1
00:00:00,000 --> 00:00:07,350
JOEL LEWIS: Hi.

2
00:00:07,350 --> 00:00:08,970
Welcome back to recitation.

3
00:00:08,970 --> 00:00:10,980
Today I wanted to talk
about something that's

4
00:00:10,980 --> 00:00:13,300
mentioned in the notes but
wasn't covered in lecture

5
00:00:13,300 --> 00:00:15,250
because of the exam review.

6
00:00:15,250 --> 00:00:19,780
So this is, the subject is
hyperbolic trig functions.

7
00:00:19,780 --> 00:00:23,111
So first I just wanted to define
them for you and graph them

8
00:00:23,111 --> 00:00:25,110
so we can get a little
bit of a feeling for what

9
00:00:25,110 --> 00:00:27,040
these functions are
like, and then I'm

10
00:00:27,040 --> 00:00:30,240
going to explain to you why
they have the words hyperbolic

11
00:00:30,240 --> 00:00:32,440
and trig in their names.

12
00:00:32,440 --> 00:00:34,950
So these are some
interesting functions.

13
00:00:34,950 --> 00:00:39,360
They're not, they don't--
aren't quite as important

14
00:00:39,360 --> 00:00:43,470
as your usual, sort of
circular trig functions.

15
00:00:43,470 --> 00:00:47,050
But yeah, so let me introduce
them and let me jump

16
00:00:47,050 --> 00:00:48,520
in just with their definition.

17
00:00:48,520 --> 00:00:51,340
So there are two
most important ones.

18
00:00:51,340 --> 00:00:53,150
Just like a regular
trigonometric functions

19
00:00:53,150 --> 00:00:55,200
there's the sine and
the cosine and then

20
00:00:55,200 --> 00:00:57,440
you can write the other
four trigonometric functions

21
00:00:57,440 --> 00:00:58,370
in terms of them.

22
00:00:58,370 --> 00:01:00,370
So for hyperbolic
trig functions we

23
00:01:00,370 --> 00:01:03,260
have the hyperbolic cosine
and the hyperbolic sine.

24
00:01:03,260 --> 00:01:06,760
So the notation here,
we write c o s h.

25
00:01:06,760 --> 00:01:08,970
So the h for hyperbolic.

26
00:01:08,970 --> 00:01:10,560
So hyperbolic cosine.

27
00:01:10,560 --> 00:01:13,530
And usually we
pronounce this "cosh."

28
00:01:13,530 --> 00:01:15,880
And similarly, for
the hyperbolic sine

29
00:01:15,880 --> 00:01:18,500
we write s i n h,
for hyperbolic sine,

30
00:01:18,500 --> 00:01:20,180
except in the reverse order.

31
00:01:20,180 --> 00:01:23,920
And we usually pronounce this
"sinch," so in American English

32
00:01:23,920 --> 00:01:25,630
as if there were an
extra c in there.

33
00:01:25,630 --> 00:01:27,080
Sinch.

34
00:01:27,080 --> 00:01:27,580
OK.

35
00:01:27,580 --> 00:01:30,650
So these functions have
fairly simple definitions

36
00:01:30,650 --> 00:01:33,570
in terms of the exponential
function, e to the x.

37
00:01:33,570 --> 00:01:39,050
So cosh of x is defined to be e
to the x plus e to the minus x

38
00:01:39,050 --> 00:01:40,470
divided by 2.

39
00:01:40,470 --> 00:01:45,730
And sinh of x is defined to be
e to x minus e to the minus x

40
00:01:45,730 --> 00:01:47,030
divided by 2.

41
00:01:47,030 --> 00:01:50,070
So if you remember what
your graph of e to the x

42
00:01:50,070 --> 00:01:52,390
looks like, and your
graph of e to the minus x,

43
00:01:52,390 --> 00:01:57,111
it's not hard to see that the
graphs of cosh x and sinh x

44
00:01:57,111 --> 00:01:58,360
should look sort of like this.

