WEBVTT 00:00:01.530 --> 00:00:06.074 In this video, we'll define the hyperbolic functions 00:00:06.074 --> 00:00:10.050 hyperbolic sign, hyperbolic cosine and Hyperbolic Tangent. 00:00:11.220 --> 00:00:13.508 We'll define hyperbolic cosine 00:00:13.508 --> 00:00:17.510 first. We write hyperbolic cosine like this. 00:00:18.670 --> 00:00:24.982 We pronounce it Cosh 00:00:24.982 --> 00:00:30.807 X. And it's defined using the exponential functions 00:00:30.807 --> 00:00:34.060 like this. Cos X equals. 00:00:34.560 --> 00:00:38.459 Each the X Plus E to the 00:00:38.459 --> 00:00:42.810 minus X. All divided by two. 00:00:44.450 --> 00:00:48.578 Hyperbolic sign is written like this. 00:00:49.700 --> 00:00:55.856 There are two ways of pronouncing 00:00:55.856 --> 00:01:01.610 hyperbolic sign. In this video I'll call it shine, but you will 00:01:01.610 --> 00:01:03.458 come across some people who call 00:01:03.458 --> 00:01:08.206 it's inch. And Shine X is defined with the exponential 00:01:08.206 --> 00:01:09.914 function, again like this. 00:01:10.630 --> 00:01:16.974 We have each the X minus this time, each the minus X all 00:01:16.974 --> 00:01:18.438 divided by two. 00:01:20.400 --> 00:01:27.456 Now we can use our knowledge of the graphs of E to the X 00:01:27.456 --> 00:01:32.496 to workout the graphs of Shine X and Koch X. 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 00:01:37.792 --> 00:01:38.960 minus X over 2. 00:01:39.990 --> 00:01:43.588 First, we'll workout what cautious 0 is. 00:01:44.470 --> 00:01:48.345 A quickly remind you that 00:01:48.345 --> 00:01:55.336 quarterbacks. Is each the X plus E to the minus X over 2? 00:01:56.050 --> 00:01:59.770 So Koch of 00:01:59.770 --> 00:02:05.686 0. Is eat 0, which is one. 00:02:06.000 --> 00:02:12.162 Plus E to the zero again, which is one all divided by two. 00:02:12.170 --> 00:02:14.634 Which is 2 over 2, which is one. 00:02:15.800 --> 00:02:19.820 So we can Mark Koch 0 onto a graph here. 00:02:22.060 --> 00:02:25.350 Now we can workout. What happens is X gets big. 00:02:26.110 --> 00:02:33.502 We can split up how we write quash X into this form. We have 00:02:33.502 --> 00:02:40.366 each the X over 2 plus E to the minus X over 2. 00:02:41.260 --> 00:02:45.740 Now we already have the graphs of E to the X over 2 and E to 00:02:45.740 --> 00:02:49.940 the minus X over 2 here, and we can see that as X gets big. 00:02:51.020 --> 00:02:53.540 East the X over 2 gets very very 00:02:53.540 --> 00:02:59.214 big. But each of the minus X over 2 gets very very small. 00:03:00.380 --> 00:03:04.340 So as X gets bigger, the second part of the sum. 00:03:04.890 --> 00:03:06.690 Gets very very close to 0. 00:03:07.950 --> 00:03:10.578 So the whole thing. 00:03:10.760 --> 00:03:17.357 Gets very very close to just eat the X. 00:03:17.900 --> 00:03:25.080 Over 2. So now we can rule 00:03:25.080 --> 00:03:29.420 that side of the graph. 00:03:31.220 --> 00:03:33.464 I'll just turn around the graph so I can actually draw it. 00:03:41.610 --> 00:03:47.550 Now remember that be'cause each the minus X over 2. 00:03:48.080 --> 00:03:51.932 Is always bigger than zero even though it gets very very small. 00:03:52.990 --> 00:03:55.270 Call shacks will always stay 00:03:55.270 --> 00:03:58.066 above. The graph of each of the axe over 2. 00:03:59.190 --> 00:04:03.020 The grass get closer together as X gets big, but 00:04:03.020 --> 00:04:07.616 Cosh X must always stay above E to the X over 2. 