1
00:00:01,530 --> 00:00:06,074
In this video, we'll define
the hyperbolic functions

2
00:00:06,074 --> 00:00:10,050
hyperbolic sign, hyperbolic
cosine and Hyperbolic Tangent.

3
00:00:11,220 --> 00:00:13,508
We'll define hyperbolic cosine

4
00:00:13,508 --> 00:00:17,510
first. We write hyperbolic
cosine like this.

5
00:00:18,670 --> 00:00:24,982
We pronounce
it Cosh

6
00:00:24,982 --> 00:00:30,807
X. And it's defined
using the exponential functions

7
00:00:30,807 --> 00:00:34,060
like this. Cos X equals.

8
00:00:34,560 --> 00:00:38,459
Each the X Plus E to the

9
00:00:38,459 --> 00:00:42,810
minus X. All divided by
two.

10
00:00:44,450 --> 00:00:48,578
Hyperbolic sign is
written like this.

11
00:00:49,700 --> 00:00:55,856
There are two
ways of pronouncing

12
00:00:55,856 --> 00:01:01,610
hyperbolic sign. In this video
I'll call it shine, but you will

13
00:01:01,610 --> 00:01:03,458
come across some people who call

14
00:01:03,458 --> 00:01:08,206
it's inch. And Shine X is
defined with the exponential

15
00:01:08,206 --> 00:01:09,914
function, again like this.

16
00:01:10,630 --> 00:01:16,974
We have each the X minus this
time, each the minus X all

17
00:01:16,974 --> 00:01:18,438
divided by two.

18
00:01:20,400 --> 00:01:27,456
Now we can use our knowledge of
the graphs of E to the X

19
00:01:27,456 --> 00:01:32,496
to workout the graphs of Shine X
and Koch X.

20
00:01:33,120 --> 00:01:37,792
Here I've got some graphs of E
to the X over 2 and E to the

21
00:01:37,792 --> 00:01:38,960
minus X over 2.

22
00:01:39,990 --> 00:01:43,588
First, we'll workout what
cautious 0 is.

23
00:01:44,470 --> 00:01:48,345
A quickly remind you that

24
00:01:48,345 --> 00:01:55,336
quarterbacks. Is each the X plus
E to the minus X over 2?

25
00:01:56,050 --> 00:01:59,770
So Koch of

26
00:01:59,770 --> 00:02:05,686
0. Is eat 0,
which is one.

27
00:02:06,000 --> 00:02:12,162
Plus E to the zero again, which
is one all divided by two.

28
00:02:12,170 --> 00:02:14,634
Which is 2 over 2, which is one.

29
00:02:15,800 --> 00:02:19,820
So we can Mark Koch
0 onto a graph here.

30
00:02:22,060 --> 00:02:25,350
Now we can workout. What happens
is X gets big.

31
00:02:26,110 --> 00:02:33,502
We can split up how we write
quash X into this form. We have

32
00:02:33,502 --> 00:02:40,366
each the X over 2 plus E
to the minus X over 2.

33
00:02:41,260 --> 00:02:45,740
Now we already have the graphs
of E to the X over 2 and E to

34
00:02:45,740 --> 00:02:49,940
the minus X over 2 here, and we
can see that as X gets big.

35
00:02:51,020 --> 00:02:53,540
East the X over 2 gets very very

36
00:02:53,540 --> 00:02:59,214
big. But each of the minus X
over 2 gets very very small.

37
00:03:00,380 --> 00:03:04,340
So as X gets bigger, the second
part of the sum.

38
00:03:04,890 --> 00:03:06,690
Gets very very close to 0.

39
00:03:07,950 --> 00:03:10,578
So the whole thing.

40
00:03:10,760 --> 00:03:17,357
Gets very very close to
just eat the X.

41
00:03:17,900 --> 00:03:25,080
Over 2. So
now we can rule

42
00:03:25,080 --> 00:03:29,420
that side of the
graph.

43
00:03:31,220 --> 00:03:33,464
I'll just turn around the graph
so I can actually draw it.

