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PROFESSOR: Hi.
11
00:00:30,870 --> 00:00:33,840
Today's lesson, well, I
settled for the title,
12
00:00:33,840 --> 00:00:36,440
"Circular Functions." But I
guess it could have been
13
00:00:36,440 --> 00:00:38,280
called a lot of different
things.
14
00:00:38,280 --> 00:00:40,530
It could've been called
'Trigonometry without
15
00:00:40,530 --> 00:00:41,490
Triangles'.
16
00:00:41,490 --> 00:00:44,740
It could have been called
'Trigonometry Revisited'.
17
00:00:44,740 --> 00:00:48,450
And the whole point is that much
of what today's lecture
18
00:00:48,450 --> 00:00:52,280
hinges on is a hang-up that
bothered me, and which I think
19
00:00:52,280 --> 00:00:55,220
may bother you and is worthwhile
discussing.
20
00:00:55,220 --> 00:00:57,570
I remember, when I was in
high school, I asked my
21
00:00:57,570 --> 00:00:59,860
trigonometry teacher, why
would I have to know
22
00:00:59,860 --> 00:01:00,720
trigonometry?
23
00:01:00,720 --> 00:01:03,280
And his answer was,
surveyors use it.
24
00:01:03,280 --> 00:01:06,030
And at that particular time, I
didn't know what I was going
25
00:01:06,030 --> 00:01:08,940
to be, but I knew what
I wasn't going to be.
26
00:01:08,940 --> 00:01:10,440
I wasn't going to
be a surveyor.
27
00:01:10,440 --> 00:01:13,690
And I kind of took the course
kind of lightly, and really
28
00:01:13,690 --> 00:01:17,300
got clobbered a year or two
later when I got into calculus
29
00:01:17,300 --> 00:01:18,680
and physics courses.
30
00:01:18,680 --> 00:01:22,230
So what I would like to do
today is to introduce the
31
00:01:22,230 --> 00:01:26,450
notion of what we call circular
functions, and point
32
00:01:26,450 --> 00:01:30,630
out what the connection is
between these and the
33
00:01:30,630 --> 00:01:33,680
trigonometric functions that we
learned when we studied the
34
00:01:33,680 --> 00:01:36,890
subject that we call
trigonometry, and which might
35
00:01:36,890 --> 00:01:39,890
better have been called
numerical geometry.
36
00:01:39,890 --> 00:01:41,890
Let me get to the point
right away.
37
00:01:41,890 --> 00:01:44,970
Let's imagine that I say
circular functions to you.
38
00:01:44,970 --> 00:01:47,280
I think it's rather natural
that, as soon as I say that,
39
00:01:47,280 --> 00:01:48,270
you think of a circle.
40
00:01:48,270 --> 00:01:51,480
And because you think of a
circle, let me draw a circle
41
00:01:51,480 --> 00:01:55,660
here, and let me assume that the
radius of the circle is 1.
42
00:01:55,660 --> 00:01:59,660
In other words, I have the
circle here, 'x squared' plus
43
00:01:59,660 --> 00:02:02,750
'y squared' equals 1.
44
00:02:02,750 --> 00:02:04,250
Now, the thing is this.
45
00:02:04,250 --> 00:02:05,620
When I talk about--
46
00:02:05,620 --> 00:02:08,430
And I'm assuming now that you
are familiar with the
47
00:02:08,430 --> 00:02:11,550
trigonometric functions in
the traditional sense.
48
00:02:11,550 --> 00:02:15,030
And in fact, the first section
of our supplementary notes in
49
00:02:15,030 --> 00:02:19,190
the reading material that goes
with the present lecture takes
50
00:02:19,190 --> 00:02:21,050
care of the fact that, if you
don't recall some of these
51
00:02:21,050 --> 00:02:24,210
things too well, there's ample
opportunity for refreshing
52
00:02:24,210 --> 00:02:26,580
your minds and getting
some review in here.
53
00:02:26,580 --> 00:02:28,280
But the idea is something
like this.
54
00:02:28,280 --> 00:02:31,610
When we're talking about
calculus, we talk about
55
00:02:31,610 --> 00:02:33,710
functions of a real variable.
56
00:02:33,710 --> 00:02:37,240
We are assuming that our
functions have the property
57
00:02:37,240 --> 00:02:41,890
that the domain is a set of
suitably chosen real numbers,
58
00:02:41,890 --> 00:02:45,120
and the image is a suitably
chosen set of real numbers.
59
00:02:45,120 --> 00:02:48,290
We do not think of inputs
as being angles and
60
00:02:48,290 --> 00:02:49,530
things of this type.
61
00:02:49,530 --> 00:02:52,730
And so the question is, how can
we define, for example--
62
00:02:52,730 --> 00:02:54,100
let's call it the
'sine machine'.
63
00:02:54,100 --> 00:02:55,510
Let me come down here.
64
00:02:55,510 --> 00:02:56,700
I'll call it the
'sine machine'.
65
00:02:56,700 --> 00:03:00,590
If the input is the number 't',
I want the output, say,
66
00:03:00,590 --> 00:03:01,810
to be 'sine t'.
67
00:03:01,810 --> 00:03:07,300
But you see, now I'm talking
about a number, not an angle.
68
00:03:07,300 --> 00:03:12,020
Well, one way of doing this
thing visually is the old idea
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00:03:12,020 --> 00:03:13,250
of the number line.
70
00:03:13,250 --> 00:03:17,500
Let us think of a number as
being a length, the same as we
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00:03:17,500 --> 00:03:19,700
do in coordinate geometry.
72
00:03:19,700 --> 00:03:22,680
We knock off lengths along the
x-axis and the y-axis.
73
00:03:22,680 --> 00:03:25,850
Let me think of 't'
as being a length.
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00:03:25,850 --> 00:03:32,540
As such, I can take 't' and lay
it off along my circle in
75
00:03:32,540 --> 00:03:36,250
such a way that the length
originates at 'S' and
76
00:03:36,250 --> 00:03:39,430
terminates, shall we say,
at some point 'P' whose
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00:03:39,430 --> 00:03:41,710
coordinates are 'x' and 'y'.
78
00:03:41,710 --> 00:03:43,050
Now, notice what I'm
saying here.
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00:03:43,050 --> 00:03:47,190
I lay the length off along
the circumference.
80
00:03:47,190 --> 00:03:50,360
I'll talk more about that
a little bit later.
81
00:03:50,360 --> 00:03:52,090
Now, so far, so good.
82
00:03:52,090 --> 00:03:54,970
No mention of the word "angle"
here or anything like this.
83
00:03:54,970 --> 00:03:58,340
Now, wherever t terminates-- and
again, conventions here,
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00:03:58,340 --> 00:04:01,830
if 't' is positive, I lay if
off along the circle in the
85
00:04:01,830 --> 00:04:04,580
so-called positive direction,
namely, what?
86
00:04:04,580 --> 00:04:05,880
Counter-clockwise.
87
00:04:05,880 --> 00:04:09,710
If 't' is negative, I'll lay
it off in the clockwise
88
00:04:09,710 --> 00:04:11,030
direction, et cetera.
89
00:04:11,030 --> 00:04:13,220
The usual trigonometric
conventions.
90
00:04:13,220 --> 00:04:18,579
Now what I do is is, at the
point 'P', I drop a
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00:04:18,579 --> 00:04:19,829
perpendicular.
92
00:04:19,829 --> 00:04:23,050
93
00:04:23,050 --> 00:04:30,020
And I define the sine of 't'
to be the length, 'PR', and
94
00:04:30,020 --> 00:04:34,150
the cosine of 'P' to be
the length, 'OR'.
95
00:04:34,150 --> 00:04:36,500
In other words, I could
write that like this.
96
00:04:36,500 --> 00:04:40,540
I could write down that I'm
defining 'sine t' to be the
97
00:04:40,540 --> 00:04:45,010
length of 'RP' in that
direction, meaning, of course,
98
00:04:45,010 --> 00:04:48,600
that this is just a fancy way of
saying that the sine of 't'
99
00:04:48,600 --> 00:04:52,200
will just be the y-coordinate
of the point at which the
100
00:04:52,200 --> 00:04:55,260
length 't' terminates
on the circle.
