:01.700 --> 00:14:05.210
divided by 'dx' would be the
same as 'dy/dx', that we
00:14:05.210 --> 00:14:11.340
wanted to avoid any ambiguity
where the same symbol could be
00:14:11.340 --> 00:14:13.400
interpreted in two different
ways to give
00:14:13.400 --> 00:14:14.840
two different answers.
00:14:14.840 --> 00:14:18.010
By the way, again, there is
nothing sacred about our
00:14:18.010 --> 00:14:21.000
choice of why we pick
circular functions.
00:14:21.000 --> 00:14:23.690
We could have picked hyperbolic
functions.
00:14:23.690 --> 00:14:27.410
Namely, why couldn't we have
started, say, with one branch
00:14:27.410 --> 00:14:31.510
of the hyperbola, 'x squared'
minus 'y squared' equals 1.
00:14:31.510 --> 00:14:35.790
Given a length 't', why couldn't
we have measured 't'
00:14:35.790 --> 00:14:37.340
off along the hyperbola?
00:14:37.340 --> 00:14:41.070
Say this way if 't' is positive,
the other way if 't'
00:14:41.070 --> 00:14:42.060
is negative.
00:14:42.060 --> 00:14:43.960
And then what we could
have done is drop the
00:14:43.960 --> 00:14:47.290
perpendicular again.
00:14:47.290 --> 00:14:48.930
And we could have
defined what?
00:14:48.930 --> 00:14:52.800
The y-coordinate to
be the hyperbolic.
00:14:52.800 --> 00:14:55.400
Well, we couldn't call it cosine
anymore because it
00:14:55.400 --> 00:14:57.510
would be confused with the
circular functions.
00:14:57.510 --> 00:15:00.430
We could have invented a name,
as we later will, called the
00:15:00.430 --> 00:15:02.810
'hyperbolic cosine'.
00:15:02.810 --> 00:15:05.000
I won't go into any more
detail on this.
00:15:05.000 --> 00:15:07.210
See, this is an abbreviation
for hyperbolic
00:15:07.210 --> 00:15:10.450
cosine, meaning this--
00:15:10.450 --> 00:15:13.390
I'm sorry, I got
this backwards.
00:15:13.390 --> 00:15:16.890
Call the x-coordinate the
hyperbolic cosine, the
00:15:16.890 --> 00:15:20.630
y-coordinate the hyperbolic
sine.
00:15:20.630 --> 00:15:22.330
You don't have to know anything
about advanced
00:15:22.330 --> 00:15:23.680
mathematics to see this.
00:15:23.680 --> 00:15:27.460
All I'm saying is, I could just
as easily have taken any
00:15:27.460 --> 00:15:32.130
geometric figure, marked off
lengths along it, taken the
00:15:32.130 --> 00:15:36.130
x-coordinates and the
y-coordinates, and seen what
00:15:36.130 --> 00:15:37.860
relationships they obey.
00:15:37.860 --> 00:15:41.090
You see, as such, there's
nothing sacred about working
00:15:41.090 --> 00:15:42.510
on a circle.
00:15:42.510 --> 00:15:45.170
Not only that, but even after
you agree to work on the
00:15:45.170 --> 00:15:47.250
circle, there are many
other ways that one
00:15:47.250 --> 00:15:47.940
could have done this.
00:15:47.940 --> 00:15:50.790
For example, someone might have
said, look it, when you
00:15:50.790 --> 00:15:54.780
take this length called 't', why
did you elect to mark it
00:15:54.780 --> 00:15:56.360
off along the circle?
00:15:56.360 --> 00:16:01.760
Why couldn't you have taken a
radius equal to 't', taken 'S'
00:16:01.760 --> 00:16:07.450
as a center, and swung an arc
that met the circle, and call
00:16:07.450 --> 00:16:10.730
this length 't'?
00:16:10.730 --> 00:16:12.960
You see, instead of measuring
along the circle, measure
00:16:12.960 --> 00:16:14.290
along the straight line.
00:16:14.290 --> 00:16:16.510
Again, you could have done
this if you wanted to.
00:16:16.510 --> 00:16:18.290
Why you would've wanted
to do this?
00:16:18.290 --> 00:16:20.500
Well, you have the same
right to do this
00:16:20.500 --> 00:16:21.550
as I had to do mine.
00:16:21.550 --> 00:16:23.650
Of course, you have to be
a little bit careful.
00:16:23.650 --> 00:16:27.350
For example, in this particular
configuration,
00:16:27.350 --> 00:16:29.420
notice that, if this is how
you're going to define your
00:16:29.420 --> 00:16:34.230
trigonometric function, your
input, your domain, has to be
00:16:34.230 --> 00:16:37.070
somewhere between 0 and 2.
00:16:37.070 --> 00:16:39.800
In other words, you cannot have
a length longer than 2,
00:16:39.800 --> 00:16:41.860
because notice that the
diameter of the
00:16:41.860 --> 00:16:43.690
circle is only 2.