44
00:03:41,610 --> 00:03:47,550
Now remember that be'cause each
the minus X over 2.

45
00:03:48,080 --> 00:03:51,932
Is always bigger than zero even
though it gets very very small.

46
00:03:52,990 --> 00:03:55,270
Call shacks will always stay

47
00:03:55,270 --> 00:03:58,066
above. The graph of each of
the axe over 2.

48
00:03:59,190 --> 00:04:03,020
The grass get closer
together as X gets big, but

49
00:04:03,020 --> 00:04:07,616
Cosh X must always stay
above E to the X over 2.

50
00:04:09,490 --> 00:04:16,574
Now we'll see what happens when
ex gets large and negative.

51
00:04:16,580 --> 00:04:22,712
Now when
ex gets

52
00:04:22,712 --> 00:04:28,934
larger negative.
We can see that each the X

53
00:04:28,934 --> 00:04:33,582
over 2 gets very very small,
but E to the minus X over 2

54
00:04:33,582 --> 00:04:34,910
gets very very big.

55
00:04:36,120 --> 00:04:39,630
So writing Cosh X again.

56
00:04:41,050 --> 00:04:48,834
As each the X over 2 plus
E to the minus X over 2.

57
00:04:49,640 --> 00:04:53,084
We can see that the first part
of the sum, this time gets very

58
00:04:53,084 --> 00:04:57,331
very small. And the second part
of the sum gets very very big.

59
00:04:58,420 --> 00:04:59,640
So the whole thing.

60
00:05:00,310 --> 00:05:04,426
As X gets larger, negative will
get closer and closer to eat.

61
00:05:04,426 --> 00:05:06,141
The minus X over 2.

62
00:05:07,150 --> 00:05:09,119
So we can draw in that side
of the graph now.

63
00:05:17,440 --> 00:05:20,818
And remember again.

64
00:05:21,520 --> 00:05:25,797
That E to the X over 2 even
though it's getting very, very

65
00:05:25,797 --> 00:05:27,442
small, will always be bigger

66
00:05:27,442 --> 00:05:30,405
than 0. So the graph will always

67
00:05:30,405 --> 00:05:32,572
stay above. East the minus X

68
00:05:32,572 --> 00:05:37,690
over 2. As X gets more negative,
the graphs get closer together.

69
00:05:38,570 --> 00:05:43,107
But the graph will always stay
above each the minus X over 2.

70
00:05:43,940 --> 00:05:45,680
So here is a graph.

71
00:05:46,340 --> 00:05:47,339
Of Cos X.

72
00:05:50,210 --> 00:05:55,098
Now you can see that this graph
is symmetric about the Y axis.

73
00:05:56,080 --> 00:05:59,470
And that tells us that cause

74
00:05:59,470 --> 00:06:06,196
checks. Is always equal
to Koch of minus

75
00:06:06,196 --> 00:06:07,004
X?

76
00:06:08,740 --> 00:06:11,386
Now let's deal with the
graph was trying X.

77
00:06:12,780 --> 00:06:17,356
This time to help me. I've got a
graph of each of the X over 2.

78
00:06:17,980 --> 00:06:22,264
And minus E to the minus
X over 2.

79
00:06:23,680 --> 00:06:26,184
Let's see what happens
when X equals 0.

80
00:06:27,600 --> 00:06:31,646
I'll just remind
you that Shine X.

81
00:06:33,260 --> 00:06:39,740
Is each the X minus E to
the minus X over 2?

82
00:06:41,890 --> 00:06:45,470
So shine. Of

83
00:06:45,470 --> 00:06:49,398
0. Is each zero?

84
00:06:50,360 --> 00:06:51,440
Which is one.

85
00:06:52,740 --> 00:06:56,310
Minus Y to 0, which is one.

86
00:06:56,960 --> 00:07:02,966
All over 2 so that zero over
2 which is 0.

87
00:07:04,140 --> 00:07:07,326
Right, so we can plot that

88
00:07:07,326 --> 00:07:12,140
point. Onto a
graph at 00.

89
00:07:14,140 --> 00:07:20,512
Now we'll see what happens
as X gets large.

