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HERBERT GROSS: Hi, always being
somebody who wants to
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00:00:33,770 --> 00:00:38,250
get something for nothing, I
figure any time in this course
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00:00:38,250 --> 00:00:40,770
that we'll study the new
function, the very next thing
11
00:00:40,770 --> 00:00:44,670
that we'll do is study
the inverse function.
12
00:00:44,670 --> 00:00:46,980
In other words, since the
inverse is just a change in
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00:00:46,980 --> 00:00:52,330
emphasis, why not just change
the emphasis to get the result
14
00:00:52,330 --> 00:00:53,390
that we want?
15
00:00:53,390 --> 00:00:55,950
So you see, today's lesson
is called the inverse
16
00:00:55,950 --> 00:00:57,450
trigonometric functions.
17
00:00:57,450 --> 00:01:00,730
And the longest part of this
lecture will be a few
18
00:01:00,730 --> 00:01:03,730
computations just to get
the feel of things.
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00:01:03,730 --> 00:01:06,870
The reason being that other than
that, everything that we
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00:01:06,870 --> 00:01:10,720
have to use will be drawn from
results that we've already
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00:01:10,720 --> 00:01:14,630
studied in our lessons called
inverse functions.
22
00:01:14,630 --> 00:01:17,200
It's still the same idea, only
a different illustration.
23
00:01:17,200 --> 00:01:20,090
Well, at any rate, let's get
on with the subject.
24
00:01:20,090 --> 00:01:25,480
Let's start with the curve
'y' equals 'sine x'.
25
00:01:25,480 --> 00:01:28,860
Now you see, when we plot the
curve 'y' equals 'sine x', it
26
00:01:28,860 --> 00:01:32,280
doesn't take us very long
to discover that this
27
00:01:32,280 --> 00:01:35,220
curve is not 1:1.
28
00:01:35,220 --> 00:01:38,550
In fact, it misses
by a long shot.
29
00:01:38,550 --> 00:01:42,420
Because for every 'y' value
between minus 1 and 1, there
30
00:01:42,420 --> 00:01:45,010
are infinitely many values
of 'x' that produce
31
00:01:45,010 --> 00:01:46,740
that value of 'y'.
32
00:01:46,740 --> 00:01:49,240
This is an oscillating
type of function.
33
00:01:49,240 --> 00:01:52,320
So you see from our rigorous
point of view, there really
34
00:01:52,320 --> 00:01:58,690
isn't much sense in defining
the inverse sine function.
35
00:01:58,690 --> 00:02:02,500
Well, in a way, this
is an artificial
36
00:02:02,500 --> 00:02:03,930
drawback that we have.
37
00:02:03,930 --> 00:02:06,990
Because you also recall that
when we talked about functions
38
00:02:06,990 --> 00:02:10,710
that were not 1:1, we could
always break them down into a
39
00:02:10,710 --> 00:02:13,240
union of 1:1 functions.
40
00:02:13,240 --> 00:02:16,550
For example, notice in the part
of the curve here which
41
00:02:16,550 --> 00:02:22,300
I've accented, namely, the
domain from minus pi over 2 to
42
00:02:22,300 --> 00:02:28,720
pi over 2, notice that on that
domain, the sine is 1:1.
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00:02:28,720 --> 00:02:35,730
And it's certainly onto the
interval from minus 1 to 1.
44
00:02:35,730 --> 00:02:40,410
In other words, every value of
the sign is taken on once and
45
00:02:40,410 --> 00:02:44,710
only once on the interval from
minus pi over 2 to pi over 2.
46
00:02:44,710 --> 00:02:48,050
So you see, if I disregarded
everything but this part of
47
00:02:48,050 --> 00:02:50,400
the curve, and how
could I do that?
48
00:02:50,400 --> 00:02:52,050
What shall I name
this function?
49
00:02:52,050 --> 00:02:53,760
Well, why not do something
like this.
50
00:02:53,760 --> 00:02:55,280
Let me invent a new name.
51
00:02:55,280 --> 00:03:00,150
Let me call this curve ''s sub
0' of x' where ''s sub 0' of
52
00:03:00,150 --> 00:03:04,370
x' is defined to be 'sine x'
provided that 'x' is in the
53
00:03:04,370 --> 00:03:08,560
closed interval from minus
pi over 2 to pi over 2.
54
00:03:08,560 --> 00:03:10,590
Again, be very, very careful.
55
00:03:10,590 --> 00:03:14,330
Recall that when we talked about
defining functions, we
56
00:03:14,330 --> 00:03:18,050
said that to specify two
functions as being equal it
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00:03:18,050 --> 00:03:20,570
was not enough that the function
machine was the same,
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00:03:20,570 --> 00:03:22,940
the inputs had to be the same.
59
00:03:22,940 --> 00:03:24,935
In other words, two functions
had to be equal.
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00:03:24,935 --> 00:03:29,190
Or to be equal, the two
functions had to have
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00:03:29,190 --> 00:03:31,030
precisely the same domain.
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00:03:31,030 --> 00:03:34,430
So notice in this sense there
is a big difference between
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00:03:34,430 --> 00:03:37,310
''s sub 0' of x' and the
function that's being
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00:03:37,310 --> 00:03:38,610
called 'sine x'.
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00:03:38,610 --> 00:03:39,840
What is the big difference?
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00:03:39,840 --> 00:03:43,680
Well, for the function that's
called 'sine x', the domain is
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00:03:43,680 --> 00:03:45,710
all real numbers.
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00:03:45,710 --> 00:03:49,530
So the function called ''s sub
0' of x', the domain is only
69
00:03:49,530 --> 00:03:53,060
the closed interval from minus
pi over 2 to pi over 2.