101
00:04:55,260 --> 00:05:00,760
And in a similar way, 'cosine t'
will be the directed length
102
00:05:00,760 --> 00:05:03,950
from 'O' to 'R', or more
conventionally, the
103
00:05:03,950 --> 00:05:05,400
x-coordinate.
104
00:05:05,400 --> 00:05:07,730
Now, notice I can do this
with any length.
105
00:05:07,730 --> 00:05:11,940
Whatever length I'm given, I
just mark this length off.
106
00:05:11,940 --> 00:05:13,120
It's a finite length.
107
00:05:13,120 --> 00:05:14,760
Eventually, it has
to terminate some
108
00:05:14,760 --> 00:05:16,000
place on the circle.
109
00:05:16,000 --> 00:05:19,590
Wherever it terminates, the
x-coordinate of the point of
110
00:05:19,590 --> 00:05:23,280
termination is called the
cosine of 't', and the
111
00:05:23,280 --> 00:05:25,960
y-coordinate is called
the sine of 't'.
112
00:05:25,960 --> 00:05:29,500
And notice that, in this way,
both the sine and the cosine
113
00:05:29,500 --> 00:05:35,246
are functions which map real
numbers into real numbers.
114
00:05:35,246 --> 00:05:37,070
So that part, I hope,
is clear.
115
00:05:37,070 --> 00:05:39,910
116
00:05:39,910 --> 00:05:43,970
Notice again, I can mimic
the usual traditional
117
00:05:43,970 --> 00:05:45,130
trigonometry.
118
00:05:45,130 --> 00:05:48,880
I can define the tangent of t
to be the number 'sine t',
119
00:05:48,880 --> 00:05:51,770
divided by the number 'cosine
t', et cetera.
120
00:05:51,770 --> 00:05:55,450
And I'll leave those details
to the reading material.
121
00:05:55,450 --> 00:05:58,870
I can ascertain rather
interesting results the same
122
00:05:58,870 --> 00:06:03,520
way as I could in regular
traditional trigonometry.
123
00:06:03,520 --> 00:06:06,650
In fact, I can get some certain
results very nicely.
124
00:06:06,650 --> 00:06:09,230
I remember, for example--
125
00:06:09,230 --> 00:06:10,550
Well, I won't even
go into these.
126
00:06:10,550 --> 00:06:14,460
But how did you talk about the
sine of 0 when one talked
127
00:06:14,460 --> 00:06:16,230
about traditional
trigonometry?
128
00:06:16,230 --> 00:06:20,220
How did you embed a 0-degree
angle into a triangle, and
129
00:06:20,220 --> 00:06:21,320
things of this type.
130
00:06:21,320 --> 00:06:24,130
Notice that in terms of
my tradition here--
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00:06:24,130 --> 00:06:26,500
and we'll summarize these
results in a minute-- but
132
00:06:26,500 --> 00:06:30,510
notice, for example, that the
sine of 0 comes out to be 0
133
00:06:30,510 --> 00:06:34,790
very nicely, because when
't' is 0, the length 0
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00:06:34,790 --> 00:06:36,630
terminates at 'S'.
135
00:06:36,630 --> 00:06:38,690
'S' is on the x-axis.
136
00:06:38,690 --> 00:06:41,990
That makes, what?
'y' equal to 0.
137
00:06:41,990 --> 00:06:46,900
Notice also that, if the radius
of my circle is 1, the
138
00:06:46,900 --> 00:06:49,200
circumference is 2 pi.
139
00:06:49,200 --> 00:06:52,530
So for example, what I usually
think of a 90-degree angle
140
00:06:52,530 --> 00:06:55,250
would be the length pi/2.
141
00:06:55,250 --> 00:06:58,370
And without making any fuss over
this, again, leaving most
142
00:06:58,370 --> 00:07:01,450
of the details to the reading
and to the simplicity of just
143
00:07:01,450 --> 00:07:05,420
plugging these things in, we
arrive at these rather
144
00:07:05,420 --> 00:07:07,430
familiar results.
145
00:07:07,430 --> 00:07:12,460
We also get, very quickly, in
addition to these results,
146
00:07:12,460 --> 00:07:15,950
things like the fundamental
result that we always like
147
00:07:15,950 --> 00:07:17,520
with trigonometric functions.
148
00:07:17,520 --> 00:07:21,890
That's 'sine squared t' plus
'cosine squared t' is 1.
149
00:07:21,890 --> 00:07:23,310
And how do we know that?
150
00:07:23,310 --> 00:07:27,620
Remember that 'cosine t' was
just another name for the
151
00:07:27,620 --> 00:07:29,280
x-coordinate at which the point
152
00:07:29,280 --> 00:07:30,670
terminated on the circle.
153
00:07:30,670 --> 00:07:32,610
In other words, notice
that 'cosine
154
00:07:32,610 --> 00:07:34,970
squared t' is 'x squared'.
155
00:07:34,970 --> 00:07:37,510
'Sine squared t'
is 'y squared'.
156
00:07:37,510 --> 00:07:41,310
The x-coordinate and the
y-coordinate are related by
157
00:07:41,310 --> 00:07:43,350
the fact that, what?
158
00:07:43,350 --> 00:07:47,930
The sum of the squares to be on
the circle is equal to 1.
159
00:07:47,930 --> 00:07:53,380
We could even graph 'sine t'
without any problem at all.
160
00:07:53,380 --> 00:07:58,280
Namely, we observe that when
't' is 0, 'sine t' is 0.
161
00:07:58,280 --> 00:08:04,250
Notice that as we go along the
circle, the sine increases up
162
00:08:04,250 --> 00:08:10,510
until we get to pi/2, at which
it peaks at 1, then decreases
163
00:08:10,510 --> 00:08:13,330
at pi, back down to 0.
164
00:08:13,330 --> 00:08:15,910
And if that's giving you trouble
to follow, let's
165
00:08:15,910 --> 00:08:18,480
simply come back to our diagram
to make sure that we
166
00:08:18,480 --> 00:08:20,580
understand this.
167
00:08:20,580 --> 00:08:25,200
In other words, all we're saying
is, as 't' gets longer,
168
00:08:25,200 --> 00:08:31,240
its y-coordinate increases from
0 to a maximum of 1, when
169
00:08:31,240 --> 00:08:33,000
the particle was over here.
170
00:08:33,000 --> 00:08:37,340
Then, as 't' goes from to pi/2
to pi, the length of the
171
00:08:37,340 --> 00:08:42,770
y-coordinate decreases until
it again becomes 0.
172
00:08:42,770 --> 00:08:46,740
And again, without making much
more ado over this, we get the
173
00:08:46,740 --> 00:08:51,490
usual curve that we associate
with the sine function even
174
00:08:51,490 --> 00:08:54,860
when we thought of it
as a traditional
175
00:08:54,860 --> 00:08:56,240
trigonometric problem.
176
00:08:56,240 --> 00:08:59,470
But the major point that I want
you to see right now--
177
00:08:59,470 --> 00:09:01,800
and we won't worry about
why I want to do this--
178
00:09:01,800 --> 00:09:05,650
I can define the trigonometric
functions in such a way that
179
00:09:05,650 --> 00:09:09,150
their domains are real numbers
rather than angles.
180
00:09:09,150 --> 00:09:13,700
And in fact, this is the main
reason why people invented the
181
00:09:13,700 --> 00:09:15,780
notion of radian measure.
182
00:09:15,780 --> 00:09:18,940
Let me see if I can't make that
a little bit clearer,
183
00:09:18,940 --> 00:09:20,780
once and for all.
184
00:09:20,780 --> 00:09:22,270
You see, the question is this.
185
00:09:22,270 --> 00:09:23,990
Let's suppose I'm talking--
186
00:09:23,990 --> 00:09:26,060
Oh, let me give you some
letters over here.
187
00:09:26,060 --> 00:09:28,580
We'll put a 'Q' over here.