90
00:07:21,780 --> 00:07:26,424
Like before, we can split up the
way we write Shine X.

91
00:07:27,540 --> 00:07:30,610
So we write Shine X.

92
00:07:30,610 --> 00:07:37,370
As each the X over
2 minus E to the

93
00:07:37,370 --> 00:07:40,074
minus X over 2.

94
00:07:41,820 --> 00:07:43,640
Now, using the same logic as we

95
00:07:43,640 --> 00:07:48,335
did before. And remembering what
the graphs of E to the X&E to

96
00:07:48,335 --> 00:07:50,610
the minus X over 2 looked like.

97
00:07:51,960 --> 00:07:53,880
We can see that.

98
00:07:55,230 --> 00:07:56,910
The first part of the some here.

99
00:07:57,470 --> 00:08:00,470
Gets very, very big as X
gets large.

100
00:08:02,180 --> 00:08:03,776
But the second part of the sum.

101
00:08:04,930 --> 00:08:07,569
Gets very, very small as X gets

102
00:08:07,569 --> 00:08:10,410
large. So again.

103
00:08:11,000 --> 00:08:17,750
Floor Jacks.
The whole of Shine

104
00:08:17,750 --> 00:08:21,270
X becomes closer and

105
00:08:21,270 --> 00:08:26,470
closer. To eat the
X over 2.

106
00:08:26,610 --> 00:08:33,510
So now we
can draw in

107
00:08:33,510 --> 00:08:39,260
that half of
the graph.

108
00:08:40,920 --> 00:08:46,764
No,
just

109
00:08:46,764 --> 00:08:49,686
this

110
00:08:49,686 --> 00:08:55,958
time.
The graph will stay below each

111
00:08:55,958 --> 00:08:57,530
the X over 2.

112
00:08:58,760 --> 00:09:00,375
And that's because we're taking

113
00:09:00,375 --> 00:09:03,359
away. E to the minus X over 2.

114
00:09:04,930 --> 00:09:09,750
Now let's see what happens when
ex gets larger negative.

115
00:09:11,340 --> 00:09:18,850
Again, just so you can
see what's happening, I'll write

116
00:09:18,850 --> 00:09:22,605
up are split up form

117
00:09:22,605 --> 00:09:28,453
shine X. I
remember as

118
00:09:28,453 --> 00:09:31,235
ex got

119
00:09:31,235 --> 00:09:35,200
larger negative. Each of

120
00:09:35,200 --> 00:09:37,518
the X. Got very very small.

121
00:09:38,270 --> 00:09:41,218
For the first part of the son
gets very very small.

122
00:09:41,960 --> 00:09:44,390
Eat the minus X over 2.

123
00:09:45,020 --> 00:09:46,228
Got very very big.

124
00:09:47,150 --> 00:09:52,418
So this whole thing as X gets
larger, negative gets closer and

125
00:09:52,418 --> 00:09:56,960
closer to. Just the second bit
of the sum, because the first

126
00:09:56,960 --> 00:09:58,784
bit is getting closer to 0.

127
00:09:59,340 --> 00:10:03,248
So for large negative.

128
00:10:03,470 --> 00:10:09,850
X.

129
00:10:10,460 --> 00:10:17,270
Sean X. Gets
very, very close to.

130
00:10:18,040 --> 00:10:21,530
Minus each the minus X.

131
00:10:22,130 --> 00:10:23,500
Over 2.

132
00:10:25,480 --> 00:10:26,992
So we can draw the second bit of

133
00:10:26,992 --> 00:10:33,820
a graph and now. Now again,
the graph must

134
00:10:33,820 --> 00:10:40,420
stay above minus
E to the

135
00:10:40,420 --> 00:10:43,720
minus X over

136
00:10:43,720 --> 00:10:50,664
2. Be'cause each the X over
2 is never 0 even though it gets

137
00:10:50,664 --> 00:10:51,858
very very small.

138
00:10:52,540 --> 00:10:57,164
So that adds a bit on to minus E
to the minus X over 2. It keeps

139
00:10:57,164 --> 00:10:58,252
the graph above it.