45
00:01:58,360 --> 00:02:02,200
So for cosh x, so we see as
x gets big, so e to the minus

46
00:02:02,200 --> 00:02:04,150
x is going to 0.

47
00:02:04,150 --> 00:02:05,220
It's not very important.

48
00:02:05,220 --> 00:02:08,210
So it mostly is driven
by this e to the x part.

49
00:02:08,210 --> 00:02:13,570
And as x gets negative and
big, then this is going to 0

50
00:02:13,570 --> 00:02:16,140
and this is getting
larger and larger.

51
00:02:16,140 --> 00:02:18,420
So we got something
that looks like this.

52
00:02:18,420 --> 00:02:20,647
So it looks a little
bit, in this picture

53
00:02:20,647 --> 00:02:22,230
it looks a little
bit like a parabola,

54
00:02:22,230 --> 00:02:24,586
but the growth here is
exponential at both sides.

55
00:02:24,586 --> 00:02:26,710
So in fact, this is growing
much, much, much faster

56
00:02:26,710 --> 00:02:29,270
than, say, 1 plus x squared.

57
00:02:29,270 --> 00:02:30,730
So it's a much steeper curve.

58
00:02:30,730 --> 00:02:33,010
OK.

59
00:02:33,010 --> 00:02:35,060
And it reaches its
minimum here at x

60
00:02:35,060 --> 00:02:38,780
equals 0-- it has the
value 1 plus 1 over 2.

61
00:02:38,780 --> 00:02:42,460
So its minimum there
is at x equals 0,

62
00:02:42,460 --> 00:02:44,872
it has its minimum value 1.

63
00:02:44,872 --> 00:02:47,080
For sinh, OK, so we're taking
the difference of them.

64
00:02:47,080 --> 00:02:52,370
So it's similar when x
is positive and large,

65
00:02:52,370 --> 00:02:55,230
e to the x is big,
and e to the minus

66
00:02:55,230 --> 00:02:58,480
x is pretty small,
almost negligible.

67
00:02:58,480 --> 00:03:01,260
So we got exponential
growth off that side.

68
00:03:01,260 --> 00:03:05,230
When x becomes
negative and large,

69
00:03:05,230 --> 00:03:09,060
e to the x is going to 0, e to
the minus x is becoming large,

70
00:03:09,060 --> 00:03:11,380
but it's-- we've got
a minus sign here.

71
00:03:11,380 --> 00:03:14,880
So as x goes to minus
infinity, this curve

72
00:03:14,880 --> 00:03:16,700
goes also to minus infinity.

73
00:03:16,700 --> 00:03:21,300
And again, the growth here
is exponential in both cases.

74
00:03:21,300 --> 00:03:24,400
And if you were curious, say
about what the slope there

75
00:03:24,400 --> 00:03:26,970
at the origin is, you could
quickly take a derivative

76
00:03:26,970 --> 00:03:29,760
and check that that's passing
through the origin with slope 1

77
00:03:29,760 --> 00:03:31,140
there.

78
00:03:31,140 --> 00:03:31,640
OK.

79
00:03:31,640 --> 00:03:33,730
So this is a sort
of basic picture

80
00:03:33,730 --> 00:03:36,810
of what these curves look like.

81
00:03:36,810 --> 00:03:42,260
They have some nice properties,
and let me talk about them.

82
00:03:42,260 --> 00:03:46,614
So for example,
one nice thing you

83
00:03:46,614 --> 00:03:48,030
might notice about
these functions

84
00:03:48,030 --> 00:03:49,988
is that it's easy to
compute their derivatives.

85
00:03:49,988 --> 00:03:50,830
Right?

86
00:03:50,830 --> 00:04:01,640
So if we look at d/dx of cosh
x, in order to compute that,

87
00:04:01,640 --> 00:04:04,440
well, just look at the
definition of cosh.