00:04:09.490 --> 00:04:16.574 Now we'll see what happens when ex gets large and negative. 00:04:16.580 --> 00:04:22.712 Now when ex gets 00:04:22.712 --> 00:04:28.934 larger negative. We can see that each the X 00:04:28.934 --> 00:04:33.582 over 2 gets very very small, but E to the minus X over 2 00:04:33.582 --> 00:04:34.910 gets very very big. 00:04:36.120 --> 00:04:39.630 So writing Cosh X again. 00:04:41.050 --> 00:04:48.834 As each the X over 2 plus E to the minus X over 2. 00:04:49.640 --> 00:04:53.084 We can see that the first part of the sum, this time gets very 00:04:53.084 --> 00:04:57.331 very small. And the second part of the sum gets very very big. 00:04:58.420 --> 00:04:59.640 So the whole thing. 00:05:00.310 --> 00:05:04.426 As X gets larger, negative will get closer and closer to eat. 00:05:04.426 --> 00:05:06.141 The minus X over 2. 00:05:07.150 --> 00:05:09.119 So we can draw in that side of the graph now. 00:05:17.440 --> 00:05:20.818 And remember again. 00:05:21.520 --> 00:05:25.797 That E to the X over 2 even though it's getting very, very 00:05:25.797 --> 00:05:27.442 small, will always be bigger 00:05:27.442 --> 00:05:30.405 than 0. So the graph will always 00:05:30.405 --> 00:05:32.572 stay above. East the minus X 00:05:32.572 --> 00:05:37.690 over 2. As X gets more negative, the graphs get closer together. 00:05:38.570 --> 00:05:43.107 But the graph will always stay above each the minus X over 2. 00:05:43.940 --> 00:05:45.680 So here is a graph. 00:05:46.340 --> 00:05:47.339 Of Cos X. 00:05:50.210 --> 00:05:55.098 Now you can see that this graph is symmetric about the Y axis. 00:05:56.080 --> 00:05:59.470 And that tells us that cause 00:05:59.470 --> 00:06:06.196 checks. Is always equal to Koch of minus 00:06:06.196 --> 00:06:07.004 X? 00:06:08.740 --> 00:06:11.386 Now let's deal with the graph was trying X. 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. 00:06:17.980 --> 00:06:22.264 And minus E to the minus X over 2. 00:06:23.680 --> 00:06:26.184 Let's see what happens when X equals 0. 00:06:27.600 --> 00:06:31.646 I'll just remind you that Shine X. 00:06:33.260 --> 00:06:39.740 Is each the X minus E to the minus X over 2? 00:06:41.890 --> 00:06:45.470 So shine. Of 00:06:45.470 --> 00:06:49.398 0. Is each zero? 00:06:50.360 --> 00:06:51.440 Which is one. 00:06:52.740 --> 00:06:56.310 Minus Y to 0, which is one. 00:06:56.960 --> 00:07:02.966 All over 2 so that zero over 2 which is 0. 00:07:04.140 --> 00:07:07.326 Right, so we can plot that 00:07:07.326 --> 00:07:12.140 point. Onto a graph at 00. 00:07:14.140 --> 00:07:20.512 Now we'll see what happens as X gets large. 00:07:21.780 --> 00:07:26.424 Like before, we can split up the way we write Shine X. 00:07:27.540 --> 00:07:30.610 So we write Shine X. 00:07:30.610 --> 00:07:37.370 As each the X over 2 minus E to the 00:07:37.370 --> 00:07:40.074 minus X over 2. 00:07:41.820 --> 00:07:43.640 Now, using the same logic as we 00:07:43.640 --> 00:07:48.335 did before. And remembering what the graphs of E to the X&E to 00:07:48.335 --> 00:07:50.610 the minus X over 2 looked like. 00:07:51.960 --> 00:07:53.880 We can see that. 00:07:55.230 --> 00:07:56.910 The first part of the some here. 00:07:57.470 --> 00:08:00.470 Gets very, very big as X gets large. 00:08:02.180 --> 00:08:03.776 But the second part of the sum. 00:08:04.930 --> 00:08:07.569 Gets very, very small as X gets 00:08:07.569 --> 00:08:10.410 large. So again. 