00:16:43.690 --> 00:16:46.680
And therefore, if 't' were
greater than 2, when you swung
00:16:46.680 --> 00:16:49.000
an arc from the point 'S',
it wouldn't meet
00:16:49.000 --> 00:16:49.900
the circle at all.
00:16:49.900 --> 00:16:51.890
Well, that's no great
handicap.
00:16:51.890 --> 00:16:53.610
It's no great disaster.
00:16:53.610 --> 00:16:55.350
You still have the right
to make up whatever
00:16:55.350 --> 00:16:56.860
functions you want.
00:16:56.860 --> 00:17:00.170
I will try to make it clearer
why we chose these circular
00:17:00.170 --> 00:17:03.190
functions from a physical point
of view as we go along.
00:17:03.190 --> 00:17:06.790
What I thought I'd like to do
now is, having motivated, that
00:17:06.790 --> 00:17:09.400
we can invent the trigonometric
functions in
00:17:09.400 --> 00:17:13.609
terms of numbers definitions
along this circle.
00:17:13.609 --> 00:17:17.000
And coupled with the fact that,
in radian measure, you
00:17:17.000 --> 00:17:20.150
can have a very nice
identification between what's
00:17:20.150 --> 00:17:23.490
happening pictorially and what's
happening analytically,
00:17:23.490 --> 00:17:26.079
to show, for example, that in
terms of our subject called
00:17:26.079 --> 00:17:30.610
calculus, that we're pretty much
home free once we learn
00:17:30.610 --> 00:17:32.280
these basic ideas.
00:17:32.280 --> 00:17:35.170
You see, the important point
is that, in a manner of
00:17:35.170 --> 00:17:38.980
speaking, we have finished
differential calculus.
00:17:38.980 --> 00:17:41.900
We know what all the recipes
are We know what properties
00:17:41.900 --> 00:17:42.870
things have.
00:17:42.870 --> 00:17:46.180
So all of the rules that we
learned will apply to any
00:17:46.180 --> 00:17:48.600
particular type of function
that we're talking about.
00:17:48.600 --> 00:17:51.390
For example, let's suppose
we define 'f of
00:17:51.390 --> 00:17:54.120
x' to be 'sine x'.
00:17:54.120 --> 00:17:56.760
And we want to find the
derivative of 'sine x'.
00:17:56.760 --> 00:18:00.830
Notice that 'f prime of x'
evaluated at any number 'x1'
00:18:00.830 --> 00:18:02.780
has already been
defined for us.
00:18:02.780 --> 00:18:06.770
It's the limit as 'delta x'
approaches 0, 'f of 'x1 plus
00:18:06.770 --> 00:18:11.100
delta x'', minus 'f of
x1' over 'delta x'.
00:18:11.100 --> 00:18:13.520
This is true for any
function 'f'.
00:18:13.520 --> 00:18:17.935
In particular, if 'f of x' is
'sine x', all we get is what?
00:18:17.935 --> 00:18:20.880
That the derivative is the limit
as 'delta x' approaches
00:18:20.880 --> 00:18:25.510
0, sine of 'x1 plus delta
x' minus sine of
00:18:25.510 --> 00:18:27.190
'x1' over 'delta x'.
00:18:27.190 --> 00:18:29.910
Now you see, on this
particular score,
00:18:29.910 --> 00:18:31.450
nobody can fault us.
00:18:31.450 --> 00:18:34.160
This is still the basic
definition.
00:18:34.160 --> 00:18:36.750
All that happens computationally
is that, if
00:18:36.750 --> 00:18:39.200
we're not familiar with our
new functions called the
00:18:39.200 --> 00:18:42.740
trigonometric functions, we
might not know how to express
00:18:42.740 --> 00:18:47.060
sine of 'x1 plus delta x' in
a more convenient form.
00:18:47.060 --> 00:18:49.080
What do we mean by a more
convenient form?
00:18:49.080 --> 00:18:51.710
Well, notice again, as is always
the case when we take a
00:18:51.710 --> 00:18:55.750
derivative, as delta x
approaches 0, our numerator
00:18:55.750 --> 00:19:01.100
becomes 'sine x1' minus 'sine
x1', which is 0/0.
00:19:01.100 --> 00:19:05.250
And we're back to our familiar
taboo form of 0/0.
00:19:05.250 --> 00:19:07.770
Somehow or other, we're going
to have to make a refinement
00:19:07.770 --> 00:19:10.360
on our numerator that
will allow us to get
00:19:10.360 --> 00:19:13.000
rid of a 0/0 form.