88
00:04:04,440 --> 00:04:07,450
So it's really just a sum of
two exponential functions.

89
00:04:07,450 --> 00:04:10,130
Exponential functions are
easy to take the derivatives.

90
00:04:10,130 --> 00:04:12,490
Take the derivative of e to
the x, you get e to the x.

91
00:04:12,490 --> 00:04:15,390
Take the derivative of e to
the minus x, well, OK, so

92
00:04:15,390 --> 00:04:17,790
it's a little chain rule, so
you get a minus 1 in front.

93
00:04:17,790 --> 00:04:23,407
So the derivative of cosh x is
e to the x minus e to the minus

94
00:04:23,407 --> 00:04:25,310
x over 2.

95
00:04:25,310 --> 00:04:26,810
But we have a name for this.

96
00:04:26,810 --> 00:04:31,250
This is actually just sinh x.

97
00:04:31,250 --> 00:04:33,550
So the derivative
of cosh is sinh,

98
00:04:33,550 --> 00:04:42,370
and the derivative
of sinh, well, OK.

99
00:04:42,370 --> 00:04:45,117
You look at the same
thing, take this formula,

100
00:04:45,117 --> 00:04:45,950
take its derivative.

101
00:04:45,950 --> 00:04:48,737
Well, e to the x, take its
derivative, you get e to the x.

102
00:04:48,737 --> 00:04:50,320
e to the minus x,
take its derivative,

103
00:04:50,320 --> 00:04:53,220
you get minus e to the minus
x, so those two minus signs

104
00:04:53,220 --> 00:04:54,860
cancel out and become a plus.

105
00:04:54,860 --> 00:05:01,660
So this is e to the x plus
e to the minus x over 2,

106
00:05:01,660 --> 00:05:03,860
which is cosh x.

107
00:05:03,860 --> 00:05:05,942
So here you have
some behavior that's

108
00:05:05,942 --> 00:05:08,400
a little bit reminiscent of
the behavior of trig functions.

109
00:05:08,400 --> 00:05:08,980
Right?

110
00:05:08,980 --> 00:05:11,510
For trig functions, if you
take the derivative of sine

111
00:05:11,510 --> 00:05:12,514
you get cosine.

112
00:05:12,514 --> 00:05:14,180
And if you take the
derivative of cosine

113
00:05:14,180 --> 00:05:16,390
you almost get back sine,
but you get minus sine.

114
00:05:16,390 --> 00:05:19,236
So here you don't have
that extra negative sign

115
00:05:19,236 --> 00:05:19,902
floating around.

116
00:05:19,902 --> 00:05:20,690
Right?

117
00:05:20,690 --> 00:05:22,560
So you, when you take
the derivative of cosh

118
00:05:22,560 --> 00:05:24,550
you get sinh on the nose.

119
00:05:24,550 --> 00:05:27,080
No minus sign needed.

120
00:05:27,080 --> 00:05:29,630
So that's interesting.

121
00:05:29,630 --> 00:05:33,150
But the real reason
that these have

122
00:05:33,150 --> 00:05:39,200
the words trig in their name is
actually a little bit deeper.

123
00:05:39,200 --> 00:05:42,810
So let me come over here
and draw a couple pictures.

124
00:05:42,810 --> 00:05:46,960
So the normal trig
functions-- what sometimes we

125
00:05:46,960 --> 00:05:50,270
call the circular trig functions
if we want to distinguish them

126
00:05:50,270 --> 00:05:53,730
from the hyperbolic trig
functions-- they're closely--

127
00:05:53,730 --> 00:05:55,420
so circular trig
functions, they're

128
00:05:55,420 --> 00:05:57,560
closely related to
the unit circle.

129
00:05:57,560 --> 00:06:02,110
So the unit circle
has equation x squared

130
00:06:02,110 --> 00:06:05,790
plus y squared equals 1.