140
00:10:59,580 --> 00:11:06,236
Now you know tist that shine X
for LG X is close to eat the X

141
00:11:06,236 --> 00:11:12,436
over 2. And Koch X also remember
was close to E to the X over 2

142
00:11:12,436 --> 00:11:14,704
for LG X. So that must mean.

143
00:11:15,610 --> 00:11:21,195
Floor Jacks.
Khashab

144
00:11:21,195 --> 00:11:27,274
X. And Shine
X are very close together.

145
00:11:27,820 --> 00:11:35,070
And we can see that
if I show you a

146
00:11:35,070 --> 00:11:40,145
graph of Cynex and koshek
spotted together.

147
00:11:41,920 --> 00:11:45,160
Got a slightly different scale
this time so it looks a bit

148
00:11:45,160 --> 00:11:47,320
wider. But there is caution X.

149
00:11:48,100 --> 00:11:51,736
And they're showing X, which
gets very, very close to quash X

150
00:11:51,736 --> 00:11:52,948
as X gets large.

151
00:11:56,400 --> 00:12:01,300
Now we'll define
hyperbolic tangent.

152
00:12:03,170 --> 00:12:07,042
We can define hyperbolic tangent
in terms of shine and Koch.

153
00:12:10,240 --> 00:12:12,968
We write Hyperbolic Tangent.

154
00:12:13,890 --> 00:12:19,954
Like this? And again, there are
two possible pronunciations for

155
00:12:19,954 --> 00:12:23,150
this. Some people call it touch.

156
00:12:23,990 --> 00:12:26,042
But I'm in this video,
we'll call it fan.

157
00:12:27,520 --> 00:12:30,646
And we define it to be.

158
00:12:30,830 --> 00:12:36,648
Sean X.
Over Cosh X.

159
00:12:37,270 --> 00:12:40,588
And again, there's an
exponential function definition

160
00:12:40,588 --> 00:12:45,328
which we can work out by looking
at this definition.

161
00:12:46,350 --> 00:12:53,310
Because this is Sean X which is
E to the X minus E to the

162
00:12:53,310 --> 00:12:55,166
minus X over 2.

163
00:12:55,830 --> 00:13:02,783
Divided by Cosh X, which is E to
the X Plus E to the minus X over

164
00:13:02,783 --> 00:13:10,035
2. And that's when you work it
out is if the X minus E to the

165
00:13:10,035 --> 00:13:12,070
minus X all divided by.

166
00:13:12,640 --> 00:13:17,104
Each the X Plus E
to the minus X.

167
00:13:19,120 --> 00:13:22,871
Now this thing doesn't look very
useful for working out the

168
00:13:22,871 --> 00:13:26,963
graph. It's a bit complicated,
so this time will use the graph

169
00:13:26,963 --> 00:13:30,373
of shine and Koch to workout the
graphs of ban.

170
00:13:31,100 --> 00:13:38,396
Now here I've got
a set of axes

171
00:13:38,396 --> 00:13:44,780
which I'll use to
draw down onto.

172
00:13:45,870 --> 00:13:50,700
I've got an asymptotes here at Y
equals 1 and one at White was

173
00:13:50,700 --> 00:13:53,115
minus one and you'll see why in

174
00:13:53,115 --> 00:13:58,940
a minute. I remember
that 4X large.

175
00:14:02,040 --> 00:14:05,484
We saw that

176
00:14:05,484 --> 00:14:10,748
Cosh X. Was
very, very close.

177
00:14:11,960 --> 00:14:15,809
To Shine X.

178
00:14:15,810 --> 00:14:19,050
So that must mean

179
00:14:19,050 --> 00:14:22,260
that. Shine X.

180
00:14:22,940 --> 00:14:27,788
Over Cosh
X.

181
00:14:29,030 --> 00:14:32,420
Must be very close to one.

182
00:14:32,940 --> 00:14:35,970
So as X

183
00:14:35,970 --> 00:14:42,410
gets large. We should expect
Linix to get very close to one.