188
00:09:28,580 --> 00:09:32,030
Let's talk about angle, 'QOS'.
189
00:09:32,030 --> 00:09:35,330
190
00:09:35,330 --> 00:09:36,560
That's a right angle.
191
00:09:36,560 --> 00:09:39,790
It's 1/4 of a rotation
of the circle.
192
00:09:39,790 --> 00:09:43,570
Now, the question that I have
in mind is, if something is
193
00:09:43,570 --> 00:09:46,250
1/4 of a rotation, why do you
need two different ways of
194
00:09:46,250 --> 00:09:47,310
saying that?
195
00:09:47,310 --> 00:09:52,260
Why do we have to say it's 90
degrees or pi/2 radians, and
196
00:09:52,260 --> 00:09:56,270
bring in a new measure when we
already have another way of
197
00:09:56,270 --> 00:09:59,250
measuring circles, angles
of circles?
198
00:09:59,250 --> 00:10:01,020
The idea is something
like this.
199
00:10:01,020 --> 00:10:05,430
Let's again mimic the idea of
taking the length 't' and
200
00:10:05,430 --> 00:10:07,820
laying it off along the
circle like this.
201
00:10:07,820 --> 00:10:11,190
Now, here's the idea.
202
00:10:11,190 --> 00:10:15,040
Remember, the radius of
this circle is 1.
203
00:10:15,040 --> 00:10:20,660
So notice that 'PR', in other
words, this y-coordinate, is
204
00:10:20,660 --> 00:10:23,750
what we call, by definition,
the sine of 't'.
205
00:10:23,750 --> 00:10:28,030
In other words, just above we
said that 'sine t' was the
206
00:10:28,030 --> 00:10:31,250
length 'R' to 'P' in
that direction.
207
00:10:31,250 --> 00:10:34,160
Now, the point is, I said
disregard traditional
208
00:10:34,160 --> 00:10:36,860
trigonometry, but we can't
really disregard it.
209
00:10:36,860 --> 00:10:38,410
It exists.
210
00:10:38,410 --> 00:10:41,480
For the person who's had
traditional trigonometry, how
211
00:10:41,480 --> 00:10:47,580
would he tend to look at this
length divided by this length?
212
00:10:47,580 --> 00:10:49,930
He would think of that
as being what?
213
00:10:49,930 --> 00:10:52,610
It's side opposite
over hypotenuse.
214
00:10:52,610 --> 00:10:54,620
That also suggests sine.
215
00:10:54,620 --> 00:10:55,890
And the sine of what?
216
00:10:55,890 --> 00:10:59,570
Well, the sine of what
angle this is.
217
00:10:59,570 --> 00:11:00,940
Now, the thing is this.
218
00:11:00,940 --> 00:11:04,530
Somehow or other, to avoid
ambiguity, if we could have
219
00:11:04,530 --> 00:11:07,520
called whatever measure this
angle was measured in terms
220
00:11:07,520 --> 00:11:11,530
of, if we could have called
that unit 't', then notice
221
00:11:11,530 --> 00:11:15,550
that the sine of the angle 't'
would have been numerically
222
00:11:15,550 --> 00:11:18,760
the same as the sine
of the number 't'.
223
00:11:18,760 --> 00:11:22,140
And again, if this seems like a
hard point to understand, we
224
00:11:22,140 --> 00:11:24,950
explore this in great
detail in our notes.
225
00:11:24,950 --> 00:11:26,670
But the idea is this.
226
00:11:26,670 --> 00:11:30,760
You see, somehow or other, if
'sine t' is going to have two
227
00:11:30,760 --> 00:11:33,620
different meanings, we would
like to make sure that we pick
228
00:11:33,620 --> 00:11:37,470
the kind of a unit where it
makes no difference whether
229
00:11:37,470 --> 00:11:40,130
you're thinking of 't' as being
a number or thinking of
230
00:11:40,130 --> 00:11:41,960
't' as being a length.
231
00:11:41,960 --> 00:11:45,370
For example, suppose I now
invent the word "radian" to
232
00:11:45,370 --> 00:11:47,310
mean the following.
233
00:11:47,310 --> 00:11:49,910
An angle is said to
have 't' radians.
234
00:11:49,910 --> 00:11:53,920
If, when made the central angle
of a unit circle, a
235
00:11:53,920 --> 00:11:59,180
circle whose radius is 1, it
subtends an arc whose length
236
00:11:59,180 --> 00:12:02,270
is 't' units of length.
237
00:12:02,270 --> 00:12:05,570
See, in other words, I would
define the measure called
238
00:12:05,570 --> 00:12:09,690
radians so that an angle of
't' radians intercepts the
239
00:12:09,690 --> 00:12:11,490
length 't' over here.
240
00:12:11,490 --> 00:12:16,210
In that way, 'sine t' is
unambiguous whether you're
241
00:12:16,210 --> 00:12:19,005
talking about an angle
or a length.
242
00:12:19,005 --> 00:12:22,860
For example, when I say
the sine of pi/1
243
00:12:22,860 --> 00:12:26,150
radians, what do I mean?
244
00:12:26,150 --> 00:12:29,280
I mean the angle which is the
sine of the angle which
245
00:12:29,280 --> 00:12:34,350
intercepts a length, an
arc, pi/2 units long.
246
00:12:34,350 --> 00:12:36,570
Well, see, pi/2 is
this length.
247
00:12:36,570 --> 00:12:40,630
I'm now talking about
this angle here.
248
00:12:40,630 --> 00:12:43,640
And the sine, therefore, of
pi/2 radians, in terms of
249
00:12:43,640 --> 00:12:46,530
classical trigonometry, is 1.
250
00:12:46,530 --> 00:12:51,040
But that's also what the sine
of a number pi/2 was.
251
00:12:51,040 --> 00:12:53,890
This explains the convention
that one says when one uses
252
00:12:53,890 --> 00:12:56,240
radians, you can leave
the label off.
253
00:12:56,240 --> 00:12:59,870
All we're saying is that, if
we had used degrees, there
254
00:12:59,870 --> 00:13:01,900
would have been an ambiguity.
255
00:13:01,900 --> 00:13:08,120
Certainly, the sine of 3
degrees is not the same
256
00:13:08,120 --> 00:13:10,490
as the sine of 3.
257
00:13:10,490 --> 00:13:13,970
You see, 3 degrees is a
rather small angle.
258
00:13:13,970 --> 00:13:19,550
But 3 is a rather great length
when you're talking about the
259
00:13:19,550 --> 00:13:20,740
arc of the unit circle here.
260
00:13:20,740 --> 00:13:26,830
Remember, 1/2 circle is pi units
long, so 3 would be just
261
00:13:26,830 --> 00:13:28,000
about this long.
262
00:13:28,000 --> 00:13:32,180
In other words, notice that 3
radians and 3 degrees are
263
00:13:32,180 --> 00:13:33,240
entirely different things.
264
00:13:33,240 --> 00:13:34,280
But the beauty is what?
265
00:13:34,280 --> 00:13:38,130
That if we agreed to use radian
measure, then we have
266
00:13:38,130 --> 00:13:41,230
no ambiguity when we talk
about the sine.
267
00:13:41,230 --> 00:13:44,670
The sine of the number 't' will
equal the sine of the
268
00:13:44,670 --> 00:13:46,170
angle 't' radians.
269
00:13:46,170 --> 00:13:49,490
The cosine of a number 't' will
equal the cosine of the
270
00:13:49,490 --> 00:13:51,360
angle 't' radians.
271
00:13:51,360 --> 00:13:53,960
In a certain sense, it was
analogous to when we talked
272
00:13:53,960 --> 00:13:57,460
about the derivative 'dy/dx',
then wanted to define
273
00:13:57,460 --> 00:14:01,700
differentials 'dy' and 'dx'
separately, so that 'dy'
274
00:14:01,700 --> 00:14:05,210
divided by 'dx' would be the
same as 'dy/dx', that we
275
00:14:05,210 --> 00:14:11,340
wanted to avoid any ambiguity
where the same symbol could be
276
00:14:11,340 --> 00:14:13,400
interpreted in two different
ways to give
277
00:14:13,400 --> 00:14:14,840
two different answers.