131
00:06:05,790 --> 00:06:06,820
It's a circle.

132
00:06:06,820 --> 00:06:09,600
Well, close enough, right?

133
00:06:09,600 --> 00:06:12,870
And what is the
nice relationship

134
00:06:12,870 --> 00:06:14,860
between this circle
and the trig functions?

135
00:06:14,860 --> 00:06:18,500
Well, if you choose any
point on this circle,

136
00:06:18,500 --> 00:06:21,990
then there exists
some value of t

137
00:06:21,990 --> 00:06:23,980
such that this point
has coordinates

138
00:06:23,980 --> 00:06:28,890
cosine t comma sine t.

139
00:06:28,890 --> 00:06:31,080
Now it happens
that the value of t

140
00:06:31,080 --> 00:06:33,640
is actually the angle
that that radius makes

141
00:06:33,640 --> 00:06:35,407
with the positive axis.

142
00:06:35,407 --> 00:06:37,240
But not going to worry
about that right now.

143
00:06:37,240 --> 00:06:40,830
It's not the key idea of import.

144
00:06:40,830 --> 00:06:45,330
So as t varies through
the real numbers,

145
00:06:45,330 --> 00:06:48,500
the point cosine t,
sine t, that varies

146
00:06:48,500 --> 00:06:50,130
and it just goes
around this curve.

147
00:06:50,130 --> 00:06:52,530
So it traces out
this circle exactly.

148
00:06:52,530 --> 00:06:55,550
So the hyperbolic
trig functions show up

149
00:06:55,550 --> 00:06:58,670
in a very similar situation.

150
00:06:58,670 --> 00:07:00,170
But instead of
looking at the unit

151
00:07:00,170 --> 00:07:02,060
circle, what we
want to look at is

152
00:07:02,060 --> 00:07:04,567
the unit rectangular hyperbola.

153
00:07:04,567 --> 00:07:05,650
So what do I mean by that?

154
00:07:05,650 --> 00:07:07,790
Well, so instead of
taking the equation

155
00:07:07,790 --> 00:07:11,240
x squared plus y squared
equals 1, which gives a circle,

156
00:07:11,240 --> 00:07:13,340
I'm going to look at a
very similar equation that

157
00:07:13,340 --> 00:07:14,310
gives a hyperbola.

158
00:07:14,310 --> 00:07:19,950
So this is the equation
x squared minus y squared

159
00:07:19,950 --> 00:07:21,690
equals 1.

160
00:07:21,690 --> 00:07:28,970
So if you if you graph this
equation, what you'll see

161
00:07:28,970 --> 00:07:32,250
is that, well, it passes
through the point (1, 0).

162
00:07:32,250 --> 00:07:38,690
And then we've got
one branch here,

163
00:07:38,690 --> 00:07:40,650
we've got a little
asymptote there.

164
00:07:45,120 --> 00:07:47,550
So it's got a right
branch like that,

165
00:07:47,550 --> 00:07:51,760
and also it's symmetric
across the y-axis.

166
00:07:51,760 --> 00:08:00,570
So there's a symmetric
left branch here.

167
00:08:00,570 --> 00:08:03,750
So this is the graph
of the equation

168
00:08:03,750 --> 00:08:05,890
x squared minus y
squared equals 1.

169
00:08:05,890 --> 00:08:09,200
So it's this hyperbola.

170
00:08:09,200 --> 00:08:14,110
Now what I claim is
that cosh and sinh have

171
00:08:14,110 --> 00:08:18,050
the same relationship to this
hyperbola as cosine and sine

172
00:08:18,050 --> 00:08:19,020
have to the circle.

173
00:08:19,020 --> 00:08:21,180
Well, so I'm fudging
a little bit.

174
00:08:21,180 --> 00:08:24,440
So it turns out it's only the
right half of the hyperbola.

175
00:08:24,440 --> 00:08:25,610
So what do I mean by that?