00:08:11.000 --> 00:08:17.750 Floor Jacks. The whole of Shine 00:08:17.750 --> 00:08:21.270 X becomes closer and 00:08:21.270 --> 00:08:26.470 closer. To eat the X over 2. 00:08:26.610 --> 00:08:33.510 So now we can draw in 00:08:33.510 --> 00:08:39.260 that half of the graph. 00:08:40.920 --> 00:08:46.764 No, just 00:08:46.764 --> 00:08:49.686 this 00:08:49.686 --> 00:08:55.958 time. The graph will stay below each 00:08:55.958 --> 00:08:57.530 the X over 2. 00:08:58.760 --> 00:09:00.375 And that's because we're taking 00:09:00.375 --> 00:09:03.359 away. E to the minus X over 2. 00:09:04.930 --> 00:09:09.750 Now let's see what happens when ex gets larger negative. 00:09:11.340 --> 00:09:18.850 Again, just so you can see what's happening, I'll write 00:09:18.850 --> 00:09:22.605 up are split up form 00:09:22.605 --> 00:09:28.453 shine X. I remember as 00:09:28.453 --> 00:09:31.235 ex got 00:09:31.235 --> 00:09:35.200 larger negative. Each of 00:09:35.200 --> 00:09:37.518 the X. Got very very small. 00:09:38.270 --> 00:09:41.218 For the first part of the son gets very very small. 00:09:41.960 --> 00:09:44.390 Eat the minus X over 2. 00:09:45.020 --> 00:09:46.228 Got very very big. 00:09:47.150 --> 00:09:52.418 So this whole thing as X gets larger, negative gets closer and 00:09:52.418 --> 00:09:56.960 closer to. Just the second bit of the sum, because the first 00:09:56.960 --> 00:09:58.784 bit is getting closer to 0. 00:09:59.340 --> 00:10:03.248 So for large negative. 00:10:03.470 --> 00:10:09.850 X. 00:10:10.460 --> 00:10:17.270 Sean X. Gets very, very close to. 00:10:18.040 --> 00:10:21.530 Minus each the minus X. 00:10:22.130 --> 00:10:23.500 Over 2. 00:10:25.480 --> 00:10:26.992 So we can draw the second bit of 00:10:26.992 --> 00:10:33.820 a graph and now. Now again, the graph must 00:10:33.820 --> 00:10:40.420 stay above minus E to the 00:10:40.420 --> 00:10:43.720 minus X over 00:10:43.720 --> 00:10:50.664 2. Be'cause each the X over 2 is never 0 even though it gets 00:10:50.664 --> 00:10:51.858 very very small. 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 00:10:57.164 --> 00:10:58.252 the graph above it. 00:10:59.580 --> 00:11:06.236 Now you know tist that shine X for LG X is close to eat the X 00:11:06.236 --> 00:11:12.436 over 2. And Koch X also remember was close to E to the X over 2 00:11:12.436 --> 00:11:14.704 for LG X. So that must mean. 00:11:15.610 --> 00:11:21.195 Floor Jacks. Khashab 00:11:21.195 --> 00:11:27.274 X. And Shine X are very close together. 00:11:27.820 --> 00:11:35.070 And we can see that if I show you a 00:11:35.070 --> 00:11:40.145 graph of Cynex and koshek spotted together. 00:11:41.920 --> 00:11:45.160 Got a slightly different scale this time so it looks a bit 00:11:45.160 --> 00:11:47.320 wider. But there is caution X. 00:11:48.100 --> 00:11:51.736 And they're showing X, which gets very, very close to quash X 00:11:51.736 --> 00:11:52.948 as X gets large. 00:11:56.400 --> 00:12:01.300 Now we'll define hyperbolic tangent. 00:12:03.170 --> 00:12:07.042 We can define hyperbolic tangent in terms of shine and Koch. 00:12:10.240 --> 00:12:12.968 We write Hyperbolic Tangent. 00:12:13.890 --> 00:12:19.954 Like this? And again, there are two possible pronunciations for 00:12:19.954 --> 00:12:23.150 this. Some people call it touch. 