278
00:14:14,840 --> 00:14:18,010
By the way, again, there is
nothing sacred about our
279
00:14:18,010 --> 00:14:21,000
choice of why we pick
circular functions.
280
00:14:21,000 --> 00:14:23,690
We could have picked hyperbolic
functions.
281
00:14:23,690 --> 00:14:27,410
Namely, why couldn't we have
started, say, with one branch
282
00:14:27,410 --> 00:14:31,510
of the hyperbola, 'x squared'
minus 'y squared' equals 1.
283
00:14:31,510 --> 00:14:35,790
Given a length 't', why couldn't
we have measured 't'
284
00:14:35,790 --> 00:14:37,340
off along the hyperbola?
285
00:14:37,340 --> 00:14:41,070
Say this way if 't' is positive,
the other way if 't'
286
00:14:41,070 --> 00:14:42,060
is negative.
287
00:14:42,060 --> 00:14:43,960
And then what we could
have done is drop the
288
00:14:43,960 --> 00:14:47,290
perpendicular again.
289
00:14:47,290 --> 00:14:48,930
And we could have
defined what?
290
00:14:48,930 --> 00:14:52,800
The y-coordinate to
be the hyperbolic.
291
00:14:52,800 --> 00:14:55,400
Well, we couldn't call it cosine
anymore because it
292
00:14:55,400 --> 00:14:57,510
would be confused with the
circular functions.
293
00:14:57,510 --> 00:15:00,430
We could have invented a name,
as we later will, called the
294
00:15:00,430 --> 00:15:02,810
'hyperbolic cosine'.
295
00:15:02,810 --> 00:15:05,000
I won't go into any more
detail on this.
296
00:15:05,000 --> 00:15:07,210
See, this is an abbreviation
for hyperbolic
297
00:15:07,210 --> 00:15:10,450
cosine, meaning this--
298
00:15:10,450 --> 00:15:13,390
I'm sorry, I got
this backwards.
299
00:15:13,390 --> 00:15:16,890
Call the x-coordinate the
hyperbolic cosine, the
300
00:15:16,890 --> 00:15:20,630
y-coordinate the hyperbolic
sine.
301
00:15:20,630 --> 00:15:22,330
You don't have to know anything
about advanced
302
00:15:22,330 --> 00:15:23,680
mathematics to see this.
303
00:15:23,680 --> 00:15:27,460
All I'm saying is, I could just
as easily have taken any
304
00:15:27,460 --> 00:15:32,130
geometric figure, marked off
lengths along it, taken the
305
00:15:32,130 --> 00:15:36,130
x-coordinates and the
y-coordinates, and seen what
306
00:15:36,130 --> 00:15:37,860
relationships they obey.
307
00:15:37,860 --> 00:15:41,090
You see, as such, there's
nothing sacred about working
308
00:15:41,090 --> 00:15:42,510
on a circle.
309
00:15:42,510 --> 00:15:45,170
Not only that, but even after
you agree to work on the
310
00:15:45,170 --> 00:15:47,250
circle, there are many
other ways that one
311
00:15:47,250 --> 00:15:47,940
could have done this.
312
00:15:47,940 --> 00:15:50,790
For example, someone might have
said, look it, when you
313
00:15:50,790 --> 00:15:54,780
take this length called 't', why
did you elect to mark it
314
00:15:54,780 --> 00:15:56,360
off along the circle?
315
00:15:56,360 --> 00:16:01,760
Why couldn't you have taken a
radius equal to 't', taken 'S'
316
00:16:01,760 --> 00:16:07,450
as a center, and swung an arc
that met the circle, and call
317
00:16:07,450 --> 00:16:10,730
this length 't'?
318
00:16:10,730 --> 00:16:12,960
You see, instead of measuring
along the circle, measure
319
00:16:12,960 --> 00:16:14,290
along the straight line.
320
00:16:14,290 --> 00:16:16,510
Again, you could have done
this if you wanted to.
321
00:16:16,510 --> 00:16:18,290
Why you would've wanted
to do this?
322
00:16:18,290 --> 00:16:20,500
Well, you have the same
right to do this
323
00:16:20,500 --> 00:16:21,550
as I had to do mine.
324
00:16:21,550 --> 00:16:23,650
Of course, you have to be
a little bit careful.
325
00:16:23,650 --> 00:16:27,350
For example, in this particular
configuration,
326
00:16:27,350 --> 00:16:29,420
notice that, if this is how
you're going to define your
327
00:16:29,420 --> 00:16:34,230
trigonometric function, your
input, your domain, has to be
328
00:16:34,230 --> 00:16:37,070
somewhere between 0 and 2.
329
00:16:37,070 --> 00:16:39,800
In other words, you cannot have
a length longer than 2,
330
00:16:39,800 --> 00:16:41,860
because notice that the
diameter of the
331
00:16:41,860 --> 00:16:43,690
circle is only 2.
332
00:16:43,690 --> 00:16:46,680
And therefore, if 't' were
greater than 2, when you swung
333
00:16:46,680 --> 00:16:49,000
an arc from the point 'S',
it wouldn't meet
334
00:16:49,000 --> 00:16:49,900
the circle at all.
335
00:16:49,900 --> 00:16:51,890
Well, that's no great
handicap.
336
00:16:51,890 --> 00:16:53,610
It's no great disaster.
337
00:16:53,610 --> 00:16:55,350
You still have the right
to make up whatever
338
00:16:55,350 --> 00:16:56,860
functions you want.
339
00:16:56,860 --> 00:17:00,170
I will try to make it clearer
why we chose these circular
340
00:17:00,170 --> 00:17:03,190
functions from a physical point
of view as we go along.
341
00:17:03,190 --> 00:17:06,790
What I thought I'd like to do
now is, having motivated, that
342
00:17:06,790 --> 00:17:09,400
we can invent the trigonometric
functions in
343
00:17:09,400 --> 00:17:13,609
terms of numbers definitions
along this circle.
344
00:17:13,609 --> 00:17:17,000
And coupled with the fact that,
in radian measure, you
345
00:17:17,000 --> 00:17:20,150
can have a very nice
identification between what's
346
00:17:20,150 --> 00:17:23,490
happening pictorially and what's
happening analytically,
347
00:17:23,490 --> 00:17:26,079
to show, for example, that in
terms of our subject called
348
00:17:26,079 --> 00:17:30,610
calculus, that we're pretty much
home free once we learn
349
00:17:30,610 --> 00:17:32,280
these basic ideas.
350
00:17:32,280 --> 00:17:35,170
You see, the important point
is that, in a manner of
351
00:17:35,170 --> 00:17:38,980
speaking, we have finished
differential calculus.
352
00:17:38,980 --> 00:17:41,900
We know what all the recipes
are We know what properties
353
00:17:41,900 --> 00:17:42,870
things have.
354
00:17:42,870 --> 00:17:46,180
So all of the rules that we
learned will apply to any
355
00:17:46,180 --> 00:17:48,600
particular type of function
that we're talking about.
356
00:17:48,600 --> 00:17:51,390
For example, let's suppose
we define 'f of
357
00:17:51,390 --> 00:17:54,120
x' to be 'sine x'.
358
00:17:54,120 --> 00:17:56,760
And we want to find the
derivative of 'sine x'.
359
00:17:56,760 --> 00:18:00,830
Notice that 'f prime of x'
evaluated at any number 'x1'
360
00:18:00,830 --> 00:18:02,780
has already been
defined for us.
361
00:18:02,780 --> 00:18:06,770
It's the limit as 'delta x'
approaches 0, 'f of 'x1 plus
362
00:18:06,770 --> 00:18:11,100
delta x'', minus 'f of
x1' over 'delta x'.
363
00:18:11,100 --> 00:18:13,520
This is true for any
function 'f'.
364
00:18:13,520 --> 00:18:17,935
In particular, if 'f of x' is
'sine x', all we get is what?
