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