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00:03:53,060 --> 00:03:55,600
Now again, there's nothing
sacred about picking this
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00:03:55,600 --> 00:03:56,770
particular interval.
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00:03:56,770 --> 00:04:00,220
What I could have done was
define say, another curve,
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00:04:00,220 --> 00:04:02,590
which I'll call ''s
sub 1' of x'.
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00:04:02,590 --> 00:04:07,160
''s sub 1' of x' will be
the function 'sine x'.
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00:04:07,160 --> 00:04:12,310
But now, the domain will be
shall we say, from 3 pi over
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00:04:12,310 --> 00:04:16,820
2, or from pi over
2 to 3 pi over 2.
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00:04:16,820 --> 00:04:21,980
In other words, notice that this
portion of our sine curve
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00:04:21,980 --> 00:04:26,110
is also 1:1, and runs the full
gamut of values that the sine
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00:04:26,110 --> 00:04:29,390
can take on from 1 to minus 1.
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00:04:29,390 --> 00:04:31,490
And by the way, I
don't have to be
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00:04:31,490 --> 00:04:33,140
prejudiced this way either.
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00:04:33,140 --> 00:04:37,490
I could just have easily have
worked with negative values.
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00:04:37,490 --> 00:04:40,590
In other words, I could have
invented say another curve,
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00:04:40,590 --> 00:04:45,740
which I'll call ''s sub minus 1'
of x', where this subscript
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00:04:45,740 --> 00:04:48,070
simply means I'm going in
the opposite direction.
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00:04:48,070 --> 00:04:51,650
''s sub minus 1' of x' might
just has well have been what?
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00:04:51,650 --> 00:04:53,180
The curve 'sine x'.
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00:04:53,180 --> 00:04:55,770
But now again, how are all of
these things differing?
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00:04:55,770 --> 00:04:58,360
Only in the choice
of the domain.
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00:04:58,360 --> 00:05:04,280
See now the domain would be
from minus 3 pi over 2 to
91
00:05:04,280 --> 00:05:05,860
minus pi over 2.
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00:05:05,860 --> 00:05:11,730
Again, in terms of the picture,
this portion here
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00:05:11,730 --> 00:05:12,740
would be fine.
94
00:05:12,740 --> 00:05:19,460
You see what I can do is break
up this curve that fails to be
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00:05:19,460 --> 00:05:21,500
single valued by-- not
single valued.
96
00:05:21,500 --> 00:05:24,060
Fails to be 1:1 by a
long shot, into a
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00:05:24,060 --> 00:05:26,500
union of 1:1 curves.
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00:05:26,500 --> 00:05:29,190
In fact, if I wanted to use
some fancy mathematical
99
00:05:29,190 --> 00:05:31,000
language, which I'll
write down just to
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00:05:31,000 --> 00:05:32,100
look impressive here.
101
00:05:32,100 --> 00:05:34,730
But if it bothers you,
ignore it completely.
102
00:05:34,730 --> 00:05:40,070
I guess what I'm really saying
is, I could have defined 'sine
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00:05:40,070 --> 00:05:47,930
x' to be the union ''s sub n'
of x' as 'n' goes from minus
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00:05:47,930 --> 00:05:49,950
infinity to infinity.
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00:05:49,950 --> 00:05:50,920
Meaning what?
106
00:05:50,920 --> 00:05:54,050
Take all of these pieces with
these integral subscripts and
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00:05:54,050 --> 00:05:58,090
form their union, and that
infinite union gives you back
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00:05:58,090 --> 00:06:00,300
the entire curve.
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00:06:00,300 --> 00:06:02,050
Well, what am I making
all this issue about
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00:06:02,050 --> 00:06:03,330
in the first place?
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00:06:03,330 --> 00:06:06,500
You see, we can't have an
inverse function unless our
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00:06:06,500 --> 00:06:08,030
original function is 1:1.
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00:06:10,620 --> 00:06:12,210
In fact, let's see
what does happen.
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00:06:12,210 --> 00:06:14,580
Remember how we invert
the function?
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00:06:14,580 --> 00:06:16,870
In other words, how do you get
from the graph of 'y' equals
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00:06:16,870 --> 00:06:20,660
'f of x' to the graph of 'y'
equals 'f inverse of x'?
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00:06:20,660 --> 00:06:23,490
Recalling from our previous
lecture on this topic, what
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00:06:23,490 --> 00:06:26,700
you essentially do, well, if
you're good at visualizing,
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00:06:26,700 --> 00:06:29,720
you just reflect with respect
to the 45 degree line.
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00:06:29,720 --> 00:06:32,150
If you're not so good
at visualizing, what
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00:06:32,150 --> 00:06:33,400
you do is, is what?
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00:06:33,400 --> 00:06:37,170
You first rotate this thing
through 90 degrees and then
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00:06:37,170 --> 00:06:38,950
flop the thing over.
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00:06:38,950 --> 00:06:42,220
And I've taken the liberty
of doing that over here.
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00:06:42,220 --> 00:06:47,060
I take my curve and again, this
would be very effective
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00:06:47,060 --> 00:06:49,780
if I had overlays and wanted
to slide these for you.
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00:06:49,780 --> 00:06:52,880
I think it's something that
you can see on your own.
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00:06:52,880 --> 00:06:54,450
So all I'm doing is what?
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00:06:54,450 --> 00:06:57,590
I rotate this thing through
90 degrees and then
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00:06:57,590 --> 00:07:01,130
flop over the result.