184
00:14:43,810 --> 00:14:46,640
Now for large negative X.

185
00:14:47,350 --> 00:14:55,002
We
saw

186
00:14:55,002 --> 00:14:58,828
that.

187
00:14:59,760 --> 00:15:06,098
Call shacks Was very close
to eat a minus X.

188
00:15:06,640 --> 00:15:07,960
Over 2.

189
00:15:09,530 --> 00:15:10,760
And Shine X.

190
00:15:14,340 --> 00:15:19,920
Was very close to minus E to the
minus X over 2.

191
00:15:21,200 --> 00:15:23,195
So you can see here that Shine

192
00:15:23,195 --> 00:15:25,882
X. Is very very close to minus

193
00:15:25,882 --> 00:15:31,857
Cosh X. So that must
mean that Shine X.

194
00:15:32,490 --> 00:15:34,968
Over Cosh X.

195
00:15:35,610 --> 00:15:41,497
Is very very close
to minus one.

196
00:15:42,470 --> 00:15:44,636
So as X gets larger, negative.

197
00:15:45,300 --> 00:15:48,612
We should expect the graph to
get very very close to minus

198
00:15:48,612 --> 00:15:54,118
one. Now just to finish off,
will see what happens when X

199
00:15:54,118 --> 00:16:00,264
equals 0. So
for X

200
00:16:00,264 --> 00:16:03,300
equals 0.

201
00:16:04,180 --> 00:16:12,440
Thanks.
Which is shine.

202
00:16:12,460 --> 00:16:15,079
Over Cos X.

203
00:16:16,430 --> 00:16:20,216
Will be well shine of X.

204
00:16:21,140 --> 00:16:23,275
Was zero, so that would be 0.

205
00:16:24,230 --> 00:16:27,458
Call Shabaks was one.

206
00:16:27,500 --> 00:16:32,164
So fun of X when X equals
0 is just zero.

207
00:16:33,210 --> 00:16:35,418
Now, putting these three bits of

208
00:16:35,418 --> 00:16:41,628
information together. We can
plot 00 on that.

209
00:16:43,210 --> 00:16:44,770
And his ex got large.

210
00:16:45,590 --> 00:16:49,754
Thanks, got close to one so we
can see this thing getting

211
00:16:49,754 --> 00:16:50,795
closer to one.

212
00:16:53,390 --> 00:16:56,336
And his ex got large and

213
00:16:56,336 --> 00:17:01,339
negative. Thanks, got close to
minus one so we can draw this.

214
00:17:02,210 --> 00:17:04,530
Going down toward minus one.

215
00:17:06,920 --> 00:17:10,850
But remember that.

216
00:17:11,630 --> 00:17:15,020
These never actually reach
one or minus one. They just

217
00:17:15,020 --> 00:17:19,088
get very very close to them
as X gets large reserves gets

218
00:17:19,088 --> 00:17:19,766
more negative.

219
00:17:20,860 --> 00:17:23,569
Also notice that.

220
00:17:24,200 --> 00:17:27,330
Fun. Of

221
00:17:27,330 --> 00:17:30,984
ex. Is equal to

222
00:17:30,984 --> 00:17:34,779
minus fan? Of minus X.

223
00:17:35,530 --> 00:17:41,542
Now hyperbolic functions
has my densities

224
00:17:41,542 --> 00:17:47,554
which are very
similar to trig

225
00:17:47,554 --> 00:17:53,566
identities, but not
quite the same.

226
00:17:54,700 --> 00:17:57,989
In this video I'll show you and
prove for you, too, and then

227
00:17:57,989 --> 00:18:01,025
I'll show you some more at the
end, which you control yourself.

228
00:18:01,710 --> 00:18:05,224
The first one will prove is this

229
00:18:05,224 --> 00:18:07,890
one. It's caution.

230
00:18:08,540 --> 00:18:11,604
X squared

231
00:18:11,604 --> 00:18:15,470
minus shine. X

232
00:18:15,470 --> 00:18:18,140
squared Equals 1.