176
00:08:25,610 --> 00:08:28,210
Well, here's what
I'd like to do.

177
00:08:28,210 --> 00:08:33,040
Set x equals-- so we're going
to introduce a new variable,

178
00:08:33,040 --> 00:08:41,090
u-- I'm going to set x equal
cosh u and y equals sinh u.

179
00:08:41,090 --> 00:08:43,540
And I'm going to
look at the quantity

180
00:08:43,540 --> 00:08:44,850
x squared minus y squared.

181
00:08:44,850 --> 00:08:48,770
So x squared minus y squared.

182
00:08:51,481 --> 00:08:53,480
So this is, so we use
most of the same notations

183
00:08:53,480 --> 00:08:55,020
for hyperbolic trig
functions that we

184
00:08:55,020 --> 00:08:56,270
do for regular trig functions.

185
00:08:56,270 --> 00:09:04,820
So this is cosh squared
u minus sinh squared u.

186
00:09:04,820 --> 00:09:07,246
And now we can plug in the
formulas for cosh and sinh

187
00:09:07,246 --> 00:09:08,110
that we have.

188
00:09:08,110 --> 00:09:16,690
So this is equal to e to the u
plus e to the minus u over 2,

189
00:09:16,690 --> 00:09:22,150
quantity squared, minus e to
the u minus e to the minus

190
00:09:22,150 --> 00:09:25,990
u over 2, quantity squared.

191
00:09:25,990 --> 00:09:28,650
And now we can expand
out both of these factors

192
00:09:28,650 --> 00:09:33,070
and-- both of these squares,
rather, and put them together.

193
00:09:33,070 --> 00:09:37,540
So over 2 squared is
over 4 and we square this

194
00:09:37,540 --> 00:09:39,860
and we get e to the 2u.

195
00:09:39,860 --> 00:09:43,480
OK, so then we get 2 times e to
the u times e to the minus u.

196
00:09:43,480 --> 00:09:48,170
But e to the u times e to the
minus u is just 1, so plus 2.

197
00:09:48,170 --> 00:09:59,170
Plus e to the minus 2u minus
e to the 2u minus 2 plus e

198
00:09:59,170 --> 00:10:03,840
to the minus 2u-- so same
thing over here-- over 4.

199
00:10:03,840 --> 00:10:07,320
OK, so the e to the 2u's cancel
and the e to the minus 2u's

200
00:10:07,320 --> 00:10:09,580
cancel and we're left
with 2 minus minus 2.

201
00:10:09,580 --> 00:10:10,420
That's 4.

202
00:10:10,420 --> 00:10:13,860
So this is 4 over 4,
so this is equal to 1.

203
00:10:13,860 --> 00:10:14,360
OK.

204
00:10:14,360 --> 00:10:19,060
So if x is equal to cosh u
and y is equal to sinh u,

205
00:10:19,060 --> 00:10:21,970
then x squared minus y
squared is equal to 1.

206
00:10:21,970 --> 00:10:26,270
So if we choose a point
(cosh u, sinh u) for some u,

207
00:10:26,270 --> 00:10:28,850
that point lies
on this hyperbola.

208
00:10:28,850 --> 00:10:29,844
That's what this says.

209
00:10:29,844 --> 00:10:35,540
That this point-- OK, so
the point (cosh u, sinh u)

210
00:10:35,540 --> 00:10:38,150
is somewhere on this hyperbola.

211
00:10:38,150 --> 00:10:41,040
And what's also true is the
sort of reverse statement.

212
00:10:41,040 --> 00:10:45,980
If you look at all such points,
if you let u vary and look--

213
00:10:45,980 --> 00:10:47,640
through the real
numbers and you ask

214
00:10:47,640 --> 00:10:50,210
what happens to this
point (cosh u, sinh u),

215
00:10:50,210 --> 00:10:52,620
the answers is that it
traces out the right half

216
00:10:52,620 --> 00:10:53,700
of this hyperbola.