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00:07:01,130 --> 00:07:06,850
And now this curve will be the
graph 'y' equals inverse sine
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00:07:06,850 --> 00:07:09,350
of 'x' provided you can
have an inverse.
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00:07:09,350 --> 00:07:12,290
You see in the old math, this
was fine because multi-valued
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00:07:12,290 --> 00:07:14,290
functions didn't bother
us at all.
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00:07:14,290 --> 00:07:18,010
However, what is clear is notice
that in this particular
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00:07:18,010 --> 00:07:22,810
sense, the accented curve from
before becomes this little
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00:07:22,810 --> 00:07:23,750
region here.
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00:07:23,750 --> 00:07:27,790
And notice that the function
that I defined to be 's sub 0'
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00:07:27,790 --> 00:07:28,860
does have an inverse.
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00:07:28,860 --> 00:07:32,190
In other words, in terms of my
new coordinate system, I can
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00:07:32,190 --> 00:07:34,240
define what?
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00:07:34,240 --> 00:07:40,160
The curve 'y' equals the
inverse of 's sub 0'.
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00:07:40,160 --> 00:07:41,990
The idea being what?
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00:07:41,990 --> 00:07:44,250
That I cannot talk about
the inverse sine.
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00:07:44,250 --> 00:07:46,860
Now the reason I bring this up
is you'll notice that in this
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00:07:46,860 --> 00:07:51,070
text, in most every text, people
do use this notation.
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00:07:51,070 --> 00:07:52,890
And they get around it
in a rather cute way.
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00:07:52,890 --> 00:07:55,750
As you read the text you will
notice the phrase called
149
00:07:55,750 --> 00:07:57,070
principal values.
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00:07:57,070 --> 00:07:58,900
Let me show you what
I mean by that.
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00:08:01,470 --> 00:08:03,850
What happens in the typical text
is they'll say look it.
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00:08:03,850 --> 00:08:07,850
By 'y' equals 'inverse sine
x', we mean that 'x'
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00:08:07,850 --> 00:08:09,860
equals 'sine y'.
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00:08:09,860 --> 00:08:13,990
But that 'y' is restricted to be
between the range of minus
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00:08:13,990 --> 00:08:16,010
pi over 2 and pi over 2.
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00:08:16,010 --> 00:08:18,450
This is called the range
of principal values.
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00:08:18,450 --> 00:08:21,500
Now you see, this sometimes
causes people to be a little
158
00:08:21,500 --> 00:08:24,890
bit upset because why do you
have to make this restriction
159
00:08:24,890 --> 00:08:27,210
since nothing here seems
to indicate that?
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00:08:27,210 --> 00:08:30,310
All I hope that you can see
from my discussion--
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00:08:30,310 --> 00:08:32,950
and this will also be written up
in our supplementary notes,
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00:08:32,950 --> 00:08:36,600
so if you need some
reinforcement from what I'm
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00:08:36,600 --> 00:08:39,500
doing in the-- not from what I'm
doing the lecture, but in
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00:08:39,500 --> 00:08:42,320
addition to what I'm doing in
the lecture you can get this
165
00:08:42,320 --> 00:08:43,370
from the notes too.
166
00:08:43,370 --> 00:08:46,080
But what I want you to see is
that what the average book
167
00:08:46,080 --> 00:08:49,620
defines to be 'y' equals inverse
'sine x', and then
168
00:08:49,620 --> 00:08:53,100
puts in the so-called principal
values, simply turns
169
00:08:53,100 --> 00:08:57,560
out to be what I'm calling
''s sub 0' inverse'.
170
00:08:57,560 --> 00:09:02,780
After all, what was 'y' equals
''s sub 0' inverse x'?
171
00:09:02,780 --> 00:09:04,820
That would be rewritten how?
172
00:09:04,820 --> 00:09:11,200
That's equivalent to saying 'x'
equals ''s sub 0' of y'.
173
00:09:11,200 --> 00:09:15,610
But ''s sub 0' of y' was
defined to be what?
174
00:09:15,610 --> 00:09:22,270
'Sine y' provided that 'y' was
in the range from minus pi
175
00:09:22,270 --> 00:09:25,870
over 2 to plus pi over 2.
176
00:09:25,870 --> 00:09:28,260
So far so good.
177
00:09:28,260 --> 00:09:30,606
At least that's one
man's opinion.
178
00:09:30,606 --> 00:09:34,370
What I'd like to show you now
is again, the beauty of what
179
00:09:34,370 --> 00:09:36,010
inverse functions means.
180
00:09:36,010 --> 00:09:39,620
That from this point on, I can
now, for example, in terms of
181
00:09:39,620 --> 00:09:43,920
calculus, get every single
calculus result I need about
182
00:09:43,920 --> 00:09:47,450
derivatives of the inverse
trigonometric functions just
183
00:09:47,450 --> 00:09:51,050
by restating them in terms
of the ordinary
184
00:09:51,050 --> 00:09:52,390
trigonometric functions.
185
00:09:52,390 --> 00:09:56,190
For example, suppose somebody
were to say to me, find the
186
00:09:56,190 --> 00:10:02,100
derivative of inverse sine of
'x' with respect to 'x'.
187
00:10:02,100 --> 00:10:04,430
Again, keeping in mind now what
this means, otherwise I
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00:10:04,430 --> 00:10:05,670
don't have a function.
189
00:10:05,670 --> 00:10:07,510
I have to have this thing
single-valued.
190
00:10:07,510 --> 00:10:08,860
This thing does it for me.
191
00:10:08,860 --> 00:10:10,770
Look it, do I know
how to find the
192
00:10:10,770 --> 00:10:13,060
derivative of the sine function?