365
00:18:17,935 --> 00:18:20,880
That the derivative is the limit
as 'delta x' approaches
366
00:18:20,880 --> 00:18:25,510
0, sine of 'x1 plus delta
x' minus sine of
367
00:18:25,510 --> 00:18:27,190
'x1' over 'delta x'.
368
00:18:27,190 --> 00:18:29,910
Now you see, on this
particular score,
369
00:18:29,910 --> 00:18:31,450
nobody can fault us.
370
00:18:31,450 --> 00:18:34,160
This is still the basic
definition.
371
00:18:34,160 --> 00:18:36,750
All that happens computationally
is that, if
372
00:18:36,750 --> 00:18:39,200
we're not familiar with our
new functions called the
373
00:18:39,200 --> 00:18:42,740
trigonometric functions, we
might not know how to express
374
00:18:42,740 --> 00:18:47,060
sine of 'x1 plus delta x' in
a more convenient form.
375
00:18:47,060 --> 00:18:49,080
What do we mean by a more
convenient form?
376
00:18:49,080 --> 00:18:51,710
Well, notice again, as is always
the case when we take a
377
00:18:51,710 --> 00:18:55,750
derivative, as delta x
approaches 0, our numerator
378
00:18:55,750 --> 00:19:01,100
becomes 'sine x1' minus 'sine
x1', which is 0/0.
379
00:19:01,100 --> 00:19:05,250
And we're back to our familiar
taboo form of 0/0.
380
00:19:05,250 --> 00:19:07,770
Somehow or other, we're going
to have to make a refinement
381
00:19:07,770 --> 00:19:10,360
on our numerator that
will allow us to get
382
00:19:10,360 --> 00:19:13,000
rid of a 0/0 form.
383
00:19:13,000 --> 00:19:16,600
Well, to make a long story
short, if we happen to know
384
00:19:16,600 --> 00:19:18,760
the addition formula
for the sine--
385
00:19:18,760 --> 00:19:22,350
in other words, 'sine 'x1 plus
delta x'' is ''sine x1'
386
00:19:22,350 --> 00:19:26,950
'cosine delta x'', plus ''sine
delta x' 'cosine x1''--
387
00:19:26,950 --> 00:19:32,080
then we subtract off 'sine x1'
and divide by 'delta x', and
388
00:19:32,080 --> 00:19:34,040
then we factor and
collect terms.
389
00:19:34,040 --> 00:19:34,940
We see what?
390
00:19:34,940 --> 00:19:36,820
Without any knowledge
of calculus at
391
00:19:36,820 --> 00:19:38,300
all, but just what?
392
00:19:38,300 --> 00:19:40,800
By our definition of derivative,
just by our
393
00:19:40,800 --> 00:19:44,150
definition, coupled with
properties of the
394
00:19:44,150 --> 00:19:47,510
trigonometric functions, we wind
up with the fact that 'f
395
00:19:47,510 --> 00:19:50,440
prime of x1' is this
particular limit.
396
00:19:50,440 --> 00:19:53,360
Now certainly, our limit
theorems don't change.
397
00:19:53,360 --> 00:19:55,180
The limit of a sum is
still going to be
398
00:19:55,180 --> 00:19:57,570
the sum of the limits.
399
00:19:57,570 --> 00:19:59,470
The limit of a product
will still be the
400
00:19:59,470 --> 00:20:00,680
product of the limits.
401
00:20:00,680 --> 00:20:04,680
So all in all, what we have to
sort of do is figure out what
402
00:20:04,680 --> 00:20:05,940
these limits will be.
403
00:20:05,940 --> 00:20:09,880
Certainly, as 'delta x'
approaches 0, this will stay
404
00:20:09,880 --> 00:20:11,870
'cosine x1'.
405
00:20:11,870 --> 00:20:16,570
Certainly this will stay 'sine
x1', because 'x1' is a fixed
406
00:20:16,570 --> 00:20:19,040
number that doesn't depend
on 'delta x'.
407
00:20:19,040 --> 00:20:22,320
But notice, rather
interestingly, that both of my
408
00:20:22,320 --> 00:20:26,980
expressions in parentheses
happen to take on that 0/0
409
00:20:26,980 --> 00:20:28,580
form if we're not careful.
410
00:20:28,580 --> 00:20:34,250
Namely, if you replace 'delta
x' by 0, sine 0 is 0, 0/0 is
411
00:20:34,250 --> 00:20:37,840
0, and we run into trouble here
if we replace 'delta x'
412
00:20:37,840 --> 00:20:40,130
by 0, which of course
we can't do.
413
00:20:40,130 --> 00:20:43,480
This is the same definition
of limit as we had before.
414
00:20:43,480 --> 00:20:46,450
'Delta x' gets arbitrarily
close to 0, but never is
415
00:20:46,450 --> 00:20:47,810
allowed to get there.
416
00:20:47,810 --> 00:20:51,340
Well, you see, if nothing else,
this motivates why we
417
00:20:51,340 --> 00:20:54,330
would like to learn this
particular type of limit.
418
00:20:54,330 --> 00:20:56,680
In other words, what we would
like to know is, how do you--
419
00:20:56,680 --> 00:20:59,400
the 'delta x' symbol here
isn't that important.
420
00:20:59,400 --> 00:21:01,370
'Delta x' just stands
for any number.
421
00:21:01,370 --> 00:21:03,760
Notice that what we would like
to know is, if you take the
422
00:21:03,760 --> 00:21:07,490
sine of something over that same
something, and take the
423
00:21:07,490 --> 00:21:10,570
limit as that same something
goes to 0, we would like to
424
00:21:10,570 --> 00:21:12,320
know what that becomes.
425
00:21:12,320 --> 00:21:14,980
In a similar way, we would like
to know how to handle
426
00:21:14,980 --> 00:21:17,480
this quotient here, because
notice that when 'delta x' is
427
00:21:17,480 --> 00:21:20,040
0, cosine 0 is 1.
428
00:21:20,040 --> 00:21:22,460
This is 1 minus 1 over 0.
429
00:21:22,460 --> 00:21:25,200
It's another 0/0 form.
430
00:21:25,200 --> 00:21:28,556
So the problem that we're
confronted with is that, what
431
00:21:28,556 --> 00:21:31,140
we would like to do is to figure
out how to handle the
432
00:21:31,140 --> 00:21:35,360
limit of 'sine t' over 't'
as 't' approaches 0.
433
00:21:35,360 --> 00:21:38,440
Now, what's 't' here?
't' is a number.
434
00:21:38,440 --> 00:21:38,900
Remember that.
435
00:21:38,900 --> 00:21:40,630
This is the big pitch
I've been making.
436
00:21:40,630 --> 00:21:42,170
We're thinking of
't' as a number.
437
00:21:42,170 --> 00:21:45,650
If, on the other hand, you feel
more comfortable thinking
438
00:21:45,650 --> 00:21:47,800
in terms of traditional
trigonometry--
439
00:21:47,800 --> 00:21:50,480
and let's face it, the more
background you've had in
440
00:21:50,480 --> 00:21:54,780
traditional trigonometry, the
more comfortable you're going
441
00:21:54,780 --> 00:21:56,480
to feel using it.
442
00:21:56,480 --> 00:21:59,620
Let's simply agree to do this,
that if it bothers you to
443
00:21:59,620 --> 00:22:02,700
think of this as a length
divided by a length, et
444
00:22:02,700 --> 00:22:05,500
cetera, and that this is a
length or a number, let's
445
00:22:05,500 --> 00:22:10,460
agree that we will go back to
angles but use radian measure.
446
00:22:10,460 --> 00:22:11,510
Why?
447
00:22:11,510 --> 00:22:15,970
Because if the angle is measured
in radians, the sine
448
00:22:15,970 --> 00:22:21,220
of the angle 't' radians is the
same as the number, the
449
00:22:21,220 --> 00:22:24,050
sine, of the number 't'.
450
00:22:24,050 --> 00:22:27,220
Well again, here's how this
problem is tackled.
451
00:22:27,220 --> 00:22:30,250
What we do is we mark off the
angle of 't' radians.
452
00:22:30,250 --> 00:22:32,260
Remember that we have
the unit circle.