217
00:10:53,700 --> 00:10:56,805
If you go back to the
graph of y equals cosh x,

218
00:10:56,805 --> 00:10:59,750
you'll see that the hyperbolic
cosine function is always

219
00:10:59,750 --> 00:11:00,580
positive.

220
00:11:00,580 --> 00:11:04,070
So we can't-- over here, we
can't trace out this left

221
00:11:04,070 --> 00:11:05,610
branch where x is negative.

222
00:11:05,610 --> 00:11:07,700
Although it's easy
enough to say what does

223
00:11:07,700 --> 00:11:09,950
trace out this left branch.

224
00:11:09,950 --> 00:11:12,280
Since it's just
the mirror image,

225
00:11:12,280 --> 00:11:18,740
this is traced out by
minus cosh u comma sinh u.

226
00:11:22,220 --> 00:11:24,520
So there's a-- so the
hyperbolic trig functions have

227
00:11:24,520 --> 00:11:27,870
the same relationship to
this branch of this hyperbola

228
00:11:27,870 --> 00:11:31,120
that the regular trig
functions have to the circle.

229
00:11:31,120 --> 00:11:35,070
So there's where the words
hyperbolic and trig functions

230
00:11:35,070 --> 00:11:36,120
come from.

231
00:11:36,120 --> 00:11:39,230
So let me say one
more thing about them,

232
00:11:39,230 --> 00:11:41,830
which is that we
saw that they have

233
00:11:41,830 --> 00:11:45,060
this analogy with
regular trig functions.

234
00:11:45,060 --> 00:11:45,560
Right?

235
00:11:45,560 --> 00:11:47,820
So instead of satisfying
cosine squared

236
00:11:47,820 --> 00:11:52,350
plus sine squared equals 1, they
satisfy cosh squared minus sinh

237
00:11:52,350 --> 00:11:53,590
squared equals 1.

238
00:11:53,590 --> 00:11:56,770
And instead of satisfying
the derivative of sine

239
00:11:56,770 --> 00:12:00,250
equals cosine and the derivative
of cosine equals minus sine,

240
00:12:00,250 --> 00:12:03,150
they satisfy derivative
of cosh equals

241
00:12:03,150 --> 00:12:05,990
sinh and derivative
of sinh equals cosh.

242
00:12:05,990 --> 00:12:07,690
So similar relationships.

243
00:12:07,690 --> 00:12:09,680
Not exactly the
same, but similar.

244
00:12:09,680 --> 00:12:13,010
So this is true of a lot
of trig relationships,

245
00:12:13,010 --> 00:12:16,960
that there's a corresponding
formula for the hyperbolic trig

246
00:12:16,960 --> 00:12:17,560
functions.

247
00:12:17,560 --> 00:12:22,170
So one example of such
a formula is your--

248
00:12:22,170 --> 00:12:24,790
for example, your angle
addition formulas.

249
00:12:24,790 --> 00:12:28,160
So I'm going to just leave
this is an exercise for you.

250
00:12:28,160 --> 00:12:31,482
So let me, I guess
I'll just stick it

251
00:12:31,482 --> 00:12:33,440
in this funny little
piece of board right here.

252
00:12:36,284 --> 00:12:36,825
So, exercise.

253
00:12:41,530 --> 00:13:03,580
Find sinh of x plus y and cosh
of x plus y in terms of sinh

254
00:13:03,580 --> 00:13:14,380
x, cosh x, sinh y, and cosh y.

255
00:13:14,380 --> 00:13:17,890
So in other words, find
the corresponding formula

256
00:13:17,890 --> 00:13:20,070
to the angle addition
formula in that case

257
00:13:20,070 --> 00:13:21,980
of the hyperbolic
trig functions.

258
00:13:21,980 --> 00:13:23,637
So I'll leave you with that.