00:19:13.000 --> 00:19:16.600
Well, to make a long story
short, if we happen to know
00:19:16.600 --> 00:19:18.760
the addition formula
for the sine--
00:19:18.760 --> 00:19:22.350
in other words, 'sine 'x1 plus
delta x'' is ''sine x1'
00:19:22.350 --> 00:19:26.950
'cosine delta x'', plus ''sine
delta x' 'cosine x1''--
00:19:26.950 --> 00:19:32.080
then we subtract off 'sine x1'
and divide by 'delta x', and
00:19:32.080 --> 00:19:34.040
then we factor and
collect terms.
00:19:34.040 --> 00:19:34.940
We see what?
00:19:34.940 --> 00:19:36.820
Without any knowledge
of calculus at
00:19:36.820 --> 00:19:38.300
all, but just what?
00:19:38.300 --> 00:19:40.800
By our definition of derivative,
just by our
00:19:40.800 --> 00:19:44.150
definition, coupled with
properties of the
00:19:44.150 --> 00:19:47.510
trigonometric functions, we wind
up with the fact that 'f
00:19:47.510 --> 00:19:50.440
prime of x1' is this
particular limit.
00:19:50.440 --> 00:19:53.360
Now certainly, our limit
theorems don't change.
00:19:53.360 --> 00:19:55.180
The limit of a sum is
still going to be
00:19:55.180 --> 00:19:57.570
the sum of the limits.
00:19:57.570 --> 00:19:59.470
The limit of a product
will still be the
00:19:59.470 --> 00:20:00.680
product of the limits.
00:20:00.680 --> 00:20:04.680
So all in all, what we have to
sort of do is figure out what
00:20:04.680 --> 00:20:05.940
these limits will be.
00:20:05.940 --> 00:20:09.880
Certainly, as 'delta x'
approaches 0, this will stay
00:20:09.880 --> 00:20:11.870
'cosine x1'.
00:20:11.870 --> 00:20:16.570
Certainly this will stay 'sine
x1', because 'x1' is a fixed
00:20:16.570 --> 00:20:19.040
number that doesn't depend
on 'delta x'.
00:20:19.040 --> 00:20:22.320
But notice, rather
interestingly, that both of my
00:20:22.320 --> 00:20:26.980
expressions in parentheses
happen to take on that 0/0
00:20:26.980 --> 00:20:28.580
form if we're not careful.
00:20:28.580 --> 00:20:34.250
Namely, if you replace 'delta
x' by 0, sine 0 is 0, 0/0 is
00:20:34.250 --> 00:20:37.840
0, and we run into trouble here
if we replace 'delta x'
00:20:37.840 --> 00:20:40.130
by 0, which of course
we can't do.
00:20:40.130 --> 00:20:43.480
This is the same definition
of limit as we had before.
00:20:43.480 --> 00:20:46.450
'Delta x' gets arbitrarily
close to 0, but never is
00:20:46.450 --> 00:20:47.810
allowed to get there.
00:20:47.810 --> 00:20:51.340
Well, you see, if nothing else,
this motivates why we
00:20:51.340 --> 00:20:54.330
would like to learn this
particular type of limit.
00:20:54.330 --> 00:20:56.680
In other words, what we would
like to know is, how do you--
00:20:56.680 --> 00:20:59.400
the 'delta x' symbol here
isn't that important.
00:20:59.400 --> 00:21:01.370
'Delta x' just stands
for any number.
00:21:01.370 --> 00:21:03.760
Notice that what we would like
to know is, if you take the
00:21:03.760 --> 00:21:07.490
sine of something over that same
something, and take the
00:21:07.490 --> 00:21:10.570
limit as that same something
goes to 0, we would like to
00:21:10.570 --> 00:21:12.320
know what that becomes.
00:21:12.320 --> 00:21:14.980
In a similar way, we would like
to know how to handle
00:21:14.980 --> 00:21:17.480
this quotient here, because
notice that when 'delta x' is
00:21:17.480 --> 00:21:20.040
0, cosine 0 is 1.
00:21:20.040 --> 00:21:22.460
This is 1 minus 1 over 0.
00:21:22.460 --> 00:21:25.200
It's another 0/0 form.
00:21:25.200 --> 00:21:28.556
So the problem that we're
confronted with is that, what
00:21:28.556 --> 00:21:31.140
we would like to do is to figure
out how to handle the
00:21:31.140 --> 00:21:35.360
limit of 'sine t' over 't'
as 't' approaches 0.
00:21:35.360 --> 00:21:38.440
Now, what's 't' here?
't' is a number.
00:21:38.440 --> 00:21:38.900
Remember that.
00:21:38.900 --> 00:21:40.630
This is the big pitch
I've been making.
00:21:40.630 --> 00:21:42.170
We're thinking of
't' as a number.
00:21:42.170 --> 00:21:45.650
If, on the other hand, you feel
more comfortable thinking
00:21:45.650 --> 00:21:47.800
in terms of traditional
trigonometry--
00:21:47.800 --> 00:21:50.480
and let's face it, the more
background you've had in
00:21:50.480 --> 00:21:54.780
traditional trigonometry, the
more comfortable you're going
00:21:54.780 --> 00:21:56.480
to feel using it.