00:12:23.990 --> 00:12:26.042 But I'm in this video, we'll call it fan. 00:12:27.520 --> 00:12:30.646 And we define it to be. 00:12:30.830 --> 00:12:36.648 Sean X. Over Cosh X. 00:12:37.270 --> 00:12:40.588 And again, there's an exponential function definition 00:12:40.588 --> 00:12:45.328 which we can work out by looking at this definition. 00:12:46.350 --> 00:12:53.310 Because this is Sean X which is E to the X minus E to the 00:12:53.310 --> 00:12:55.166 minus X over 2. 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 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 00:13:10.035 --> 00:13:12.070 minus X all divided by. 00:13:12.640 --> 00:13:17.104 Each the X Plus E to the minus X. 00:13:19.120 --> 00:13:22.871 Now this thing doesn't look very useful for working out the 00:13:22.871 --> 00:13:26.963 graph. It's a bit complicated, so this time will use the graph 00:13:26.963 --> 00:13:30.373 of shine and Koch to workout the graphs of ban. 00:13:31.100 --> 00:13:38.396 Now here I've got a set of axes 00:13:38.396 --> 00:13:44.780 which I'll use to draw down onto. 00:13:45.870 --> 00:13:50.700 I've got an asymptotes here at Y equals 1 and one at White was 00:13:50.700 --> 00:13:53.115 minus one and you'll see why in 00:13:53.115 --> 00:13:58.940 a minute. I remember that 4X large. 00:14:02.040 --> 00:14:05.484 We saw that 00:14:05.484 --> 00:14:10.748 Cosh X. Was very, very close. 00:14:11.960 --> 00:14:15.809 To Shine X. 00:14:15.810 --> 00:14:19.050 So that must mean 00:14:19.050 --> 00:14:22.260 that. Shine X. 00:14:22.940 --> 00:14:27.788 Over Cosh X. 00:14:29.030 --> 00:14:32.420 Must be very close to one. 00:14:32.940 --> 00:14:35.970 So as X 00:14:35.970 --> 00:14:42.410 gets large. We should expect Linix to get very close to one. 00:14:43.810 --> 00:14:46.640 Now for large negative X. 00:14:47.350 --> 00:14:55.002 We saw 00:14:55.002 --> 00:14:58.828 that. 00:14:59.760 --> 00:15:06.098 Call shacks Was very close to eat a minus X. 00:15:06.640 --> 00:15:07.960 Over 2. 00:15:09.530 --> 00:15:10.760 And Shine X. 00:15:14.340 --> 00:15:19.920 Was very close to minus E to the minus X over 2. 00:15:21.200 --> 00:15:23.195 So you can see here that Shine 00:15:23.195 --> 00:15:25.882 X. Is very very close to minus 00:15:25.882 --> 00:15:31.857 Cosh X. So that must mean that Shine X. 00:15:32.490 --> 00:15:34.968 Over Cosh X. 00:15:35.610 --> 00:15:41.497 Is very very close to minus one. 00:15:42.470 --> 00:15:44.636 So as X gets larger, negative. 00:15:45.300 --> 00:15:48.612 We should expect the graph to get very very close to minus 00:15:48.612 --> 00:15:54.118 one. Now just to finish off, will see what happens when X 00:15:54.118 --> 00:16:00.264 equals 0. So for X 00:16:00.264 --> 00:16:03.300 equals 0. 00:16:04.180 --> 00:16:12.440 Thanks. Which is shine. 00:16:12.460 --> 00:16:15.079 Over Cos X. 00:16:16.430 --> 00:16:20.216 Will be well shine of X. 00:16:21.140 --> 00:16:23.275 Was zero, so that would be 0. 00:16:24.230 --> 00:16:27.458 Call Shabaks was one. 00:16:27.500 --> 00:16:32.164 So fun of X when X equals 0 is just zero. 00:16:33.210 --> 00:16:35.418 Now, putting these three bits of 00:16:35.418 --> 00:16:41.628 information together. We can plot 00 on that. 00:16:43.210 --> 00:16:44.770 And his ex got large. 