WEBVTT 00:00:02.350 --> 00:00:08.974 This video is going to look at three knew trig functions. Cosec 00:00:08.974 --> 00:00:10.630 Zack and caught. 00:00:11.270 --> 00:00:14.591 However, they're not entirely knew, because they are derived 00:00:14.591 --> 00:00:18.650 from the three that we know about already sign calls and 00:00:18.650 --> 00:00:21.460 tan. So let's have a look. 00:00:22.470 --> 00:00:29.700 The first one that we want to have a look at is cosec that is 00:00:29.700 --> 00:00:36.448 defined to be one over sine. So one over sine Theta is equal to. 00:00:36.448 --> 00:00:41.750 Now to give it its full name it is the cosecant. 00:00:41.750 --> 00:00:49.057 Home theater But we shorten that till cosec theater. 00:00:49.890 --> 00:00:57.027 Second one. Follows the same line one over 'cause 00:00:57.027 --> 00:01:04.599 Theater and it's full name is the secant of Theta. But again, 00:01:04.599 --> 00:01:12.171 we shorten that to set theater and the final one, one over 00:01:12.171 --> 00:01:19.884 10 theater. Equals and it's full name is the cotangent of 00:01:19.884 --> 00:01:25.518 Theta, and again we shorten that to caught theater. 00:01:26.190 --> 00:01:31.706 Now, why do we need these? Well, first of all, they will help us 00:01:31.706 --> 00:01:34.858 to solve trig equations. Secondly, there involved in 00:01:34.858 --> 00:01:38.798 identity's and 3rd they come up when we do calculus, 00:01:38.798 --> 00:01:40.374 particularly when we do 00:01:40.374 --> 00:01:45.830 integration. Let's just have a look at one example of where 00:01:45.830 --> 00:01:48.524 they might occur in terms of 00:01:48.524 --> 00:01:54.038 basic identity's. So the basic trig identity that we've got, 00:01:54.038 --> 00:01:59.798 the sine squared Theta Plus Cost Square theater, equals 1. 00:02:00.520 --> 00:02:07.180 And if I choose to divide everything on both sides of this 00:02:07.180 --> 00:02:12.730 identity by Cos squared, then I'll have sine squared Theta 00:02:12.730 --> 00:02:18.280 over cost Square theater plus cost Square theater over Cos 00:02:18.280 --> 00:02:22.720 squared Theta equals one over cost squared Theta. 00:02:23.220 --> 00:02:29.434 And so this one is sign over cause all squared. So that gives 00:02:29.434 --> 00:02:34.692 us stand square theater plus cost squared into Cos squared is 00:02:34.692 --> 00:02:40.428 one equals and then one over cost squared is one over cause 00:02:40.428 --> 00:02:43.296 all squared. So that is set 00:02:43.296 --> 00:02:48.218 squared Theta. So there's one of our new trick functions 00:02:48.218 --> 00:02:53.090 popping up in an identity this time, and there is a similar 00:02:53.090 --> 00:02:57.556 one that we can get if we divide throughout by sine 00:02:57.556 --> 00:03:02.834 squared, and if we do that, we end up with one plus cot 00:03:02.834 --> 00:03:06.894 squared Theta is equal to cosec squared Theta, so there 00:03:06.894 --> 00:03:10.548 the other two trig functions that we've just introduced 00:03:10.548 --> 00:03:11.766 again pop up. 00:03:13.110 --> 00:03:18.115 Let's have a look at what might happen when we reached the later 00:03:18.115 --> 00:03:20.040 stages of solving a trig 00:03:20.040 --> 00:03:27.279 equation. So let's take cot squared Theta equals 00:03:27.279 --> 00:03:35.095 3 four theater between 360 and 0 degrees. 00:03:35.670 --> 00:03:40.926 Well, we begin to solve this by taking the square root. So 00:03:40.926 --> 00:03:46.182 caught theater equals Route 3 or minus Route 3. Remember, we take 00:03:46.182 --> 00:03:52.314 a square root, it has to be plus or minus. Now we might think, 00:03:52.314 --> 00:03:58.008 well, let's just look this up in some tables or let's take our 00:03:58.008 --> 00:04:02.826 Calculator, but do we really need to? We know what caught 00:04:02.826 --> 00:04:05.454 theater is. It's one over Tan 00:04:05.454 --> 00:04:10.320 Theta. And that's Route 3 or minus Route 3. 