00:21:56.480 --> 00:21:59.620
Let's simply agree to do this,
that if it bothers you to
00:21:59.620 --> 00:22:02.700
think of this as a length
divided by a length, et
00:22:02.700 --> 00:22:05.500
cetera, and that this is a
length or a number, let's
00:22:05.500 --> 00:22:10.460
agree that we will go back to
angles but use radian measure.
00:22:10.460 --> 00:22:11.510
Why?
00:22:11.510 --> 00:22:15.970
Because if the angle is measured
in radians, the sine
00:22:15.970 --> 00:22:21.220
of the angle 't' radians is the
same as the number, the
00:22:21.220 --> 00:22:24.050
sine, of the number 't'.
00:22:24.050 --> 00:22:27.220
Well again, here's how this
problem is tackled.
00:22:27.220 --> 00:22:30.250
What we do is we mark off the
angle of 't' radians.
00:22:30.250 --> 00:22:32.260
Remember that we have
the unit circle.
00:22:32.260 --> 00:22:37.510
And what we very cleverly do
is we catch our wedge, our
00:22:37.510 --> 00:22:40.840
circular wedge, between
two right triangles.
00:22:40.840 --> 00:22:44.400
Again, without making a big
issue over this, notice that
00:22:44.400 --> 00:22:49.060
this length is 'sine t', this
length is 'cosine t', so the
00:22:49.060 --> 00:22:52.710
area of the small triangle
is 'sine t' times
00:22:52.710 --> 00:22:55.650
'cosine t' over 2.
00:22:55.650 --> 00:22:59.500
See, 'sine t' times
'cosine t' over 2.
00:22:59.500 --> 00:23:02.630
Now, on the other hand, since
that's caught in our wedge,
00:23:02.630 --> 00:23:04.650
what is the area of our wedge?
00:23:04.650 --> 00:23:08.480
Well, since the area of the
entire circle is pi-- see, pi
00:23:08.480 --> 00:23:10.180
'R squared' and 'R' is 1--
00:23:10.180 --> 00:23:12.870
since the area of the entire
circle is pi--
00:23:12.870 --> 00:23:14.490
and we're taking what?
00:23:14.490 --> 00:23:17.290
't' of the 2pi.
00:23:17.290 --> 00:23:19.980
So there are two pi radians
in a circle.
00:23:19.980 --> 00:23:24.230
So the sector of the circle that
we have, it's 't/2pi' of
00:23:24.230 --> 00:23:25.390
the entire circle.
00:23:25.390 --> 00:23:28.490
And by the way, this is done
more rigorously and carried
00:23:28.490 --> 00:23:30.030
out in detail in the notes.
00:23:30.030 --> 00:23:33.080
Let me point out that, if we
insisted on working with
00:23:33.080 --> 00:23:36.450
degrees, instead of
't/2pi', we just
00:23:36.450 --> 00:23:40.310
would have had 't/360'.
00:23:40.310 --> 00:23:43.170
Because, you see, if we're
dealing with degrees, the
00:23:43.170 --> 00:23:47.110
entire angle and measure of the
circle is 360 degrees, and
00:23:47.110 --> 00:23:49.060
we would have had 't/360'.
00:23:49.060 --> 00:23:50.740
But here we've used the
fact that we're
00:23:50.740 --> 00:23:53.630
dealing with radians.
00:23:53.630 --> 00:23:56.670
And finally, the bigger
triangle, which includes the
00:23:56.670 --> 00:24:01.460
wedge, has, as its base, 1,
so that's the radius.
00:24:01.460 --> 00:24:05.410
And since the tangent is side
opposite over side adjacent,
00:24:05.410 --> 00:24:07.520
this length is 'tangent t'.
00:24:07.520 --> 00:24:09.980
And so what we have is what?
00:24:09.980 --> 00:24:15.200
That ''sine t' 'cosine t/2' must
be less than this, which
00:24:15.200 --> 00:24:20.220
in turn must be less than this,
multiplying through by 2
00:24:20.220 --> 00:24:25.380
and dividing through
by 'sine t'.
00:24:25.380 --> 00:24:27.780
And by the way, this hinges on
the fact that 't' is positive.
00:24:27.780 --> 00:24:30.740
Again, in our notes, we treat
the case where 't' is negative
00:24:30.740 --> 00:24:32.390
to arrive at the same result.
00:24:32.390 --> 00:24:37.000
Remembering that 'tan t' is
'sine t' over 'cosine t', we
00:24:37.000 --> 00:24:39.000
wind up with this result.
00:24:39.000 --> 00:24:44.240
And now, observing that
as 't' approaches 0,
00:24:44.240 --> 00:24:46.000
this approaches 1.
00:24:46.000 --> 00:24:48.270
This also approaches 1.
