0:00:00.970,0:00:05.568 In this video, we're going to be[br]solving whole collection of 0:00:05.568,0:00:09.330 trigonometric equations now be[br]cause it's the technique of 0:00:09.330,0:00:13.510 solving the equation and in[br]ensuring that we get enough 0:00:13.510,0:00:17.272 solutions, that's important and[br]not actually looking up the 0:00:17.272,0:00:21.452 angle. All of these are designed[br]around certain special angles, 0:00:21.452,0:00:26.886 so I'm just going to list at the[br]very beginning here the special 0:00:26.886,0:00:31.484 angles and their sines, cosines,[br]and tangents that are going to 0:00:31.484,0:00:33.468 form. The basis of what[br]we're doing. 0:00:37.830,0:00:41.951 So the special angles that we're[br]going to have a look at our 0:00:41.951,0:00:49.444 zero. 30 4560[br]and 90 there in degrees. 0:00:49.444,0:00:55.236 If we're thinking about radians,[br]then there's zero. 0:00:55.940,0:00:59.348 Pie by 6. 0:01:00.130,0:01:03.718 Pie by 4. 0:01:04.670,0:01:05.738 Pie by three. 0:01:06.370,0:01:09.090 And Π by 2. 0:01:10.220,0:01:15.786 Trig ratios we're going to be[br]looking at are the sign. 0:01:15.790,0:01:21.104 The cosine. On the tangent[br]of each of these. 0:01:22.460,0:01:25.850 Sign of 0 is 0. 0:01:26.690,0:01:29.666 The sign of 30 is 1/2. 0:01:30.370,0:01:34.338 Sign of 45 is one over Route 2. 0:01:34.850,0:01:41.714 The sign of 60 is Route 3 over 2[br]and the sign of 90 is one. 0:01:43.440,0:01:45.260 Cosine of 0 is one. 0:01:45.990,0:01:52.665 Cosine of 30 is Route 3 over 2[br]cosine of 45 is one over Route 0:01:52.665,0:01:58.895 2, the cosine of 60 is 1/2, and[br]the cosine of 90 is 0. 0:01:59.570,0:02:06.096 The town of 0 is 0 the[br]town of 30 is one over 0:02:06.096,0:02:12.622 Route 3 that Anna 45 is[br]110 of 60 is Route 3 and 0:02:12.622,0:02:17.140 the town of 90 degrees is[br]infinite, it's undefined. 0:02:18.630,0:02:21.738 It's these that we're going to[br]be looking at and working with. 0:02:22.560,0:02:25.900 Let's look at our first 0:02:25.900,0:02:30.546 equation then. We're going to[br]begin with some very simple 0:02:30.546,0:02:37.844 ones. So we take sign of[br]X is equal to nought .5. Now 0:02:37.844,0:02:43.751 invariably when we get an[br]equation we get a range of 0:02:43.751,0:02:45.899 values along with it. 0:02:46.610,0:02:52.714 So in this case will take X is[br]between North and 360. So what 0:02:52.714,0:02:56.638 we're looking for is all the[br]values of X. 0:02:57.200,0:03:01.460 Husain gives us N[br].5. 0:03:03.970,0:03:10.830 Let's sketch a graph of[br]sine X over this range. 0:03:13.610,0:03:16.165 And sign looks like that[br]with 90. 0:03:17.310,0:03:18.500 180 0:03:20.120,0:03:27.152 270 and 360 and ranging between[br]one 4 sign 90 and minus 0:03:27.152,0:03:30.668 one for the sign of 270. 0:03:31.360,0:03:36.610 Sign of X is nought .5. So[br]we go there. 0:03:37.890,0:03:38.720 And there. 0:03:39.780,0:03:44.410 So there's our first angle, and[br]there's our second angle. 0:03:45.900,0:03:52.517 We know the first one is 30[br]degrees because sign of 30 is 0:03:52.517,0:03:58.625 1/2, so our first angle is 30[br]degrees. This curve is symmetric 0:03:58.625,0:04:05.242 and so because were 30 degrees[br]in from there, this one's got to 0:04:05.242,0:04:08.296 be 30 degrees back from there. 0:04:08.810,0:04:12.610 That would make it 0:04:12.610,0:04:17.080 150. There are no more answers[br]because within this range as we 0:04:17.080,0:04:18.360 go along this line. 0:04:18.890,0:04:23.453 It doesn't cross the curve at[br]any other points. 0:04:23.460,0:04:30.004 Let's have a look[br]at a cosine cause 0:04:30.004,0:04:36.548 of X is minus[br]nought .5 and the 0:04:36.548,0:04:43.092 range for this X[br]between North and 360. 0:04:43.820,0:04:49.112 So again, let's have a look at a[br]graph of the function. 0:04:50.070,0:04:53.899 Involved in the equation,[br]the cosine graph. 0:04:56.510,0:05:00.094 Looks like that. One and 0:05:00.094,0:05:04.819 minus one.