233
00:18:19,420 --> 00:18:24,316
Not prove this will substitute
in the exponential definitions

234
00:18:24,316 --> 00:18:26,492
for Koch and shine.

235
00:18:27,480 --> 00:18:34,620
So we have the costs. X was each
the X Plus E to the minus X over

236
00:18:34,620 --> 00:18:37,560
2, and we want that all squared.

237
00:18:38,730 --> 00:18:44,745
We take away shine X squared,
which was E to the X minus E to

238
00:18:44,745 --> 00:18:47,552
the minus X over 2 all squared.

239
00:18:49,860 --> 00:18:52,576
Now notice that in both bits of

240
00:18:52,576 --> 00:18:55,354
this. We have the same

241
00:18:55,354 --> 00:18:58,665
denominator squared. So we can
take that out of the factor.

242
00:18:59,810 --> 00:19:01,218
So we have that.

243
00:19:01,870 --> 00:19:03,448
As a quarter.

244
00:19:04,170 --> 00:19:06,818
Of the numerator squared.

245
00:19:07,780 --> 00:19:12,960
So we have each of the X Plus E
to the minus X squared.

246
00:19:13,570 --> 00:19:20,412
Minus eat the AX minus E
to the minus X squared.

247
00:19:22,150 --> 00:19:26,416
Now we want to expand out
these squared brackets.

248
00:19:28,410 --> 00:19:31,518
So that is going to be.

249
00:19:31,660 --> 00:19:34,720
Well, these brackets.

250
00:19:35,650 --> 00:19:40,546
Are going to be each the X times
E to the X which is E to the 2X.

251
00:19:41,400 --> 00:19:47,510
And because each the X Times E
to the minus X is one.

252
00:19:48,300 --> 00:19:50,538
And we get that happening twice.

253
00:19:51,140 --> 00:19:53,210
We get a plus two here.

254
00:19:53,760 --> 00:19:57,577
And each the minus X Times E to
the minus X.

255
00:19:58,250 --> 00:20:00,010
Is each the minus 2X?

256
00:20:02,910 --> 00:20:07,370
And I will take away this
bracket squared which working

257
00:20:07,370 --> 00:20:10,046
out before we can see is.

258
00:20:10,640 --> 00:20:12,050
Each the two weeks again.

259
00:20:12,720 --> 00:20:16,640
And this time we get a minus two

260
00:20:16,640 --> 00:20:18,880
here. And we get.

261
00:20:19,680 --> 00:20:23,180
Plus, if the minus 2X.

262
00:20:24,600 --> 00:20:28,400
Now we can remove these

263
00:20:28,400 --> 00:20:32,380
inner brackets. And we get a

264
00:20:32,380 --> 00:20:35,140
quarter. Of.

265
00:20:36,250 --> 00:20:42,310
Just take out the first brackets
first, E to the two X +2, plus E

266
00:20:42,310 --> 00:20:43,926
to the minus 2X.

267
00:20:44,540 --> 00:20:46,520
Put a minus here, so that's

268
00:20:46,520 --> 00:20:48,748
minus. Each the 2X.

269
00:20:50,150 --> 00:20:56,870
+2. Minus.
E to the minus 2X.

270
00:20:57,490 --> 00:21:02,027
And of course, a lot of things
that are going to cancel here.

271
00:21:02,027 --> 00:21:06,564
We have E to the two X minus
either 2X, so these go.

272
00:21:08,190 --> 00:21:12,684
We have plus E to the minus
two X here and minus it again

273
00:21:12,684 --> 00:21:13,968
there, so they go.

274
00:21:15,680 --> 00:21:21,936
And then we just have 2 + 2, so
the whole thing turns out to be

275
00:21:21,936 --> 00:21:25,064
1/4 of 2 + 2, which is 4.

276
00:21:25,070 --> 00:21:26,258
Which is one.

277
00:21:27,520 --> 00:21:30,413
So there we have the proof
that cause squared X minus

278
00:21:30,413 --> 00:21:33,306
sign squared X, which is
all that is equal to 1.

279
00:21:34,340 --> 00:21:42,290
And one more density or
proof you now is this

280
00:21:42,290 --> 00:21:45,470
one. It's shine 2X.