193
00:10:13,060 --> 00:10:14,150
The answer is yes, I do.
194
00:10:14,150 --> 00:10:15,040
We've already done that.
195
00:10:15,040 --> 00:10:17,830
That was the last assignment,
in fact.
196
00:10:17,830 --> 00:10:24,240
In other words, from 'x'
equals 'sine y',
197
00:10:24,240 --> 00:10:26,750
can I find 'dx dy'?
198
00:10:26,750 --> 00:10:27,850
Certainly.
199
00:10:27,850 --> 00:10:34,550
'dx dy' is 'cosine y'.
200
00:10:34,550 --> 00:10:35,560
Now, look it.
201
00:10:35,560 --> 00:10:39,160
We didn't want 'dx dy',
we wanted 'dy dx'.
202
00:10:41,680 --> 00:10:44,530
But since we're on a 1:1 strip
here where the inverse
203
00:10:44,530 --> 00:10:47,520
function does exist, notice that
the relationship between
204
00:10:47,520 --> 00:10:50,930
'dx dy' and 'dy dx'
is that they are
205
00:10:50,930 --> 00:10:52,130
reciprocals of one another.
206
00:10:52,130 --> 00:10:54,640
That was exactly one of the
nice properties of this
207
00:10:54,640 --> 00:10:56,400
notation that we've
talked about on
208
00:10:56,400 --> 00:10:57,950
several occasions before.
209
00:10:57,950 --> 00:11:04,930
In other words, 'dy dx'
is '1 over cosine y'.
210
00:11:04,930 --> 00:11:08,310
By the way, I like this better
than saying things like the
211
00:11:08,310 --> 00:11:10,340
reciprocal of cosine
of secant.
212
00:11:10,340 --> 00:11:14,010
I think when you print a book
people like to use secant
213
00:11:14,010 --> 00:11:17,400
instead of 1 over the cosine
simply because you can get
214
00:11:17,400 --> 00:11:19,530
everything on one line and
don't have to write
215
00:11:19,530 --> 00:11:20,710
fractions this way.
216
00:11:20,710 --> 00:11:23,730
But that is not a major
concern here.
217
00:11:23,730 --> 00:11:27,500
What I do want you to see is
given that 'y' equals 'inverse
218
00:11:27,500 --> 00:11:30,910
sine of x', can I find dy dx?
219
00:11:30,910 --> 00:11:32,010
The answer is yes.
220
00:11:32,010 --> 00:11:33,560
And by the way, notice
that this is
221
00:11:33,560 --> 00:11:35,180
perfectly well defined.
222
00:11:35,180 --> 00:11:38,850
Namely, for a given value of
'y', 'cosine y' you see a
223
00:11:38,850 --> 00:11:40,410
single value.
224
00:11:40,410 --> 00:11:43,800
For a given number, it
has only one cosine.
225
00:11:43,800 --> 00:11:44,640
You see, I don't need
226
00:11:44,640 --> 00:11:47,680
single-valuedness to get this result.
227
00:11:47,680 --> 00:11:50,060
The place I have to be careful
about inverse functions
228
00:11:50,060 --> 00:11:54,140
existing is that since there
are many different x-values
229
00:11:54,140 --> 00:11:59,350
that correspond to the same
y-value, if I don't specify--
230
00:11:59,350 --> 00:12:01,580
you see, if I have a
multi-valued function, if I
231
00:12:01,580 --> 00:12:04,880
don't specify what branch I'm
on the trouble will not come
232
00:12:04,880 --> 00:12:08,940
in when I'm looking for 'dy dx'
at a given value of 'y'.
233
00:12:08,940 --> 00:12:12,420
It's that 99 times out of 100,
when you're looking for 'dy
234
00:12:12,420 --> 00:12:16,070
dx', 'y' is given as
a function of 'x'.
235
00:12:16,070 --> 00:12:19,900
In other words, you want to be
able to convert this in an
236
00:12:19,900 --> 00:12:23,720
ambiguous way into a
function of 'x'.
237
00:12:23,720 --> 00:12:25,240
That's the question
that comes up.
238
00:12:25,240 --> 00:12:27,200
There is nothing wrong
with this answer.
239
00:12:27,200 --> 00:12:32,800
In other words, I can write
down that 'dy dx' is
240
00:12:32,800 --> 00:12:35,070
'1 over cosine y'.
241
00:12:35,070 --> 00:12:37,040
But somebody can say
to me, I'd like the
242
00:12:37,040 --> 00:12:39,170
answer in terms of 'x'.
243
00:12:39,170 --> 00:12:43,400
By the way, how can I get this
answer in terms of 'x'?
244
00:12:43,400 --> 00:12:46,140
That's an interesting
question sometimes.
245
00:12:46,140 --> 00:12:52,280
Recall that 'y' equals
inverse sine of 'x'.
246
00:12:52,280 --> 00:12:53,630
The same as what?
247
00:12:53,630 --> 00:12:58,790
'x' equals 'sine y'.
248
00:12:58,790 --> 00:13:00,790
Now what identity do I know?
249
00:13:00,790 --> 00:13:04,550
I know that sine squared plus
cosine squared is 1.
250
00:13:04,550 --> 00:13:11,050
'x squared' is 'sine
squared y'.
251
00:13:11,050 --> 00:13:12,110
Now what do I know?
252
00:13:12,110 --> 00:13:18,510
I know that 'cosine squared y'
is '1 - sine squared y'.
253
00:13:18,510 --> 00:13:23,930
So 'cosine y' is plus or minus
the square root of '1 - sine
254
00:13:23,930 --> 00:13:25,100
squared y'.