00:04:10.880 --> 00:04:16.508 Now we can turn this one upside down to give us Tan Theta equals 00:04:16.508 --> 00:04:22.136 and we can think of each of these as being root 3 over one 00:04:22.136 --> 00:04:27.764 or minus Route 3 over one, and so we can turn these upside down 00:04:27.764 --> 00:04:32.588 to get one over Route 3 or minus one over Route 3. 00:04:33.470 --> 00:04:39.574 And now it's in terms of Tan Theater and this is now one of 00:04:39.574 --> 00:04:44.370 those special values of our trig functions. In fact, one over 00:04:44.370 --> 00:04:50.038 Route 3 is the tangent of 30 degrees, so we know that this 00:04:50.038 --> 00:04:54.834 has one solution that is 30 degrees. But what about the 00:04:54.834 --> 00:04:58.758 other solutions? Well, let's have a look at those. 00:05:00.080 --> 00:05:01.700 Sketch of the graph. 00:05:02.390 --> 00:05:10.000 Tan Theta 0 up to 90 from 90 up through 00:05:10.000 --> 00:05:13.044 180 up towards 270. 00:05:13.850 --> 00:05:17.028 Stopping there at 360, so that's not. 00:05:18.270 --> 00:05:22.026 9180, two, 70 00:05:22.026 --> 00:05:28.454 and 360. And the tangent of 30 is one over 00:05:28.454 --> 00:05:33.563 Route 3. So somewhere here is one over Route 3 coming down to 00:05:33.563 --> 00:05:38.672 30. So of course the next one is across there and the symmetry 00:05:38.672 --> 00:05:44.960 tells us if this is 30 on from zero. This is 30 on from 180, so 00:05:44.960 --> 00:05:49.676 the next one is 210 degrees minus one over Route 3. Well, 00:05:49.676 --> 00:05:51.641 that's going to be somewhere 00:05:51.641 --> 00:05:56.800 along here. And again, the symmetry tells us if this is 30 00:05:56.800 --> 00:06:02.485 on this way, then this one is 30 back this way. So that gives us 00:06:02.485 --> 00:06:06.275 150 degrees and we've got another value here which is 00:06:06.275 --> 00:06:11.202 going to be 30 back from there, which is going to be 330 00:06:11.202 --> 00:06:15.654 degrees. So solving equations that involve things like caught, 00:06:15.654 --> 00:06:20.910 encek and Cosec is no different to solving equations to do with 00:06:20.910 --> 00:06:26.166 sign causing tan because we just turn them into sign calls and 00:06:26.166 --> 00:06:31.860 tab to conclude this, we're just going to have a look at the 00:06:31.860 --> 00:06:34.488 graphs of these three knew trig 00:06:34.488 --> 00:06:40.430 functions. And in order to do that, we will begin each one by 00:06:40.430 --> 00:06:44.665 looking at the graph of the related trig function. So to 00:06:44.665 --> 00:06:49.670 look at Cosec, we're going to look at sign first. So what does 00:06:49.670 --> 00:06:51.210 the graph of sign? 00:06:52.480 --> 00:06:58.870 Look like. Will take one complete cycle between North and 00:06:58.870 --> 00:07:01.600 360. So 0. 00:07:02.460 --> 00:07:09.160 180, three, 160 and the peak and trough are in 00:07:09.160 --> 00:07:15.860 between 1970 and that goes from one down 2 - 00:07:15.860 --> 00:07:22.560 1 and what we're going to graph now is cosec 00:07:22.560 --> 00:07:29.260 theater, which of course is one over sine Theta. So 00:07:29.260 --> 00:07:32.610 let's set up similar axes. 00:07:32.740 --> 00:07:36.016 So mark them off, there's 90. 00:07:36.550 --> 00:07:39.310 180 270 00:07:39.960 --> 00:07:46.630 360 Now here at 90 00:07:46.630 --> 00:07:53.770 the value of sign is warm. 00:07:54.840 --> 00:08:02.160 So at 90 the value of cosec must also be one, so I'm going to 00:08:02.160 --> 00:08:08.504 market their one here at 270. The value of sign is minus one. 00:08:09.070 --> 00:08:14.452 And so at 270, the value of cosec must be one over minus 00:08:14.452 --> 00:08:16.936 one, which again is just minus 00:08:16.936 --> 00:08:19.800 one. So there are two points. 