281
00:21:46,900 --> 00:21:49,429
Equal to two.

282
00:21:51,570 --> 00:21:57,270
Sean X. Call
Josh X.

283
00:21:58,430 --> 00:22:02,824
Now with this one, I think it's
easier to start with the right

284
00:22:02,824 --> 00:22:06,880
hand side, so will do that
again. Will a substitute in the

285
00:22:06,880 --> 00:22:09,246
exponential definitions to the
right hand side.

286
00:22:10,070 --> 00:22:15,915
So 2.
Sean

287
00:22:15,915 --> 00:22:19,790
X. Cosh

288
00:22:19,790 --> 00:22:23,518
X. Is equal to.

289
00:22:24,330 --> 00:22:28,269
Two lots of.

290
00:22:29,040 --> 00:22:36,978
Each 3X minus each the minus X
over 2 times caution X which was

291
00:22:36,978 --> 00:22:43,782
E to the X Plus E to
the minus X over 2.

292
00:22:45,530 --> 00:22:50,546
Now you can see the two councils
with one of these denominators,

293
00:22:50,546 --> 00:22:52,218
so will cancel those.

294
00:22:53,020 --> 00:22:55,589
And take out this half as a

295
00:22:55,589 --> 00:23:00,920
factor. So now
we have 1/2.

296
00:23:02,390 --> 00:23:09,718
Each of the X minus E to the
minus X Times E to the X Plus

297
00:23:09,718 --> 00:23:12,008
E to the minus X.

298
00:23:12,570 --> 00:23:15,244
And if we just multiply out the

299
00:23:15,244 --> 00:23:17,508
brackets. We get a half.

300
00:23:18,360 --> 00:23:22,035
Get each the X Times E to the X,
which is each the two acts.

301
00:23:23,610 --> 00:23:26,706
Each the X Times Plus E to the

302
00:23:26,706 --> 00:23:30,218
minus X. Which is plus one.

303
00:23:31,410 --> 00:23:37,350
Minus E to the minus X times Y
to the X, which is minus one.

304
00:23:37,880 --> 00:23:40,650
And finally, minus each the

305
00:23:40,650 --> 00:23:47,180
minus 2X. So obviously
these ones cancel.

306
00:23:47,850 --> 00:23:55,014
And we get this bracket here,
which was E to the two

307
00:23:55,014 --> 00:23:58,948
X minus. Eat the minus 2X.

308
00:23:59,580 --> 00:24:01,588
All divided by two.

309
00:24:02,430 --> 00:24:04,992
And that is the definition of

310
00:24:04,992 --> 00:24:11,610
shine 2X. So that's
the second identity proved.

311
00:24:13,520 --> 00:24:19,850
Now here is more identity's
which you can prove yourself.

312
00:24:21,800 --> 00:24:27,266
The first one
is Koch 2X.

313
00:24:28,400 --> 00:24:29,870
Is equal to.

314
00:24:31,160 --> 00:24:34,760
Call shacks

315
00:24:34,760 --> 00:24:39,290
squared.
Plus

316
00:24:41,090 --> 00:24:43,529
Shine X squared.

317
00:24:44,450 --> 00:24:46,850
We also have.

318
00:24:47,390 --> 00:24:50,630
Shine of X

319
00:24:50,630 --> 00:24:52,790
Plus Y.

320
00:24:52,790 --> 00:24:55,380
Equals.

321
00:24:56,360 --> 00:25:02,180
Shine X.
Gosh why?

322
00:25:03,430 --> 00:25:06,796
Plus Shine

323
00:25:06,796 --> 00:25:11,860
Y. Koch
X.

324
00:25:14,290 --> 00:25:16,618
And also for Koch.

325
00:25:17,460 --> 00:25:20,130
X plus Y.

326
00:25:21,930 --> 00:25:23,589
We have cautious.

327
00:25:24,450 --> 00:25:31,808
X. Call
Shuai Plus Shine

328
00:25:31,808 --> 00:25:34,980
X. Shine

329
00:25:34,980 --> 00:25:38,496
Y. And

330
00:25:38,496 --> 00:25:41,988
two more.