255
00:13:25,100 --> 00:13:26,470
That's plus or minus the square
256
00:13:26,470 --> 00:13:28,660
root of '1 - x squared'.
257
00:13:28,660 --> 00:13:32,710
Notice by the way, in this
notation that 'y' wasn't any
258
00:13:32,710 --> 00:13:36,000
old number. 'y' had to
be in what range?
259
00:13:36,000 --> 00:13:43,610
'y' had to be between minus
pi over 2 and pi over 2.
260
00:13:43,610 --> 00:13:46,490
And as long as that's the case,
notice that in that
261
00:13:46,490 --> 00:13:48,990
range the cosine is positive.
262
00:13:48,990 --> 00:13:53,110
That makes the negative
sign redundant.
263
00:13:53,110 --> 00:13:56,380
And by the way, that negative
sign would not be redundant if
264
00:13:56,380 --> 00:13:59,960
we hadn't restricted our range
to making the curve
265
00:13:59,960 --> 00:14:01,930
single-valued and 1:1.
266
00:14:01,930 --> 00:14:04,170
You see, it's the restriction
that 'y' has to be between
267
00:14:04,170 --> 00:14:07,860
minus pi over 2 and pi over 2
that makes this the positive
268
00:14:07,860 --> 00:14:09,160
square root.
269
00:14:09,160 --> 00:14:11,800
In other words then, what
we see is what?
270
00:14:11,800 --> 00:14:22,880
That therefore 'dy dx' is equal
to '1 over cosine y'.
271
00:14:22,880 --> 00:14:29,850
That's 1 over the square root
of '1 - 'x squared''.
272
00:14:29,850 --> 00:14:31,940
And that's a rather interesting
result.
273
00:14:31,940 --> 00:14:34,240
It's a rather straightforward
result.
274
00:14:34,240 --> 00:14:38,120
I would like to make a few
comments about this thing.
275
00:14:38,120 --> 00:14:41,560
And one of them is what I want
to summarize with later too.
276
00:14:41,560 --> 00:14:43,960
Suppose you are given the
problem of saying, I would
277
00:14:43,960 --> 00:14:47,980
like to trace a curve, or plot
a curve, and all I know about
278
00:14:47,980 --> 00:14:51,680
that curve is that its
derivative at any point is 1
279
00:14:51,680 --> 00:14:55,190
over the square root of 1
minus the square of its
280
00:14:55,190 --> 00:14:56,690
x-coordinate.
281
00:14:56,690 --> 00:15:02,320
Do you have to know any
trigonometry to understand
282
00:15:02,320 --> 00:15:03,240
that problem?
283
00:15:03,240 --> 00:15:07,470
In other words, notice that
this expression in no way
284
00:15:07,470 --> 00:15:10,270
utilizes trigonometry.
285
00:15:10,270 --> 00:15:13,890
Yet to solve this problem, it
appears from what we've shown
286
00:15:13,890 --> 00:15:14,840
is that what?
287
00:15:14,840 --> 00:15:18,730
That given that the slope is
this, the curve itself turns
288
00:15:18,730 --> 00:15:23,150
out to be 'y' equals
'sine inverse x'.
289
00:15:23,150 --> 00:15:26,100
And that is an inverse
trigonometric function.
290
00:15:26,100 --> 00:15:29,790
And so here's a very important
reason as to why the
291
00:15:29,790 --> 00:15:33,110
trigonometric functions
are that important.
292
00:15:33,110 --> 00:15:34,730
The inverse trigonometric
functions.
293
00:15:34,730 --> 00:15:37,860
Namely, inverse trigonometric
functions can
294
00:15:37,860 --> 00:15:39,900
wind up as being what?
295
00:15:39,900 --> 00:15:43,400
The inverse derivative of
a function which is
296
00:15:43,400 --> 00:15:45,690
non-trigonometric.
297
00:15:45,690 --> 00:15:48,860
And that's reason enough
to study these things.
298
00:15:48,860 --> 00:15:52,020
Again, as I told you in my
last lecture, I was sadly
299
00:15:52,020 --> 00:15:55,070
mistaken when I thought that
trigonometry belonged solely
300
00:15:55,070 --> 00:15:56,270
to the surveyor.
301
00:15:56,270 --> 00:15:58,190
I mean, look at all the
different places that this
302
00:15:58,190 --> 00:15:59,570
material comes up in.
303
00:15:59,570 --> 00:16:01,520
And if you don't like practical
applications, at
304
00:16:01,520 --> 00:16:04,220
least observe that all we're
really doing here is
305
00:16:04,220 --> 00:16:08,260
discussing inverse functions in
terms of function itself.
306
00:16:08,260 --> 00:16:10,850
In other words, the result that
we've shown now is that
307
00:16:10,850 --> 00:16:14,560
the integral of 1 over the
square root of '1 - 'x
308
00:16:14,560 --> 00:16:19,680
squared'' is 'inverse sine
x' plus a constant.
309
00:16:19,680 --> 00:16:22,420
By the way, since there is such
a connection between the
310
00:16:22,420 --> 00:16:25,880
inverse trig functions and the
regular trig functions, and
311
00:16:25,880 --> 00:16:29,720
since the trig functions lend
themselves to geometry rather
312
00:16:29,720 --> 00:16:33,850
nicely, I thought I would like
to show you here a rather nice
313
00:16:33,850 --> 00:16:38,930
device that allows you say, to
solve problems like this if
314
00:16:38,930 --> 00:16:41,040
you weren't given the
answer in advance.