00:16:45.590 --> 00:16:49.754 Thanks, got close to one so we can see this thing getting 00:16:49.754 --> 00:16:50.795 closer to one. 00:16:53.390 --> 00:16:56.336 And his ex got large and 00:16:56.336 --> 00:17:01.339 negative. Thanks, got close to minus one so we can draw this. 00:17:02.210 --> 00:17:04.530 Going down toward minus one. 00:17:06.920 --> 00:17:10.850 But remember that. 00:17:11.630 --> 00:17:15.020 These never actually reach one or minus one. They just 00:17:15.020 --> 00:17:19.088 get very very close to them as X gets large reserves gets 00:17:19.088 --> 00:17:19.766 more negative. 00:17:20.860 --> 00:17:23.569 Also notice that. 00:17:24.200 --> 00:17:27.330 Fun. Of 00:17:27.330 --> 00:17:30.984 ex. Is equal to 00:17:30.984 --> 00:17:34.779 minus fan? Of minus X. 00:17:35.530 --> 00:17:41.542 Now hyperbolic functions has my densities 00:17:41.542 --> 00:17:47.554 which are very similar to trig 00:17:47.554 --> 00:17:53.566 identities, but not quite the same. 00:17:54.700 --> 00:17:57.989 In this video I'll show you and prove for you, too, and then 00:17:57.989 --> 00:18:01.025 I'll show you some more at the end, which you control yourself. 00:18:01.710 --> 00:18:05.224 The first one will prove is this 00:18:05.224 --> 00:18:07.890 one. It's caution. 00:18:08.540 --> 00:18:11.604 X squared 00:18:11.604 --> 00:18:15.470 minus shine. X 00:18:15.470 --> 00:18:18.140 squared Equals 1. 00:18:19.420 --> 00:18:24.316 Not prove this will substitute in the exponential definitions 00:18:24.316 --> 00:18:26.492 for Koch and shine. 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 00:18:34.620 --> 00:18:37.560 2, and we want that all squared. 00:18:38.730 --> 00:18:44.745 We take away shine X squared, which was E to the X minus E to 00:18:44.745 --> 00:18:47.552 the minus X over 2 all squared. 00:18:49.860 --> 00:18:52.576 Now notice that in both bits of 00:18:52.576 --> 00:18:55.354 this. We have the same 00:18:55.354 --> 00:18:58.665 denominator squared. So we can take that out of the factor. 00:18:59.810 --> 00:19:01.218 So we have that. 00:19:01.870 --> 00:19:03.448 As a quarter. 00:19:04.170 --> 00:19:06.818 Of the numerator squared. 00:19:07.780 --> 00:19:12.960 So we have each of the X Plus E to the minus X squared. 00:19:13.570 --> 00:19:20.412 Minus eat the AX minus E to the minus X squared. 00:19:22.150 --> 00:19:26.416 Now we want to expand out these squared brackets. 00:19:28.410 --> 00:19:31.518 So that is going to be. 00:19:31.660 --> 00:19:34.720 Well, these brackets. 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. 00:19:41.400 --> 00:19:47.510 And because each the X Times E to the minus X is one. 00:19:48.300 --> 00:19:50.538 And we get that happening twice. 00:19:51.140 --> 00:19:53.210 We get a plus two here. 00:19:53.760 --> 00:19:57.577 And each the minus X Times E to the minus X. 00:19:58.250 --> 00:20:00.010 Is each the minus 2X? 00:20:02.910 --> 00:20:07.370 And I will take away this bracket squared which working 00:20:07.370 --> 00:20:10.046 out before we can see is. 00:20:10.640 --> 00:20:12.050 Each the two weeks again. 