331
00:25:41,990 --> 00:25:46,280
Gosh. X over
2.

332
00:25:47,340 --> 00:25:51,302
All squared. Is 1

333
00:25:51,302 --> 00:25:57,128
plus. Cosh X all
divided by two.

334
00:25:58,150 --> 00:26:01,350
And finish off shine.

335
00:26:02,110 --> 00:26:03,970
X over 2.

336
00:26:04,500 --> 00:26:11,510
All squared. Is
Cosh X minus one

337
00:26:11,510 --> 00:26:14,830
all divided by two?

338
00:26:16,020 --> 00:26:20,167
Now you can see how similar
these are to trig identities.

339
00:26:20,910 --> 00:26:24,402
And in fact, the hyperbolic
functions and triggered entities

340
00:26:24,402 --> 00:26:25,954
are very closely related.

341
00:26:26,710 --> 00:26:29,947
And you can learn more about
that if you go and study complex

342
00:26:29,947 --> 00:26:37,068
numbers. And before we
finish, we look at some

343
00:26:37,068 --> 00:26:42,956
other functions which are
related to hyperbolic functions.

344
00:26:43,780 --> 00:26:45,259
Firstly, some notation.

345
00:26:45,820 --> 00:26:48,225
You've already seen me use

346
00:26:48,225 --> 00:26:55,430
notation like. Sean X
squared, which means.

347
00:26:55,460 --> 00:26:59,156
The whole of Shine X squared.

348
00:27:00,010 --> 00:27:02,122
Just remember that if you see

349
00:27:02,122 --> 00:27:09,810
something like. Sean X with a
minus one there that is not

350
00:27:09,810 --> 00:27:11,688
equal to 1.

351
00:27:12,420 --> 00:27:15,510
Over Shine X.

352
00:27:16,130 --> 00:27:19,986
This refers to the inverse
function of shine.

353
00:27:21,060 --> 00:27:27,670
If you remember that if we
have a function F.

354
00:27:29,020 --> 00:27:32,224
F minus one the inverse function
works like this.

355
00:27:33,400 --> 00:27:40,190
You have F minus one.
The inverse function of X

356
00:27:40,190 --> 00:27:46,980
gives you this thing why
where F of Y is

357
00:27:46,980 --> 00:27:49,017
equal to X.

358
00:27:49,790 --> 00:27:51,258
And the same way.

359
00:27:53,160 --> 00:27:55,938
Shine with the minus one there.

360
00:27:56,620 --> 00:27:59,780
Of X gives You Y.

361
00:28:01,650 --> 00:28:08,034
Will shine of why was
equal to X?

362
00:28:09,160 --> 00:28:12,562
We can also define inverse
functions for caution than.

363
00:28:13,540 --> 00:28:15,840
But for these we have to
restrict the domains of

364
00:28:15,840 --> 00:28:16,300
the function.

365
00:28:17,350 --> 00:28:23,603
For inverse Cosh, Cosh the minus
one of X, we need to have.

366
00:28:24,220 --> 00:28:27,230
X greater than or equal to 1.

367
00:28:28,280 --> 00:28:33,816
And for the inverse of
fan of X.

368
00:28:34,360 --> 00:28:40,808
We need X to be greater than
minus one and less than one.

369
00:28:43,560 --> 00:28:49,936
Some other functions you
might come across other

370
00:28:49,936 --> 00:28:51,530
reciprocal functions.

371
00:28:53,630 --> 00:28:55,850
These are such.

372
00:28:57,950 --> 00:29:05,790
X. Which is defined
to be one over Cosh of X.

373
00:29:06,300 --> 00:29:07,788
There are two more of these.

374
00:29:08,990 --> 00:29:15,054
This Kosach
Of X, which

375
00:29:15,054 --> 00:29:20,442
is one over
shine of X.

376
00:29:21,490 --> 00:29:26,270
And finally, there's
cough X.

377
00:29:28,910 --> 00:29:32,880
Which is one over fan.