00:24:48.270 --> 00:24:51.230
And 't' over 'sine t' is caught
between these two.
00:24:51.230 --> 00:24:54.510
We get that the limit of 't'
over 'sine t' as 't'
00:24:54.510 --> 00:24:56.690
approaches 0 is 1.
00:24:56.690 --> 00:25:00.340
Now of course, since this limit
is 1, the limit of the
00:25:00.340 --> 00:25:03.930
reciprocal of this will be
the reciprocal of this.
00:25:03.930 --> 00:25:07.290
But what's very nice about the
number 1 is that it's equal to
00:25:07.290 --> 00:25:10.500
its own reciprocal.
00:25:10.500 --> 00:25:14.050
In other words, what we've now
shown is that the limit of
00:25:14.050 --> 00:25:18.400
'sine t' over 't' as 't'
approaches 0 is 1.
00:25:18.400 --> 00:25:20.710
That, as I said before,
is done in the text.
00:25:20.710 --> 00:25:22.090
We do it in our notes.
00:25:22.090 --> 00:25:25.750
But the thing that I hope this
motivates is why we want to do
00:25:25.750 --> 00:25:27.230
this in the first place.
00:25:27.230 --> 00:25:30.640
Notice that this was a limit
that we had to compute if we
00:25:30.640 --> 00:25:34.630
wanted to compute the derivative
of the sine.
00:25:34.630 --> 00:25:38.260
Now, the next thing was, how
do we handle '1 - cosine t'
00:25:38.260 --> 00:25:40.790
over 't' as 't' approaches 0?
00:25:40.790 --> 00:25:43.930
Again, leaving the details to
you to sketch in as you see
00:25:43.930 --> 00:25:46.680
fit, let me point out simply
what the mathematics
00:25:46.680 --> 00:25:48.100
involved here is.
00:25:48.100 --> 00:25:51.820
You see, what we can handle
is 'sine t' over 't'.
00:25:51.820 --> 00:25:54.610
That means that what we would
like to do is, whenever we're
00:25:54.610 --> 00:25:57.920
given an alien form, we would
somehow or other like to
00:25:57.920 --> 00:26:00.900
figure some way of factoring
a sine t over
00:26:00.900 --> 00:26:02.580
t out of this thing.
00:26:02.580 --> 00:26:06.970
When you look at '1 - cosine
t', the identity, 'sine
00:26:06.970 --> 00:26:10.040
squared' equals '1 - 'cosine
squared t'',
00:26:10.040 --> 00:26:12.240
should suggest itself.
00:26:12.240 --> 00:26:16.130
Now, how do you get from '1 -
cosine t' to '1 - 'cosine
00:26:16.130 --> 00:26:17.170
squared t''?
00:26:17.170 --> 00:26:20.800
You have to multiply
by '1 + cosine t'.
00:26:20.800 --> 00:26:24.050
And if you multiply by '1 +
cosine t' upstairs, you must
00:26:24.050 --> 00:26:27.780
multiply by '1 + cosine
t' downstairs.
00:26:27.780 --> 00:26:30.970
By the way, the only time you
can't multiply by something is
00:26:30.970 --> 00:26:33.310
when the thing is 0.
00:26:33.310 --> 00:26:35.140
You can't put that into
the denominator.
00:26:35.140 --> 00:26:39.410
Notice that 'cosine t' is
not 0 in a neighborhood
00:26:39.410 --> 00:26:40.790
of 't' equals 0.
00:26:40.790 --> 00:26:45.170
See, 'cosine t' behaves like 1
when 't' is near 0, so this is
00:26:45.170 --> 00:26:47.870
a permissible step in this
particular problem.
00:26:47.870 --> 00:26:52.880
The point is, we now factor '1 -
'cosine squared t'' as 'sine
00:26:52.880 --> 00:26:54.920
t' times 'sine t'.
00:26:54.920 --> 00:26:56.370
See, that's 'sine squared t'.
00:26:56.370 --> 00:27:01.740
We break up our 't' times
'1 + cosine t' this way.
00:27:01.740 --> 00:27:04.880
Now we know that the limit of
a product is the product of
00:27:04.880 --> 00:27:05.710
the limits.
00:27:05.710 --> 00:27:08.050
This we already know
goes to 1.
00:27:08.050 --> 00:27:12.050
And as 't' approaches 0, from
our previous limit work on the
00:27:12.050 --> 00:27:15.140
like, notice here, the limit of
a quotient is the quotient
00:27:15.140 --> 00:27:19.360
of the limits, the numerator
goes to 0, the denominator
00:27:19.360 --> 00:27:23.900
goes to 2, because as t
approaches 0, cosine 0 is 1.
00:27:23.900 --> 00:27:28.370
At any rate, that's
0/2, which is 0.
00:27:28.370 --> 00:27:31.570
And so this limit is 0.