00:08:20.420 --> 00:08:22.940 What about this point? 00:08:23.570 --> 00:08:29.213 Here at zero the sign of 0 is 0. 00:08:30.110 --> 00:08:34.280 So the value of Cosec would be one over 0. 00:08:35.350 --> 00:08:40.225 But we're not allowed to divide by zero, but we can divide by 00:08:40.225 --> 00:08:45.100 something a little bit away. What we can see is that would be 00:08:45.100 --> 00:08:49.975 a very very tiny positive number that we were dividing by. So if 00:08:49.975 --> 00:08:54.475 we divide 1 by a very tiny positive number, the answer has 00:08:54.475 --> 00:08:59.350 to be very big, but still positive. So with a bit of curve 00:08:59.350 --> 00:09:01.975 there, let's have a look at 180. 00:09:02.650 --> 00:09:09.804 Well, at 180 sign of Theta is again 0 so cosec is one over 00:09:09.804 --> 00:09:16.958 0 at this 180 degrees. Let's go a little bit this side here of 00:09:16.958 --> 00:09:23.090 180 and the value of sign is really very small. It's very 00:09:23.090 --> 00:09:24.623 close to 0. 00:09:25.130 --> 00:09:31.740 So again, 1 divided by something very small and positive. 00:09:32.240 --> 00:09:37.844 Is again something very large and positive, so let me put in 00:09:37.844 --> 00:09:41.628 an asymptotes. And we've got a piece of curve there. 00:09:42.160 --> 00:09:47.020 Now this curve goes like that. What we're seeing is that this 00:09:47.020 --> 00:09:52.690 curve is going to come down and up like that, and it's going to 00:09:52.690 --> 00:09:57.145 do the same here, except because what we're dividing by are 00:09:57.145 --> 00:09:59.170 negative numbers, it's going to 00:09:59.170 --> 00:10:01.120 be like. That 00:10:02.030 --> 00:10:08.402 So there's our graph of cosec derived from the graph of sign. 00:10:10.910 --> 00:10:14.810 Let's take now calls feta. 00:10:15.340 --> 00:10:16.708 Do the same. 00:10:17.940 --> 00:10:25.640 Will take the graph of costita between North and 360. 00:10:25.640 --> 00:10:33.180 At the extreme, values will be minus one plus one 00:10:33.180 --> 00:10:36.950 9180 two 7360. Just make 00:10:36.950 --> 00:10:41.874 that clearer. And so let's have a look here. 00:10:42.980 --> 00:10:46.668 Mark off the same 00:10:46.668 --> 00:10:53.565 points. And we're graphing SEK this 00:10:53.565 --> 00:11:00.735 time sex theater, which is one 00:11:00.735 --> 00:11:04.320 over 'cause theater. 00:11:05.000 --> 00:11:07.405 So again, let's Mark some 00:11:07.405 --> 00:11:10.940 points. Here when theater is 0. 00:11:11.650 --> 00:11:16.420 Costita is one Soucek Theater is one over one 00:11:16.420 --> 00:11:22.780 which is one. So will mark the one there here at 180. 00:11:24.010 --> 00:11:29.080 Cost theater is minus one. Soucek Theater is 1 divided 00:11:29.080 --> 00:11:34.150 by minus one and so will mark minus one here. 00:11:35.490 --> 00:11:40.144 Here at 90 we got exactly the same problems we have before the 00:11:40.144 --> 00:11:45.872 value of Cos theater at 90 zero. So 1 / 0 is a very big number. 00:11:45.872 --> 00:11:51.242 Well, in fact we're not allowed to do it, so we have to go a 00:11:51.242 --> 00:11:56.612 little bit away from 90 to get a value of Cos Theta which is very 00:11:56.612 --> 00:12:01.624 small, close to 0 but positive. And if we divide 1 by that small 00:12:01.624 --> 00:12:05.562 positive number, the answer that we get is very big and. 00:12:05.640 --> 00:12:10.398 Positive so we have a bit of curve going up like that. What 00:12:10.398 --> 00:12:15.156 about this side of 90? Well this side of 90 where dividing by 00:12:15.156 --> 00:12:19.182 something which the value of Cos Theta is very small but 00:12:19.182 --> 00:12:23.208 definitely negative. So the answer is going to be very big 00:12:23.208 --> 00:12:28.332 in size when we divide it into one but negative. So a bit of 00:12:28.332 --> 00:12:32.724 the curve here coming down to their same problem again at 270 00:12:32.724 --> 00:12:36.750 so we can see the curve is going to go round. 