315
00:16:41,040 --> 00:16:45,000
In other words, you'll notice
that in our problem by sort of
316
00:16:45,000 --> 00:16:47,590
working backwards, we found
that the answer to this
317
00:16:47,590 --> 00:16:53,980
problem was 'inverse
sine x + c'.
318
00:16:53,980 --> 00:16:58,170
And the question is, what if
we hadn't been given this?
319
00:16:58,170 --> 00:17:01,500
Is there a way that we
could've utilized the
320
00:17:01,500 --> 00:17:04,890
knowledge of trigonometry,
classical trigonometry, to get
321
00:17:04,890 --> 00:17:06,829
a hint as to how to do this?
322
00:17:06,829 --> 00:17:10,760
And not only is there such a
way of doing this, but the
323
00:17:10,760 --> 00:17:13,140
method turns out to be so
important that later in the
324
00:17:13,140 --> 00:17:16,520
course, in the section called
'techniques of integration',
325
00:17:16,520 --> 00:17:20,010
this comes up under the very
special name of trigonometric
326
00:17:20,010 --> 00:17:21,109
substitution.
327
00:17:21,109 --> 00:17:22,550
It works something like this.
328
00:17:22,550 --> 00:17:26,160
Whenever you see the sum or the
difference of two squares,
329
00:17:26,160 --> 00:17:29,200
think of a right triangle.
330
00:17:29,200 --> 00:17:35,730
For example, in this case, if
I call the hypotenuse 1 and
331
00:17:35,730 --> 00:17:39,710
one of the sides 'x', and call
this angle theta say, the
332
00:17:39,710 --> 00:17:42,110
third side of the triangle
is the square
333
00:17:42,110 --> 00:17:46,380
root of '1 - 'x squared''.
334
00:17:46,380 --> 00:17:48,990
And now if I think of this this
way, what is the easiest
335
00:17:48,990 --> 00:17:52,850
relationship that allows me
to express theta as a
336
00:17:52,850 --> 00:17:54,700
trigonometric function
involving 'x'?
337
00:17:54,700 --> 00:17:58,315
There are many trigonometric
relationships I can reach from
338
00:17:58,315 --> 00:17:59,210
this diagram.
339
00:17:59,210 --> 00:18:02,420
But it appears that the easiest
one is the one that
340
00:18:02,420 --> 00:18:05,560
says 'sine theta' equals
'x' over 1.
341
00:18:05,560 --> 00:18:08,950
In other words, 'sine
theta' equals 'x'.
342
00:18:08,950 --> 00:18:12,300
And now you see from this using
differential notation
343
00:18:12,300 --> 00:18:13,120
and the like--
344
00:18:13,120 --> 00:18:16,520
and again, the technique will be
drilled into you and you'll
345
00:18:16,520 --> 00:18:18,750
get plenty of opportunity
for using this in the
346
00:18:18,750 --> 00:18:20,260
exercises in the text.
347
00:18:20,260 --> 00:18:24,110
But for now, I just want you to
get to see an idea of how
348
00:18:24,110 --> 00:18:27,410
the trigonometry does
come back into this.
349
00:18:27,410 --> 00:18:30,620
Notice that from this diagram
I get this relation.
350
00:18:30,620 --> 00:18:33,670
Now taking the differential of
both sides and notice that I'm
351
00:18:33,670 --> 00:18:36,270
working with the ordinary trig
functions here, which I
352
00:18:36,270 --> 00:18:38,510
allegedly know at this stage.
353
00:18:38,510 --> 00:18:42,860
I also know from this triangle
that the square root of '1 -
354
00:18:42,860 --> 00:18:46,380
'x squared'' is 'cosine
theta'.
355
00:18:46,380 --> 00:18:50,330
Therefore, if I make these
substitutions in the integral
356
00:18:50,330 --> 00:18:54,250
'dx' over the square root of '1
- 'x squared'', using the
357
00:18:54,250 --> 00:18:59,210
differential notation 'dx'
becomes replaced by 'cosine
358
00:18:59,210 --> 00:19:01,550
theta 'd theta''.
359
00:19:01,550 --> 00:19:07,370
The square root of '1 - 'x
squared'' becomes replaced by
360
00:19:07,370 --> 00:19:09,060
'cosine theta'.
361
00:19:09,060 --> 00:19:10,780
These cancel.
362
00:19:10,780 --> 00:19:14,110
The integral then turns out to
be what See the integral of 'd
363
00:19:14,110 --> 00:19:17,200
theta' is 'theta'
plus a constant.
364
00:19:17,200 --> 00:19:19,170
But what was 'theta'?
365
00:19:19,170 --> 00:19:21,220
Theta was what?
366
00:19:21,220 --> 00:19:24,870
'Sine theta' was 'x', so 'theta'
is the number whose
367
00:19:24,870 --> 00:19:26,120
sine is 'x'.
368
00:19:31,020 --> 00:19:36,600
Notice how the trigonometry
comes in and helps us to solve
369
00:19:36,600 --> 00:19:39,610
a particular problem.
370
00:19:39,610 --> 00:19:43,700
A particular problem that might
not have seemed that
371
00:19:43,700 --> 00:19:46,740
obvious if we hadn't have
used the trigonometry.
372
00:19:46,740 --> 00:19:50,000
And again, let me point out
where we've used the fact that
373
00:19:50,000 --> 00:19:51,630
we're using 1:1.
374
00:19:51,630 --> 00:19:54,460
You see, when we drew this
particular triangle, we
375
00:19:54,460 --> 00:19:57,890
assumed that 'theta' was
in the first quadrant.
376
00:19:57,890 --> 00:19:59,880
The angle could have been any
place if we're thinking of it
377
00:19:59,880 --> 00:20:00,790
as an angle.