453
00:22:32,260 --> 00:22:37,510
And what we very cleverly do
is we catch our wedge, our
454
00:22:37,510 --> 00:22:40,840
circular wedge, between
two right triangles.
455
00:22:40,840 --> 00:22:44,400
Again, without making a big
issue over this, notice that
456
00:22:44,400 --> 00:22:49,060
this length is 'sine t', this
length is 'cosine t', so the
457
00:22:49,060 --> 00:22:52,710
area of the small triangle
is 'sine t' times
458
00:22:52,710 --> 00:22:55,650
'cosine t' over 2.
459
00:22:55,650 --> 00:22:59,500
See, 'sine t' times
'cosine t' over 2.
460
00:22:59,500 --> 00:23:02,630
Now, on the other hand, since
that's caught in our wedge,
461
00:23:02,630 --> 00:23:04,650
what is the area of our wedge?
462
00:23:04,650 --> 00:23:08,480
Well, since the area of the
entire circle is pi-- see, pi
463
00:23:08,480 --> 00:23:10,180
'R squared' and 'R' is 1--
464
00:23:10,180 --> 00:23:12,870
since the area of the entire
circle is pi--
465
00:23:12,870 --> 00:23:14,490
and we're taking what?
466
00:23:14,490 --> 00:23:17,290
't' of the 2pi.
467
00:23:17,290 --> 00:23:19,980
So there are two pi radians
in a circle.
468
00:23:19,980 --> 00:23:24,230
So the sector of the circle that
we have, it's 't/2pi' of
469
00:23:24,230 --> 00:23:25,390
the entire circle.
470
00:23:25,390 --> 00:23:28,490
And by the way, this is done
more rigorously and carried
471
00:23:28,490 --> 00:23:30,030
out in detail in the notes.
472
00:23:30,030 --> 00:23:33,080
Let me point out that, if we
insisted on working with
473
00:23:33,080 --> 00:23:36,450
degrees, instead of
't/2pi', we just
474
00:23:36,450 --> 00:23:40,310
would have had 't/360'.
475
00:23:40,310 --> 00:23:43,170
Because, you see, if we're
dealing with degrees, the
476
00:23:43,170 --> 00:23:47,110
entire angle and measure of the
circle is 360 degrees, and
477
00:23:47,110 --> 00:23:49,060
we would have had 't/360'.
478
00:23:49,060 --> 00:23:50,740
But here we've used the
fact that we're
479
00:23:50,740 --> 00:23:53,630
dealing with radians.
480
00:23:53,630 --> 00:23:56,670
And finally, the bigger
triangle, which includes the
481
00:23:56,670 --> 00:24:01,460
wedge, has, as its base, 1,
so that's the radius.
482
00:24:01,460 --> 00:24:05,410
And since the tangent is side
opposite over side adjacent,
483
00:24:05,410 --> 00:24:07,520
this length is 'tangent t'.
484
00:24:07,520 --> 00:24:09,980
And so what we have is what?
485
00:24:09,980 --> 00:24:15,200
That ''sine t' 'cosine t/2' must
be less than this, which
486
00:24:15,200 --> 00:24:20,220
in turn must be less than this,
multiplying through by 2
487
00:24:20,220 --> 00:24:25,380
and dividing through
by 'sine t'.
488
00:24:25,380 --> 00:24:27,780
And by the way, this hinges on
the fact that 't' is positive.
489
00:24:27,780 --> 00:24:30,740
Again, in our notes, we treat
the case where 't' is negative
490
00:24:30,740 --> 00:24:32,390
to arrive at the same result.
491
00:24:32,390 --> 00:24:37,000
Remembering that 'tan t' is
'sine t' over 'cosine t', we
492
00:24:37,000 --> 00:24:39,000
wind up with this result.
493
00:24:39,000 --> 00:24:44,240
And now, observing that
as 't' approaches 0,
494
00:24:44,240 --> 00:24:46,000
this approaches 1.
495
00:24:46,000 --> 00:24:48,270
This also approaches 1.
496
00:24:48,270 --> 00:24:51,230
And 't' over 'sine t' is caught
between these two.
497
00:24:51,230 --> 00:24:54,510
We get that the limit of 't'
over 'sine t' as 't'
498
00:24:54,510 --> 00:24:56,690
approaches 0 is 1.
499
00:24:56,690 --> 00:25:00,340
Now of course, since this limit
is 1, the limit of the
500
00:25:00,340 --> 00:25:03,930
reciprocal of this will be
the reciprocal of this.
501
00:25:03,930 --> 00:25:07,290
But what's very nice about the
number 1 is that it's equal to
502
00:25:07,290 --> 00:25:10,500
its own reciprocal.
503
00:25:10,500 --> 00:25:14,050
In other words, what we've now
shown is that the limit of
504
00:25:14,050 --> 00:25:18,400
'sine t' over 't' as 't'
approaches 0 is 1.
505
00:25:18,400 --> 00:25:20,710
That, as I said before,
is done in the text.
506
00:25:20,710 --> 00:25:22,090
We do it in our notes.
507
00:25:22,090 --> 00:25:25,750
But the thing that I hope this
motivates is why we want to do
508
00:25:25,750 --> 00:25:27,230
this in the first place.
509
00:25:27,230 --> 00:25:30,640
Notice that this was a limit
that we had to compute if we
510
00:25:30,640 --> 00:25:34,630
wanted to compute the derivative
of the sine.
511
00:25:34,630 --> 00:25:38,260
Now, the next thing was, how
do we handle '1 - cosine t'
512
00:25:38,260 --> 00:25:40,790
over 't' as 't' approaches 0?
513
00:25:40,790 --> 00:25:43,930
Again, leaving the details to
you to sketch in as you see
514
00:25:43,930 --> 00:25:46,680
fit, let me point out simply
what the mathematics
515
00:25:46,680 --> 00:25:48,100
involved here is.
516
00:25:48,100 --> 00:25:51,820
You see, what we can handle
is 'sine t' over 't'.
517
00:25:51,820 --> 00:25:54,610
That means that what we would
like to do is, whenever we're
518
00:25:54,610 --> 00:25:57,920
given an alien form, we would
somehow or other like to
519
00:25:57,920 --> 00:26:00,900
figure some way of factoring
a sine t over
520
00:26:00,900 --> 00:26:02,580
t out of this thing.
521
00:26:02,580 --> 00:26:06,970
When you look at '1 - cosine
t', the identity, 'sine
522
00:26:06,970 --> 00:26:10,040
squared' equals '1 - 'cosine
squared t'',
523
00:26:10,040 --> 00:26:12,240
should suggest itself.
524
00:26:12,240 --> 00:26:16,130
Now, how do you get from '1 -
cosine t' to '1 - 'cosine
525
00:26:16,130 --> 00:26:17,170
squared t''?
526
00:26:17,170 --> 00:26:20,800
You have to multiply
by '1 + cosine t'.
527
00:26:20,800 --> 00:26:24,050
And if you multiply by '1 +
cosine t' upstairs, you must
528
00:26:24,050 --> 00:26:27,780
multiply by '1 + cosine
t' downstairs.
529
00:26:27,780 --> 00:26:30,970
By the way, the only time you
can't multiply by something is
530
00:26:30,970 --> 00:26:33,310
when the thing is 0.
531
00:26:33,310 --> 00:26:35,140
You can't put that into
the denominator.
532
00:26:35,140 --> 00:26:39,410
Notice that 'cosine t' is
not 0 in a neighborhood
533
00:26:39,410 --> 00:26:40,790
of 't' equals 0.
534
00:26:40,790 --> 00:26:45,170
See, 'cosine t' behaves like 1
when 't' is near 0, so this is
535
00:26:45,170 --> 00:26:47,870
a permissible step in this
particular problem.
536
00:26:47,870 --> 00:26:52,880
The point is, we now factor '1 -
'cosine squared t'' as 'sine
537
00:26:52,880 --> 00:26:54,920
t' times 'sine t'.
538
00:26:54,920 --> 00:26:56,370
See, that's 'sine squared t'.