00:12:36.760 --> 00:12:40.708 And back like that. And then here again at 360, we're going 00:12:40.708 --> 00:12:44.985 to be able to mark that point. We're going to have that one 00:12:44.985 --> 00:12:46.301 coming down at that. 00:12:46.990 --> 00:12:50.220 So there we've managed to get the graph of SEK. 00:12:51.040 --> 00:12:54.706 Out of the graph, of course. 00:12:54.710 --> 00:13:00.584 Let's now have a look at the graph of Tan Theater. 00:13:02.280 --> 00:13:09.528 These off 00:13:09.528 --> 00:13:16.776 9180, two, 00:13:16.776 --> 00:13:24.024 70 and 00:13:24.024 --> 00:13:32.257 360. And now we'll have a look at 00:13:32.257 --> 00:13:36.665 caught theater, which is one over Tan Theater. 00:13:37.390 --> 00:13:43.578 So we'll take the same graph and I'll do the same as I've done 00:13:43.578 --> 00:13:45.346 before. Mark these off. 00:13:45.860 --> 00:13:49.700 So we're using the same 00:13:49.700 --> 00:13:55.610 scale. OK, let's have a look what's happening here. This bit 00:13:55.610 --> 00:14:00.710 of curve between North and 90. We begin with something for tan 00:14:00.710 --> 00:14:03.260 that is very small but positive. 00:14:03.960 --> 00:14:09.056 Just above 0 and then it gets bigger and bigger and bigger as 00:14:09.056 --> 00:14:12.976 it rises. The value of Tan Theta rises towards Infinity. 00:14:14.150 --> 00:14:19.103 Well down here divide the value of theater is very near to zero 00:14:19.103 --> 00:14:24.056 and so tan Theta is very small but positive. So when we divide 00:14:24.056 --> 00:14:28.247 into one we're going to get something very big and positive 00:14:28.247 --> 00:14:30.152 self. But if curve there. 00:14:30.780 --> 00:14:34.927 Up here, the value of Tan Theater is enormous. It's huge. 00:14:34.927 --> 00:14:39.828 So if we divide something huge into one, the answer is going to 00:14:39.828 --> 00:14:44.729 be very nearly zero. And the closer we get to 90, the closer 00:14:44.729 --> 00:14:46.614 it would be to 0. 00:14:47.470 --> 00:14:51.747 So now if we look here, we can see we've got something very, 00:14:51.747 --> 00:14:56.024 very big, but negative. So the answer is going to be very, very 00:14:56.024 --> 00:14:59.972 small, but also negative. This is going to be coming out of 00:14:59.972 --> 00:15:02.960 that point there. Here 180. 00:15:03.510 --> 00:15:08.268 Got a problem at 180. Tan Theater is 0 one over 10 theater 00:15:08.268 --> 00:15:13.392 is there for something very very big so we can put in an acid 00:15:13.392 --> 00:15:18.150 tote and we can see we've got exactly the same problem here at 00:15:18.150 --> 00:15:23.861 360. So if I join up what I've got in the direction of what's 00:15:23.861 --> 00:15:27.678 happening, we're getting a very similar curve and repeat it over 00:15:27.678 --> 00:15:31.148 here, 'cause the curves are repeated. We're getting a very 00:15:31.148 --> 00:15:32.883 similar curve, except the other 00:15:32.883 --> 00:15:38.753 way around. So we've seen again how we can derive the graph of 00:15:38.753 --> 00:15:42.137 coffee to directly from the graph of Tan. 00:15:42.740 --> 00:15:45.998 So remember these three new functions. 00:15:47.220 --> 00:15:50.360 Co sack sack and caught. 00:15:51.700 --> 00:15:56.020 Respectively, they are one over sign, one over cosine 00:15:56.020 --> 00:15:57.940 and one over Tangent. 00:15:59.040 --> 00:16:00.960 We can use them to solve 00:16:00.960 --> 00:16:05.520 equations. But each time we can get back to using sign 00:16:05.520 --> 00:16:08.733 cause and tab to help us workout the angles.