378
00:20:00,790 --> 00:20:04,480
And knowing the sine only
determines the cosine up to a
379
00:20:04,480 --> 00:20:05,680
plus or minus.
380
00:20:05,680 --> 00:20:08,320
In other words, technically
speaking, this should've been
381
00:20:08,320 --> 00:20:10,210
plus or minus over here.
382
00:20:10,210 --> 00:20:12,850
But the fact that for principal
values 'theta' must
383
00:20:12,850 --> 00:20:16,600
be between minus pi over 2 and
pi over 2, and since the
384
00:20:16,600 --> 00:20:19,880
cosine is positive in that range
that was why we were
385
00:20:19,880 --> 00:20:23,010
able to get rid of the
negative sign.
386
00:20:23,010 --> 00:20:28,070
So again, all I want you to see
from this is the fact--
387
00:20:28,070 --> 00:20:29,890
and this is very important.
388
00:20:29,890 --> 00:20:33,120
That aside from the inverse
trigonometric functions being
389
00:20:33,120 --> 00:20:37,830
rather important, we can study
them completely by giving a
390
00:20:37,830 --> 00:20:40,200
short lecture because everything
that we have to
391
00:20:40,200 --> 00:20:43,020
know primarily came from
previous lectures.
392
00:20:43,020 --> 00:20:46,970
The hard part as is often the
case with inverse function
393
00:20:46,970 --> 00:20:51,560
notation is that you may not be
familiar with the language
394
00:20:51,560 --> 00:20:53,710
that rapidly.
395
00:20:53,710 --> 00:20:57,210
I sometimes, when I was first
learning, had to think twice
396
00:20:57,210 --> 00:20:58,890
about notation like this.
397
00:20:58,890 --> 00:21:00,730
Let me start something
new here.
398
00:21:00,730 --> 00:21:02,600
I'll start at the bottom
and work up.
399
00:21:02,600 --> 00:21:05,940
When people wrote down
identities like this, I found
400
00:21:05,940 --> 00:21:09,490
it very difficult to think
in terms of those.
401
00:21:09,490 --> 00:21:10,170
What does this say?
402
00:21:10,170 --> 00:21:13,770
It says that the 'inverse cosine
of x' is pi over 2
403
00:21:13,770 --> 00:21:16,290
minus the 'inverse sine of x'.
404
00:21:16,290 --> 00:21:19,630
And I couldn't remember why
that would be true.
405
00:21:19,630 --> 00:21:22,910
Yet the funny part is, if I
let-- see let's go back to the
406
00:21:22,910 --> 00:21:24,080
top now and start.
407
00:21:24,080 --> 00:21:28,090
If I let 'y' equal 'inverse
cosine x' and tried to draw
408
00:21:28,090 --> 00:21:30,050
what that really says--
409
00:21:30,050 --> 00:21:32,340
see again, I have to be
careful about 1:1
410
00:21:32,340 --> 00:21:32,990
and what have you.
411
00:21:32,990 --> 00:21:35,480
But if I just mechanically
translated this,
412
00:21:35,480 --> 00:21:36,520
this would say what?
413
00:21:36,520 --> 00:21:38,630
'x' equals 'cosine y'.
414
00:21:38,630 --> 00:21:40,120
'Cosine y' is 'x'.
415
00:21:40,120 --> 00:21:41,880
That's 'x' over 1.
416
00:21:41,880 --> 00:21:43,150
I've written it that way.
417
00:21:43,150 --> 00:21:44,430
Well, look it.
418
00:21:44,430 --> 00:21:47,870
If this angle is 'y', it's kind
of clear that this angle
419
00:21:47,870 --> 00:21:50,550
is 'pi over 2' minus 'y'.
420
00:21:50,550 --> 00:21:53,660
Now in this familiar
environment, how difficult is
421
00:21:53,660 --> 00:21:56,160
it for me to see that
the sum of these two
422
00:21:56,160 --> 00:21:58,330
angles is pi over 2?
423
00:21:58,330 --> 00:22:02,160
You see, in that familiar
environment it's almost so
424
00:22:02,160 --> 00:22:03,870
obvious I wonder why anybody
would want to
425
00:22:03,870 --> 00:22:05,340
point it out to me.
426
00:22:05,340 --> 00:22:08,730
Yet notice that from here,
how do I read this?
427
00:22:08,730 --> 00:22:10,290
See what's another
name for 'y'?
428
00:22:10,290 --> 00:22:13,340
In the angular system here,
'y' is that angle
429
00:22:13,340 --> 00:22:15,700
whose cosine is 'x'.
430
00:22:15,700 --> 00:22:17,370
It's 'inverse cosine x'.
431
00:22:17,370 --> 00:22:19,700
What's 'pi over 2' minus 'y'?
432
00:22:19,700 --> 00:22:24,230
'pi over 2' minus 'y' is that
angle whose sine is 'x'.
433
00:22:24,230 --> 00:22:28,470
See notice that the side that's
adjacent to 'y' is
434
00:22:28,470 --> 00:22:31,630
opposite 'pi over
2' minus 'y'.
435
00:22:31,630 --> 00:22:34,720
At any rate, 'pi over 2'
minus 'y' can be named
436
00:22:34,720 --> 00:22:36,130
'inverse sine x'.
437
00:22:36,130 --> 00:22:39,490
And so from this simple thing,
simple because the language is
438
00:22:39,490 --> 00:22:43,170
familiar to me, I get down to
the result that inverse
439
00:22:43,170 --> 00:22:46,570
'cosine x' plus 'inverse
sine x' is pi over 2.