539
00:26:56,370 --> 00:27:01,740
We break up our 't' times
'1 + cosine t' this way.
540
00:27:01,740 --> 00:27:04,880
Now we know that the limit of
a product is the product of
541
00:27:04,880 --> 00:27:05,710
the limits.
542
00:27:05,710 --> 00:27:08,050
This we already know
goes to 1.
543
00:27:08,050 --> 00:27:12,050
And as 't' approaches 0, from
our previous limit work on the
544
00:27:12,050 --> 00:27:15,140
like, notice here, the limit of
a quotient is the quotient
545
00:27:15,140 --> 00:27:19,360
of the limits, the numerator
goes to 0, the denominator
546
00:27:19,360 --> 00:27:23,900
goes to 2, because as t
approaches 0, cosine 0 is 1.
547
00:27:23,900 --> 00:27:28,370
At any rate, that's
0/2, which is 0.
548
00:27:28,370 --> 00:27:31,570
And so this limit is 0.
549
00:27:31,570 --> 00:27:35,410
Now, at the risk of giving you
a slight headache as I take
550
00:27:35,410 --> 00:27:39,000
the board down here, let
me just review what
551
00:27:39,000 --> 00:27:40,130
it was that we did.
552
00:27:40,130 --> 00:27:42,410
You see, notice that, without
any knowledge of these limits
553
00:27:42,410 --> 00:27:45,720
at all, we were able to show
that whatever the derivative
554
00:27:45,720 --> 00:27:49,110
of 'sine x' was, it was this
particular thing here.
555
00:27:49,110 --> 00:27:53,230
Now what we've done is we've
shown that this is 1, and
556
00:27:53,230 --> 00:27:55,630
we've shown that this is 0.
557
00:27:55,630 --> 00:27:59,090
And using our limit theorems,
what we now see is what?
558
00:27:59,090 --> 00:28:06,650
That if 'f of x' is 'sine x', 'f
prime of x' is 'cosine x'.
559
00:28:06,650 --> 00:28:10,370
560
00:28:10,370 --> 00:28:15,780
Let me just write that down over
here, that if 'y' equals
561
00:28:15,780 --> 00:28:23,620
'sine x', 'dy/dx'
is 'cosine x'.
562
00:28:23,620 --> 00:28:27,560
And again, notice how much of
the calculus involved here was
563
00:28:27,560 --> 00:28:28,350
nothing new.
564
00:28:28,350 --> 00:28:31,880
It goes back to the so-called
baby chapter that nobody
565
00:28:31,880 --> 00:28:36,690
likes, where we go back to
epsilons, deltas, you see
566
00:28:36,690 --> 00:28:39,430
derivatives by 'delta
x', et cetera.
567
00:28:39,430 --> 00:28:42,370
See, those recipes always
remain the same.
568
00:28:42,370 --> 00:28:45,670
What happens is, as you invent
new functions, you need a
569
00:28:45,670 --> 00:28:48,920
different degree of
computational sophistication
570
00:28:48,920 --> 00:28:50,740
to find the desired limits.
571
00:28:50,740 --> 00:28:53,880
By the way, once you get over
these hurdles, everything
572
00:28:53,880 --> 00:28:56,520
again starts to go smoothly
as before.
573
00:28:56,520 --> 00:28:58,460
For example, our chain rule.
574
00:28:58,460 --> 00:29:02,030
Suppose we have now that 'y'
equals 'sine u', where 'u' is
575
00:29:02,030 --> 00:29:04,220
some differentiable
function of 'x'.
576
00:29:04,220 --> 00:29:06,920
And we now want to
find the 'dy/dx'.
577
00:29:06,920 --> 00:29:09,160
Well, you see, the point is
that we know that the
578
00:29:09,160 --> 00:29:12,750
derivative of 'sine u' with
respect to 'u' would be
579
00:29:12,750 --> 00:29:15,510
'cosine u'.
580
00:29:15,510 --> 00:29:16,850
What we want is the derivative
of 'sine u'
581
00:29:16,850 --> 00:29:17,990
with respect to 'x'.
582
00:29:17,990 --> 00:29:21,480
And we motivate the chain rule
the same way as we did before.
583
00:29:21,480 --> 00:29:23,920
It happens to be that we're
dealing with the specific
584
00:29:23,920 --> 00:29:26,035
value called sine,
but it could've
585
00:29:26,035 --> 00:29:27,000
been any old function.
586
00:29:27,000 --> 00:29:30,580
How would you differentiate 'f
of u' with respect to 'x' if
587
00:29:30,580 --> 00:29:33,550
you know how to differentiate 'f
of u' with respect to 'u'?
588
00:29:33,550 --> 00:29:35,430
And the answer is, you would
just differentiate with
589
00:29:35,430 --> 00:29:39,060
respect to 'u', and multiply
that by a derivative of 'u'
590
00:29:39,060 --> 00:29:40,490
with respect to 'x'.
591
00:29:40,490 --> 00:29:42,350
In other words, we get
the result what?
592
00:29:42,350 --> 00:29:46,990
That since 'dy/du' is 'cosine
u', we get that the derivative
593
00:29:46,990 --> 00:29:49,570
of 'sine u' with respect
to 'x' is
594
00:29:49,570 --> 00:29:52,000
'cosine u' times 'du/dx'.
595
00:29:52,000 --> 00:29:55,690
And by the way, one rather nice
application of this is
596
00:29:55,690 --> 00:29:58,740
that it gives us a very quick
way of getting the derivative
597
00:29:58,740 --> 00:30:00,510
of 'cosine x'.
598
00:30:00,510 --> 00:30:04,870
After all, our basic identity
is that 'cosine x' is sine
599
00:30:04,870 --> 00:30:06,900
pi/2 minus 'x'.
600
00:30:06,900 --> 00:30:09,910
Again, a number or an
angle, either way.
601
00:30:09,910 --> 00:30:12,870
As long as the measurement is
in radians, it makes no
602
00:30:12,870 --> 00:30:15,360
difference whether you think of
this as being an angle or
603
00:30:15,360 --> 00:30:16,320
being a number.
604
00:30:16,320 --> 00:30:18,000
The answer will be the same.
605
00:30:18,000 --> 00:30:19,320
The idea is this.
606
00:30:19,320 --> 00:30:21,410
To take the derivative of
'cosine x' with respect to
607
00:30:21,410 --> 00:30:25,430
'x', all I have to differentiate
is sine pi/2
608
00:30:25,430 --> 00:30:27,420
minus 'x' with respect to 'x'.
609
00:30:27,420 --> 00:30:28,850
But I know how to do that.
610
00:30:28,850 --> 00:30:35,290
Namely, the derivative of sine
pi/2 minus 'x' is cosine pi/2
611
00:30:35,290 --> 00:30:39,680
minus 'x,' and by the chain
rule, times the derivative of
612
00:30:39,680 --> 00:30:41,230
this with respect to 'x'.
613
00:30:41,230 --> 00:30:43,370
Well, pi/2 is a constant.
614
00:30:43,370 --> 00:30:46,580
The derivative of 'minus
x' is minus 1.
615
00:30:46,580 --> 00:30:51,230
And then, remembering that the
cosine of pi/2 minus 'x' is
616
00:30:51,230 --> 00:30:55,110
'sine x', I now have the result
that the derivative of
617
00:30:55,110 --> 00:30:58,460
the cosine is minus the sine.
618
00:30:58,460 --> 00:31:01,070
And again, I can do all sorts
of things this way.
619
00:31:01,070 --> 00:31:04,350
If I want the derivative of a
tangent, I could write tangent
620
00:31:04,350 --> 00:31:06,040
as sine over cosine.
621
00:31:06,040 --> 00:31:07,450
Use the quotient rule.
622
00:31:07,450 --> 00:31:10,330
You see, as soon as I make one
breakthrough, all of the
623
00:31:10,330 --> 00:31:13,260
previous body of calculus
comes to my
624
00:31:13,260 --> 00:31:15,660
rescue, so to speak.