00:27:31.570 --> 00:27:35.410
Now, at the risk of giving you
a slight headache as I take
00:27:35.410 --> 00:27:39.000
the board down here, let
me just review what
00:27:39.000 --> 00:27:40.130
it was that we did.
00:27:40.130 --> 00:27:42.410
You see, notice that, without
any knowledge of these limits
00:27:42.410 --> 00:27:45.720
at all, we were able to show
that whatever the derivative
00:27:45.720 --> 00:27:49.110
of 'sine x' was, it was this
particular thing here.
00:27:49.110 --> 00:27:53.230
Now what we've done is we've
shown that this is 1, and
00:27:53.230 --> 00:27:55.630
we've shown that this is 0.
00:27:55.630 --> 00:27:59.090
And using our limit theorems,
what we now see is what?
00:27:59.090 --> 00:28:06.650
That if 'f of x' is 'sine x', 'f
prime of x' is 'cosine x'.
00:28:10.370 --> 00:28:15.780
Let me just write that down over
here, that if 'y' equals
00:28:15.780 --> 00:28:23.620
'sine x', 'dy/dx'
is 'cosine x'.
00:28:23.620 --> 00:28:27.560
And again, notice how much of
the calculus involved here was
00:28:27.560 --> 00:28:28.350
nothing new.
00:28:28.350 --> 00:28:31.880
It goes back to the so-called
baby chapter that nobody
00:28:31.880 --> 00:28:36.690
likes, where we go back to
epsilons, deltas, you see
00:28:36.690 --> 00:28:39.430
derivatives by 'delta
x', et cetera.
00:28:39.430 --> 00:28:42.370
See, those recipes always
remain the same.
00:28:42.370 --> 00:28:45.670
What happens is, as you invent
new functions, you need a
00:28:45.670 --> 00:28:48.920
different degree of
computational sophistication
00:28:48.920 --> 00:28:50.740
to find the desired limits.
00:28:50.740 --> 00:28:53.880
By the way, once you get over
these hurdles, everything
00:28:53.880 --> 00:28:56.520
again starts to go smoothly
as before.
00:28:56.520 --> 00:28:58.460
For example, our chain rule.
00:28:58.460 --> 00:29:02.030
Suppose we have now that 'y'
equals 'sine u', where 'u' is
00:29:02.030 --> 00:29:04.220
some differentiable
function of 'x'.
00:29:04.220 --> 00:29:06.920
And we now want to
find the 'dy/dx'.
00:29:06.920 --> 00:29:09.160
Well, you see, the point is
that we know that the
00:29:09.160 --> 00:29:12.750
derivative of 'sine u' with
respect to 'u' would be
00:29:12.750 --> 00:29:15.510
'cosine u'.
00:29:15.510 --> 00:29:16.850
What we want is the derivative
of 'sine u'
00:29:16.850 --> 00:29:17.990
with respect to 'x'.
00:29:17.990 --> 00:29:21.480
And we motivate the chain rule
the same way as we did before.
00:29:21.480 --> 00:29:23.920
It happens to be that we're
dealing with the specific
00:29:23.920 --> 00:29:26.035
value called sine,
but it could've
00:29:26.035 --> 00:29:27.000
been any old function.
00:29:27.000 --> 00:29:30.580
How would you differentiate 'f
of u' with respect to 'x' if
00:29:30.580 --> 00:29:33.550
you know how to differentiate 'f
of u' with respect to 'u'?
00:29:33.550 --> 00:29:35.430
And the answer is, you would
just differentiate with
00:29:35.430 --> 00:29:39.060
respect to 'u', and multiply
that by a derivative of 'u'
00:29:39.060 --> 00:29:40.490
with respect to 'x'.
00:29:40.490 --> 00:29:42.350
In other words, we get
the result what?
00:29:42.350 --> 00:29:46.990
That since 'dy/du' is 'cosine
u', we get that the derivative
00:29:46.990 --> 00:29:49.570
of 'sine u' with respect
to 'x' is
00:29:49.570 --> 00:29:52.000
'cosine u' times 'du/dx'.
00:29:52.000 --> 00:29:55.690
And by the way, one rather nice
application of this is
00:29:55.690 --> 00:29:58.740
that it gives us a very quick
way of getting the derivative
00:29:58.740 --> 00:30:00.510
of 'cosine x'.
00:30:00.510 --> 00:30:04.870
After all, our basic identity
is that 'cosine x' is sine
00:30:04.870 --> 00:30:06.900
pi/2 minus 'x'.
00:30:06.900 --> 00:30:09.910
Again, a number or an
angle, either way.
00:30:09.910 --> 00:30:12.870
As long as the measurement is
in radians, it makes no
00:30:12.870 --> 00:30:15.360
difference whether you think of
this as being an angle or
00:30:15.360 --> 00:30:16.320
being a number.
00:30:16.320 --> 00:30:18.000
The answer will be the same.
00:30:18.000 --> 00:30:19.320
The idea is this.