440
00:22:46,570 --> 00:22:50,640
From which, of course, the step
here is a triviality.
441
00:22:50,640 --> 00:22:53,980
See again, when we often deal
with inverses, one of the
442
00:22:53,980 --> 00:22:57,250
fringe benefits that we have is
that when we get stuck, we
443
00:22:57,250 --> 00:23:01,270
can always reduce the
given result.
444
00:23:01,270 --> 00:23:05,850
Reverse the terminology so to
speak and return from the
445
00:23:05,850 --> 00:23:10,450
inverse language to the
original language.
446
00:23:10,450 --> 00:23:14,280
Now again, let me point out that
once you have a result
447
00:23:14,280 --> 00:23:17,000
like this all the results
of calculus work
448
00:23:17,000 --> 00:23:18,500
the same way as before.
449
00:23:18,500 --> 00:23:21,650
For example, suppose somebody
says to me, gee, I wonder what
450
00:23:21,650 --> 00:23:25,400
the derivative of 'inverse
cosine x' is
451
00:23:25,400 --> 00:23:26,930
with respect to 'x'?
452
00:23:26,930 --> 00:23:30,220
To be sure, I could go all
through this again and mimic
453
00:23:30,220 --> 00:23:32,250
the results of getting
the derivative
454
00:23:32,250 --> 00:23:33,640
for the inverse sine.
455
00:23:33,640 --> 00:23:37,490
But notice now, by this result,
this is just the
456
00:23:37,490 --> 00:23:42,680
derivative with respect to 'x'
of pi over 2 minus the
457
00:23:42,680 --> 00:23:46,130
'inverse sine of x'.
458
00:23:46,130 --> 00:23:49,080
But this is the derivative
of a difference.
459
00:23:49,080 --> 00:23:51,290
And a derivative of a difference
is the difference
460
00:23:51,290 --> 00:23:52,430
of derivatives.
461
00:23:52,430 --> 00:23:55,720
And pi over 2 is a constant.
462
00:23:55,720 --> 00:23:58,040
So the derivative of
pi over 2 is 0.
463
00:23:58,040 --> 00:24:01,190
I already know how to
differentiate 'inverse sine of
464
00:24:01,190 --> 00:24:02,820
x', I've done that before.
465
00:24:02,820 --> 00:24:03,650
So I want what?
466
00:24:03,650 --> 00:24:05,110
Minus that derivative.
467
00:24:05,110 --> 00:24:10,760
I want 'minus 'd dx' 'inverse
sine of x''.
468
00:24:10,760 --> 00:24:13,130
And that in turn is just what?
469
00:24:13,130 --> 00:24:14,420
Well, I'll write it over here.
470
00:24:14,420 --> 00:24:17,200
It's just minus 1
over the square
471
00:24:17,200 --> 00:24:20,180
root of '1 - 'x squared''.
472
00:24:20,180 --> 00:24:25,210
And again, another result
but obtained with a
473
00:24:25,210 --> 00:24:28,620
minimum of new knowledge.
474
00:24:28,620 --> 00:24:33,360
Now, I could go on but I think
that from here on in, it's
475
00:24:33,360 --> 00:24:38,080
much easier for you to dig out
what you want on your own.
476
00:24:42,650 --> 00:24:45,200
I had some other material that
I thought I would give you,
477
00:24:45,200 --> 00:24:47,380
but I think that this will just
turn out to be a little
478
00:24:47,380 --> 00:24:49,500
bit on the boring side now.
479
00:24:49,500 --> 00:24:51,590
Not boring, but in the sense
that either you see what I'm
480
00:24:51,590 --> 00:24:53,580
driving at or you don't.
481
00:24:53,580 --> 00:24:57,480
And what I'd like you to do now
is simply to go and see
482
00:24:57,480 --> 00:25:00,540
how much of this material
on inverse trigonometric
483
00:25:00,540 --> 00:25:03,700
functions is yours now
free of charge.
484
00:25:03,700 --> 00:25:07,280
If I have to pick one thing I
want to caution you about,
485
00:25:07,280 --> 00:25:10,590
don't be upset by the language
called 'principal values' and
486
00:25:10,590 --> 00:25:11,550
what have you.
487
00:25:11,550 --> 00:25:15,090
Everything comes out in the wash
if you recognize that we
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00:25:15,090 --> 00:25:19,440
legally cannot define an inverse
function unless we
489
00:25:19,440 --> 00:25:21,840
have a 1:1 function
to begin with.
490
00:25:21,840 --> 00:25:24,290
So we must take the multi-valued
trigonometric
491
00:25:24,290 --> 00:25:29,240
function and view it as
a union of 1:1 curves.
492
00:25:29,240 --> 00:25:32,730
And then you see if you want to
pick a different principal
493
00:25:32,730 --> 00:25:34,880
value, I think if you understand
what's happening
494
00:25:34,880 --> 00:25:38,000
basically, you'll be able to
do this thing on your own.
495
00:25:38,000 --> 00:25:41,860
In any event though, I think
this is enough on what we call
496
00:25:41,860 --> 00:25:43,870
the 'inverse circular
functions'.
497
00:25:43,870 --> 00:25:45,550
So until next time, goodbye.
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00:25:48,580 --> 00:25:51,120
ANNOUNCER: Funding for the
publication of this video was
499
00:25:51,120 --> 00:25:55,830
provided by the Gabriella and
Paul Rosenbaum Foundation.
500
00:25:55,830 --> 00:26:00,010
Help OCW continue to provide
free and open access to MIT
501
00:26:00,010 --> 00:26:04,200
courses by making a donation
at ocw.mit.edu/donate.