00:20:12.720 --> 00:20:16.640 And this time we get a minus two 00:20:16.640 --> 00:20:18.880 here. And we get. 00:20:19.680 --> 00:20:23.180 Plus, if the minus 2X. 00:20:24.600 --> 00:20:28.400 Now we can remove these 00:20:28.400 --> 00:20:32.380 inner brackets. And we get a 00:20:32.380 --> 00:20:35.140 quarter. Of. 00:20:36.250 --> 00:20:42.310 Just take out the first brackets first, E to the two X +2, plus E 00:20:42.310 --> 00:20:43.926 to the minus 2X. 00:20:44.540 --> 00:20:46.520 Put a minus here, so that's 00:20:46.520 --> 00:20:48.748 minus. Each the 2X. 00:20:50.150 --> 00:20:56.870 +2. Minus. E to the minus 2X. 00:20:57.490 --> 00:21:02.027 And of course, a lot of things that are going to cancel here. 00:21:02.027 --> 00:21:06.564 We have E to the two X minus either 2X, so these go. 00:21:08.190 --> 00:21:12.684 We have plus E to the minus two X here and minus it again 00:21:12.684 --> 00:21:13.968 there, so they go. 00:21:15.680 --> 00:21:21.936 And then we just have 2 + 2, so the whole thing turns out to be 00:21:21.936 --> 00:21:25.064 1/4 of 2 + 2, which is 4. 00:21:25.070 --> 00:21:26.258 Which is one. 00:21:27.520 --> 00:21:30.413 So there we have the proof that cause squared X minus 00:21:30.413 --> 00:21:33.306 sign squared X, which is all that is equal to 1. 00:21:34.340 --> 00:21:42.290 And one more density or proof you now is this 00:21:42.290 --> 00:21:45.470 one. It's shine 2X. 00:21:46.900 --> 00:21:49.429 Equal to two. 00:21:51.570 --> 00:21:57.270 Sean X. Call Josh X. 00:21:58.430 --> 00:22:02.824 Now with this one, I think it's easier to start with the right 00:22:02.824 --> 00:22:06.880 hand side, so will do that again. Will a substitute in the 00:22:06.880 --> 00:22:09.246 exponential definitions to the right hand side. 00:22:10.070 --> 00:22:15.915 So 2. Sean 00:22:15.915 --> 00:22:19.790 X. Cosh 00:22:19.790 --> 00:22:23.518 X. Is equal to. 00:22:24.330 --> 00:22:28.269 Two lots of. 00:22:29.040 --> 00:22:36.978 Each 3X minus each the minus X over 2 times caution X which was 00:22:36.978 --> 00:22:43.782 E to the X Plus E to the minus X over 2. 00:22:45.530 --> 00:22:50.546 Now you can see the two councils with one of these denominators, 00:22:50.546 --> 00:22:52.218 so will cancel those. 00:22:53.020 --> 00:22:55.589 And take out this half as a 00:22:55.589 --> 00:23:00.920 factor. So now we have 1/2. 00:23:02.390 --> 00:23:09.718 Each of the X minus E to the minus X Times E to the X Plus 00:23:09.718 --> 00:23:12.008 E to the minus X. 00:23:12.570 --> 00:23:15.244 And if we just multiply out the 00:23:15.244 --> 00:23:17.508 brackets. We get a half. 00:23:18.360 --> 00:23:22.035 Get each the X Times E to the X, which is each the two acts. 00:23:23.610 --> 00:23:26.706 Each the X Times Plus E to the 00:23:26.706 --> 00:23:30.218 minus X. Which is plus one. 00:23:31.410 --> 00:23:37.350 Minus E to the minus X times Y to the X, which is minus one. 00:23:37.880 --> 00:23:40.650 And finally, minus each the 00:23:40.650 --> 00:23:47.180 minus 2X. So obviously these ones cancel. 00:23:47.850 --> 00:23:55.014 And we get this bracket here, which was E to the two 00:23:55.014 --> 00:23:58.948 X minus. Eat the minus 2X. 00:23:59.580 --> 00:24:01.588 All divided by two. 00:24:02.430 --> 00:24:04.992 And that is the definition of 00:24:04.992 --> 00:24:11.610 shine 2X. So that's the second identity proved. 00:24:13.520 --> 00:24:19.850 Now here is more identity's which you can prove yourself. 00:24:21.800 --> 00:24:27.266 The first one is Koch 2X. 00:24:28.400 --> 00:24:29.870 Is equal to. 00:24:31.160 --> 00:24:34.760 Call shacks 00:24:34.760 --> 00:24:39.290 squared. Plus 00:24:41.090 --> 00:24:43.529 Shine X squared. 00:24:44.450 --> 00:24:46.850 We also have. 00:24:47.390 --> 00:24:50.630 Shine of X 00:24:50.630 --> 00:24:52.790 Plus Y. 00:24:52.790 --> 00:24:55.380 Equals. 