00:30:19.320 --> 00:30:21.410
To take the derivative of
'cosine x' with respect to
00:30:21.410 --> 00:30:25.430
'x', all I have to differentiate
is sine pi/2
00:30:25.430 --> 00:30:27.420
minus 'x' with respect to 'x'.
00:30:27.420 --> 00:30:28.850
But I know how to do that.
00:30:28.850 --> 00:30:35.290
Namely, the derivative of sine
pi/2 minus 'x' is cosine pi/2
00:30:35.290 --> 00:30:39.680
minus 'x,' and by the chain
rule, times the derivative of
00:30:39.680 --> 00:30:41.230
this with respect to 'x'.
00:30:41.230 --> 00:30:43.370
Well, pi/2 is a constant.
00:30:43.370 --> 00:30:46.580
The derivative of 'minus
x' is minus 1.
00:30:46.580 --> 00:30:51.230
And then, remembering that the
cosine of pi/2 minus 'x' is
00:30:51.230 --> 00:30:55.110
'sine x', I now have the result
that the derivative of
00:30:55.110 --> 00:30:58.460
the cosine is minus the sine.
00:30:58.460 --> 00:31:01.070
And again, I can do all sorts
of things this way.
00:31:01.070 --> 00:31:04.350
If I want the derivative of a
tangent, I could write tangent
00:31:04.350 --> 00:31:06.040
as sine over cosine.
00:31:06.040 --> 00:31:07.450
Use the quotient rule.
00:31:07.450 --> 00:31:10.330
You see, as soon as I make one
breakthrough, all of the
00:31:10.330 --> 00:31:13.260
previous body of calculus
comes to my
00:31:13.260 --> 00:31:15.660
rescue, so to speak.
00:31:15.660 --> 00:31:18.990
By the way, what I'd like to do
now is point out why, from
00:31:18.990 --> 00:31:23.440
a physical point of view, we
like circular functions to be
00:31:23.440 --> 00:31:26.570
independent of angles
and the like.
00:31:26.570 --> 00:31:29.430
With the results that we've
derived so far, it's rather
00:31:29.430 --> 00:31:31.650
easy to derive one
more result.
00:31:31.650 --> 00:31:34.390
Namely, let's assume
that a particle is
00:31:34.390 --> 00:31:35.980
moving along the x-axis--
00:31:35.980 --> 00:31:38.030
I'm going to start with the
answer, sort of, and work
00:31:38.030 --> 00:31:38.900
backwards--
00:31:38.900 --> 00:31:42.410
according to the rule, 'x'
equals 'sine kt', where 't' is
00:31:42.410 --> 00:31:44.810
time and 'k' is a constant.
00:31:44.810 --> 00:31:47.680
Then its speed, 'dx/dt',
is what?
00:31:47.680 --> 00:31:51.150
It's the derivative of 'sine
kt', which is 'cosine kt',
00:31:51.150 --> 00:31:54.090
times the derivative of what's
inside with respect to 't'.
00:31:54.090 --> 00:31:56.960
In other words, it's
'k cosine kt'.
00:31:56.960 --> 00:32:00.100
The second derivative of 'x'
with respect to 't', namely,
00:32:00.100 --> 00:32:01.650
the acceleration is what?
00:32:01.650 --> 00:32:04.080
How do you differentiate
the cosine?
00:32:04.080 --> 00:32:07.850
The derivative of the cosine
is minus the sine.
00:32:07.850 --> 00:32:10.490
By the chain rule, I must
multiply by the derivative of
00:32:10.490 --> 00:32:13.830
'kt' with respect to 't', which
gives me another factor
00:32:13.830 --> 00:32:15.450
of 't' over here.
00:32:15.450 --> 00:32:24.780
Remembering that 'x' equals
'sine kt', I arrive at this
00:32:24.780 --> 00:32:27.370
particular so-called
differential equation.
00:32:27.370 --> 00:32:28.510
And what does this say?
00:32:28.510 --> 00:32:32.620
It says that 'd2x/ dt squared',
the acceleration, is
00:32:32.620 --> 00:32:35.090
proportional to the
displacement, the distance
00:32:35.090 --> 00:32:37.580
traveled, but in the
opposite direction.
00:32:37.580 --> 00:32:40.900
You see, 'k squared' can't
be negative, so 'minus 'k
00:32:40.900 --> 00:32:42.940
squared'' can't be positive.
00:32:42.940 --> 00:32:43.810
This says what?
00:32:43.810 --> 00:32:46.510
The acceleration is proportional
to the
00:32:46.510 --> 00:32:49.610
displacement, but in the
opposite direction.
00:32:49.610 --> 00:32:54.800
Does that problem require any
knowledge of angles to solve?
00:32:54.800 --> 00:32:56.470
Notice that this is a perfectly
00:32:56.470 --> 00:32:57.950
good physical problem.