625
00:31:15,660 --> 00:31:18,990
By the way, what I'd like to do
now is point out why, from
626
00:31:18,990 --> 00:31:23,440
a physical point of view, we
like circular functions to be
627
00:31:23,440 --> 00:31:26,570
independent of angles
and the like.
628
00:31:26,570 --> 00:31:29,430
With the results that we've
derived so far, it's rather
629
00:31:29,430 --> 00:31:31,650
easy to derive one
more result.
630
00:31:31,650 --> 00:31:34,390
Namely, let's assume
that a particle is
631
00:31:34,390 --> 00:31:35,980
moving along the x-axis--
632
00:31:35,980 --> 00:31:38,030
I'm going to start with the
answer, sort of, and work
633
00:31:38,030 --> 00:31:38,900
backwards--
634
00:31:38,900 --> 00:31:42,410
according to the rule, 'x'
equals 'sine kt', where 't' is
635
00:31:42,410 --> 00:31:44,810
time and 'k' is a constant.
636
00:31:44,810 --> 00:31:47,680
Then its speed, 'dx/dt',
is what?
637
00:31:47,680 --> 00:31:51,150
It's the derivative of 'sine
kt', which is 'cosine kt',
638
00:31:51,150 --> 00:31:54,090
times the derivative of what's
inside with respect to 't'.
639
00:31:54,090 --> 00:31:56,960
In other words, it's
'k cosine kt'.
640
00:31:56,960 --> 00:32:00,100
The second derivative of 'x'
with respect to 't', namely,
641
00:32:00,100 --> 00:32:01,650
the acceleration is what?
642
00:32:01,650 --> 00:32:04,080
How do you differentiate
the cosine?
643
00:32:04,080 --> 00:32:07,850
The derivative of the cosine
is minus the sine.
644
00:32:07,850 --> 00:32:10,490
By the chain rule, I must
multiply by the derivative of
645
00:32:10,490 --> 00:32:13,830
'kt' with respect to 't', which
gives me another factor
646
00:32:13,830 --> 00:32:15,450
of 't' over here.
647
00:32:15,450 --> 00:32:24,780
Remembering that 'x' equals
'sine kt', I arrive at this
648
00:32:24,780 --> 00:32:27,370
particular so-called
differential equation.
649
00:32:27,370 --> 00:32:28,510
And what does this say?
650
00:32:28,510 --> 00:32:32,620
It says that 'd2x/ dt squared',
the acceleration, is
651
00:32:32,620 --> 00:32:35,090
proportional to the
displacement, the distance
652
00:32:35,090 --> 00:32:37,580
traveled, but in the
opposite direction.
653
00:32:37,580 --> 00:32:40,900
You see, 'k squared' can't
be negative, so 'minus 'k
654
00:32:40,900 --> 00:32:42,940
squared'' can't be positive.
655
00:32:42,940 --> 00:32:43,810
This says what?
656
00:32:43,810 --> 00:32:46,510
The acceleration is proportional
to the
657
00:32:46,510 --> 00:32:49,610
displacement, but in the
opposite direction.
658
00:32:49,610 --> 00:32:54,800
Does that problem require any
knowledge of angles to solve?
659
00:32:54,800 --> 00:32:56,470
Notice that this is a perfectly
660
00:32:56,470 --> 00:32:57,950
good physical problem.
661
00:32:57,950 --> 00:33:00,940
It's known as simple
harmonic motion.
662
00:33:00,940 --> 00:33:04,310
And all I'm trying to have you
see is that, by inventing the
663
00:33:04,310 --> 00:33:08,960
circular functions in the proper
way, not only can we do
664
00:33:08,960 --> 00:33:12,660
their calculus, but even more
importantly, if we reverse
665
00:33:12,660 --> 00:33:17,220
these steps, for example, we
can show that, to solve the
666
00:33:17,220 --> 00:33:20,920
physical problem of simple
harmonic motion, we have to
667
00:33:20,920 --> 00:33:24,890
know the so-called circular
trigonometric functions.
668
00:33:24,890 --> 00:33:29,030
And this is a far cry, you see,
from using trigonometry
669
00:33:29,030 --> 00:33:32,385
in the sense that the surveyor
uses trigonometry.
670
00:33:32,385 --> 00:33:34,740
You see, this ties up with my
initial hang-up that I was
671
00:33:34,740 --> 00:33:37,110
telling you about at the
beginning of the program.
672
00:33:37,110 --> 00:33:40,270
By the way, in closing, I should
also make reference to
673
00:33:40,270 --> 00:33:42,640
something that we pointed out
in our last lecture, namely,
674
00:33:42,640 --> 00:33:44,780
inverse differentiation.
675
00:33:44,780 --> 00:33:48,200
Keep in mind, also, that as you
read the calculus of the
676
00:33:48,200 --> 00:33:51,440
trigonometric functions, that
the fact that we know that the
677
00:33:51,440 --> 00:33:55,230
derivative of sine u with
respect to 'u' was 'cosine u'
678
00:33:55,230 --> 00:33:58,560
gives us, with a switch in
emphasis, the result that the
679
00:33:58,560 --> 00:34:03,290
integral 'cosine u', 'du' is
'sine u' plus a constant.
680
00:34:03,290 --> 00:34:06,060
And in a similar way, since the
derivative of cosine is
681
00:34:06,060 --> 00:34:09,070
minus the sine, the integral
of 'sine u' with respect to
682
00:34:09,070 --> 00:34:12,590
'u' is 'minus cosine
u' plus a constant.
683
00:34:12,590 --> 00:34:13,280
Be careful.
684
00:34:13,280 --> 00:34:16,420
Notice how the sines
can screw you up.
685
00:34:16,420 --> 00:34:18,310
Namely, they're in the opposite
sense when you're
686
00:34:18,310 --> 00:34:20,929
integrating as when you
were differentiating.
687
00:34:20,929 --> 00:34:24,239
But again, these are the details
which I expect you can
688
00:34:24,239 --> 00:34:26,909
have come out in the
wash rather nicely.
689
00:34:26,909 --> 00:34:29,510
We can continue on this way,
from knowing how to
690
00:34:29,510 --> 00:34:32,389
differentiate 'sine x'
to the nth power.
691
00:34:32,389 --> 00:34:38,380
Namely, it's 'n - 1' 'x', times
the derivative of 'sine
692
00:34:38,380 --> 00:34:40,900
x', which is 'cosine x'.
693
00:34:40,900 --> 00:34:42,510
We don't want this in here.
694
00:34:42,510 --> 00:34:44,050
That's a differential form.
695
00:34:44,050 --> 00:34:46,790
Without going into any detail
here, notice that a
696
00:34:46,790 --> 00:34:50,690
modification of this shows us
that, if we differentiate
697
00:34:50,690 --> 00:34:53,630
this, we wind up with this.
698
00:34:53,630 --> 00:34:57,730
We could now take the time, if
this were the proper place, to
699
00:34:57,730 --> 00:35:01,430
develop all sorts of derivative
formulas and
700
00:35:01,430 --> 00:35:02,650
integral formulas.
701
00:35:02,650 --> 00:35:06,000
As you study your study guide,
you will notice that the
702
00:35:06,000 --> 00:35:11,000
lesson after this is concerned
with the calculus of the
703
00:35:11,000 --> 00:35:12,650
circular functions.
704
00:35:12,650 --> 00:35:16,190
My feeling is is that, with
this as background, a very
705
00:35:16,190 --> 00:35:20,390
good review of the previous part
of the course will be to
706
00:35:20,390 --> 00:35:25,350
see how much of this you can
apply on your own to these new
707
00:35:25,350 --> 00:35:28,420
functions called the
circular functions.
708
00:35:28,420 --> 00:35:31,810
Next time, we will talk, as
you may be able to guess,
709
00:35:31,810 --> 00:35:34,200
about the inverse circular
functions
710
00:35:34,200 --> 00:35:35,800
and why they're important.
711
00:35:35,800 --> 00:35:37,340
But until next time, goodbye.
712
00:35:37,340 --> 00:35:40,240
713
00:35:40,240 --> 00:35:43,440
Funding for the publication of
this video was provided by the
714
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