00:24:56.360 --> 00:25:02.180 Shine X. Gosh why? 00:25:03.430 --> 00:25:06.796 Plus Shine 00:25:06.796 --> 00:25:11.860 Y. Koch X. 00:25:14.290 --> 00:25:16.618 And also for Koch. 00:25:17.460 --> 00:25:20.130 X plus Y. 00:25:21.930 --> 00:25:23.589 We have cautious. 00:25:24.450 --> 00:25:31.808 X. Call Shuai Plus Shine 00:25:31.808 --> 00:25:34.980 X. Shine 00:25:34.980 --> 00:25:38.496 Y. And 00:25:38.496 --> 00:25:41.988 two more. 00:25:41.990 --> 00:25:46.280 Gosh. X over 2. 00:25:47.340 --> 00:25:51.302 All squared. Is 1 00:25:51.302 --> 00:25:57.128 plus. Cosh X all divided by two. 00:25:58.150 --> 00:26:01.350 And finish off shine. 00:26:02.110 --> 00:26:03.970 X over 2. 00:26:04.500 --> 00:26:11.510 All squared. Is Cosh X minus one 00:26:11.510 --> 00:26:14.830 all divided by two? 00:26:16.020 --> 00:26:20.167 Now you can see how similar these are to trig identities. 00:26:20.910 --> 00:26:24.402 And in fact, the hyperbolic functions and triggered entities 00:26:24.402 --> 00:26:25.954 are very closely related. 00:26:26.710 --> 00:26:29.947 And you can learn more about that if you go and study complex 00:26:29.947 --> 00:26:37.068 numbers. And before we finish, we look at some 00:26:37.068 --> 00:26:42.956 other functions which are related to hyperbolic functions. 00:26:43.780 --> 00:26:45.259 Firstly, some notation. 00:26:45.820 --> 00:26:48.225 You've already seen me use 00:26:48.225 --> 00:26:55.430 notation like. Sean X squared, which means. 00:26:55.460 --> 00:26:59.156 The whole of Shine X squared. 00:27:00.010 --> 00:27:02.122 Just remember that if you see 00:27:02.122 --> 00:27:09.810 something like. Sean X with a minus one there that is not 00:27:09.810 --> 00:27:11.688 equal to 1. 00:27:12.420 --> 00:27:15.510 Over Shine X. 00:27:16.130 --> 00:27:19.986 This refers to the inverse function of shine. 00:27:21.060 --> 00:27:27.670 If you remember that if we have a function F. 00:27:29.020 --> 00:27:32.224 F minus one the inverse function works like this. 00:27:33.400 --> 00:27:40.190 You have F minus one. The inverse function of X 00:27:40.190 --> 00:27:46.980 gives you this thing why where F of Y is 00:27:46.980 --> 00:27:49.017 equal to X. 00:27:49.790 --> 00:27:51.258 And the same way. 00:27:53.160 --> 00:27:55.938 Shine with the minus one there. 00:27:56.620 --> 00:27:59.780 Of X gives You Y. 00:28:01.650 --> 00:28:08.034 Will shine of why was equal to X? 00:28:09.160 --> 00:28:12.562 We can also define inverse functions for caution than. 00:28:13.540 --> 00:28:15.840 But for these we have to restrict the domains of 00:28:15.840 --> 00:28:16.300 the function. 00:28:17.350 --> 00:28:23.603 For inverse Cosh, Cosh the minus one of X, we need to have. 00:28:24.220 --> 00:28:27.230 X greater than or equal to 1. 00:28:28.280 --> 00:28:33.816 And for the inverse of fan of X. 00:28:34.360 --> 00:28:40.808 We need X to be greater than minus one and less than one. 00:28:43.560 --> 00:28:49.936 Some other functions you might come across other 00:28:49.936 --> 00:28:51.530 reciprocal functions. 00:28:53.630 --> 00:28:55.850 These are such. 00:28:57.950 --> 00:29:05.790 X. Which is defined to be one over Cosh of X. 00:29:06.300 --> 00:29:07.788 There are two more of these. 00:29:08.990 --> 00:29:15.054 This Kosach Of X, which 00:29:15.054 --> 00:29:20.442 is one over shine of X. 00:29:21.490 --> 00:29:26.270 And finally, there's cough X. 00:29:28.910 --> 00:29:32.880 Which is one over fan.