00:32:57.950 --> 00:33:00.940
It's known as simple
harmonic motion.
00:33:00.940 --> 00:33:04.310
And all I'm trying to have you
see is that, by inventing the
00:33:04.310 --> 00:33:08.960
circular functions in the proper
way, not only can we do
00:33:08.960 --> 00:33:12.660
their calculus, but even more
importantly, if we reverse
00:33:12.660 --> 00:33:17.220
these steps, for example, we
can show that, to solve the
00:33:17.220 --> 00:33:20.920
physical problem of simple
harmonic motion, we have to
00:33:20.920 --> 00:33:24.890
know the so-called circular
trigonometric functions.
00:33:24.890 --> 00:33:29.030
And this is a far cry, you see,
from using trigonometry
00:33:29.030 --> 00:33:32.385
in the sense that the surveyor
uses trigonometry.
00:33:32.385 --> 00:33:34.740
You see, this ties up with my
initial hang-up that I was
00:33:34.740 --> 00:33:37.110
telling you about at the
beginning of the program.
00:33:37.110 --> 00:33:40.270
By the way, in closing, I should
also make reference to
00:33:40.270 --> 00:33:42.640
something that we pointed out
in our last lecture, namely,
00:33:42.640 --> 00:33:44.780
inverse differentiation.
00:33:44.780 --> 00:33:48.200
Keep in mind, also, that as you
read the calculus of the
00:33:48.200 --> 00:33:51.440
trigonometric functions, that
the fact that we know that the
00:33:51.440 --> 00:33:55.230
derivative of sine u with
respect to 'u' was 'cosine u'
00:33:55.230 --> 00:33:58.560
gives us, with a switch in
emphasis, the result that the
00:33:58.560 --> 00:34:03.290
integral 'cosine u', 'du' is
'sine u' plus a constant.
00:34:03.290 --> 00:34:06.060
And in a similar way, since the
derivative of cosine is
00:34:06.060 --> 00:34:09.070
minus the sine, the integral
of 'sine u' with respect to
00:34:09.070 --> 00:34:12.590
'u' is 'minus cosine
u' plus a constant.
00:34:12.590 --> 00:34:13.280
Be careful.
00:34:13.280 --> 00:34:16.420
Notice how the sines
can screw you up.
00:34:16.420 --> 00:34:18.310
Namely, they're in the opposite
sense when you're
00:34:18.310 --> 00:34:20.929
integrating as when you
were differentiating.
00:34:20.929 --> 00:34:24.239
But again, these are the details
which I expect you can
00:34:24.239 --> 00:34:26.909
have come out in the
wash rather nicely.
00:34:26.909 --> 00:34:29.510
We can continue on this way,
from knowing how to
00:34:29.510 --> 00:34:32.389
differentiate 'sine x'
to the nth power.
00:34:32.389 --> 00:34:38.380
Namely, it's 'n - 1' 'x', times
the derivative of 'sine
00:34:38.380 --> 00:34:40.900
x', which is 'cosine x'.
00:34:40.900 --> 00:34:42.510
We don't want this in here.
00:34:42.510 --> 00:34:44.050
That's a differential form.
00:34:44.050 --> 00:34:46.790
Without going into any detail
here, notice that a
00:34:46.790 --> 00:34:50.690
modification of this shows us
that, if we differentiate
00:34:50.690 --> 00:34:53.630
this, we wind up with this.
00:34:53.630 --> 00:34:57.730
We could now take the time, if
this were the proper place, to
00:34:57.730 --> 00:35:01.430
develop all sorts of derivative
formulas and
00:35:01.430 --> 00:35:02.650
integral formulas.
00:35:02.650 --> 00:35:06.000
As you study your study guide,
you will notice that the
00:35:06.000 --> 00:35:11.000
lesson after this is concerned
with the calculus of the
00:35:11.000 --> 00:35:12.650
circular functions.
00:35:12.650 --> 00:35:16.190
My feeling is is that, with
this as background, a very
00:35:16.190 --> 00:35:20.390
good review of the previous part
of the course will be to
00:35:20.390 --> 00:35:25.350
see how much of this you can
apply on your own to these new
00:35:25.350 --> 00:35:28.420
functions called the
circular functions.
00:35:28.420 --> 00:35:31.810
Next time, we will talk, as
you may be able to guess,
00:35:31.810 --> 00:35:34.200
about the inverse circular
functions
00:35:34.200 --> 00:35:35.800
and why they're important.
00:35:35.800 --> 00:35:37.340
But until next time, goodbye.
00:35:40.240 --> 00:35:43.440
Funding for the publication of
this video was provided by the
00:35:43.440 --> 00:35:47.490
Gabriella and Paul Rosenbaum
Foundation.
00:35:47.490 --> 00:35:51.660
Help OCW continue to provide
free and open access to MIT
00:35:51.660 --> 00:35:55.860
courses by making a donation
at ocw.mit.edu/donate.