[br]This is 90. 0:05:06.300,0:05:08.160 180 0:05:09.310,0:05:15.838 270 and then here[br]at the end, 360. 0:05:17.020,0:05:19.770 Minus 9.5. 0:05:20.810,0:05:26.387 Gain across there at minus[br]9.5 and down to their and 0:05:26.387,0:05:27.908 down to their. 0:05:28.970,0:05:35.450 Now the one thing we do know is[br]that the cause of 60 is plus N 0:05:35.450,0:05:40.715 .5, and so that's there. So we[br]know there is 60. Now again, 0:05:40.715,0:05:45.980 this curve is symmetric, so if[br]that one is 30 back that way 0:05:45.980,0:05:51.650 this one must be 30 further on.[br]So I'll first angle must be 120 0:05:51.650,0:05:56.915 degrees. This one's got to be in[br]a similar position as this bit 0:05:56.915,0:05:59.345 of the curve is again symmetric. 0:05:59.380,0:06:07.216 So that's 270 and we need[br]to come back 30 degrees, so 0:06:07.216,0:06:12.512 that's 240. Now we're going to[br]have a look at an example where 0:06:12.512,0:06:16.494 we've got what we call on[br]multiple angle. So instead of 0:06:16.494,0:06:21.562 just being cause of X or sign of[br]X, it's going to be something 0:06:21.562,0:06:26.268 like sign of 2X or cause of[br]three X. So let's begin with 0:06:26.268,0:06:33.076 sign of. 2X is equal[br]to Route 3 over 2 0:06:33.076,0:06:40.046 and again will take X[br]to be between North and 0:06:40.046,0:06:40.743 360. 0:06:41.810,0:06:44.840 Now we've got 2X here. 0:06:45.520,0:06:52.501 So if we've got 2X and X[br]is between Norton 360, then the 0:06:52.501,0:06:59.482 total range that we're going to[br]be looking at is not to 722. 0:06:59.482,0:07:06.463 X is going to come between 0[br]and 720, and the sign function 0:07:06.463,0:07:12.370 is periodic. It repeats itself[br]every 360 degrees, so I'm going 0:07:12.370,0:07:16.129 to need 2 copies of the sine 0:07:16.129,0:07:22.410 curve. As the first one going up[br]to 360 and now I need a second 0:07:22.410,0:07:24.460 copy there going on till. 0:07:25.240,0:07:27.320 720 0:07:28.370,0:07:34.535 OK, so sign 2 X equals root, 3[br]over 2, but we know that the 0:07:34.535,0:07:41.933 sign of 60 is Route 3 over 2. So[br]if we put in Route 3 over 2 it's 0:07:41.933,0:07:46.865 there, then it's going to be[br]these along here as well. So 0:07:46.865,0:07:52.208 what have we got? Well, the[br]first one here we know is 60. 0:07:52.208,0:07:57.962 This point we know is 180 so[br]that one's got to be the same 0:07:57.962,0:08:04.184 distance. Back in due to the[br]symmetry 120, so we do know that 0:08:04.184,0:08:10.932 2X will be 60 or 120, but we[br]also now we've got these other 0:08:10.932,0:08:16.716 points on here, so let's just[br]count on where we are. There's 0:08:16.716,0:08:22.018 the 1st loop of the sign[br]function, the first copy, its 0:08:22.018,0:08:27.320 periodic and repeats itself[br]again. So now we need to know 0:08:27.320,0:08:29.248 where are these well. 0:08:29.270,0:08:36.302 This is an exact copy of[br]that, so this must be 60 0:08:36.302,0:08:43.334 further on. In other words, at[br]420, and this must be another 0:08:43.334,0:08:50.366 120 further on. In other words,[br]at 480. So we've got two 0:08:50.366,0:08:57.206 more answers. And it's X[br]that we actually want, not 0:08:57.206,0:09:00.641 2X. So this is 3060. 0:09:00.820,0:09:04.441 210 and finally 0:09:04.441,0:09:11.657 240. Let's have a[br]look at that with a tangent 0:09:11.657,0:09:17.587 function. This time tan or three[br]X is equal to. 0:09:18.160,0:09:24.864 Minus one and will[br]take X to be 0:09:24.864,0:09:28.216 between North and 180. 0:09:29.740,0:09:36.040 So we draw a graph[br]of the tangent function. 0:09:37.140,0:09:38.380 So we go up. 0:09:40.990,0:09:43.780 We've got that there. That's 90. 0:09:53.090,0:09:59.837 This is 180 and this is 270[br]now. It's 3X. X is between 0:09:59.837,0:10:07.103 Norton 180, so 3X can be between[br]North and 3 * 180 which is 0:10:07.103,0:10:13.850 540. So I need to get copies[br]of this using the periodicity of 0:10:13.850,0:10:20.597 the tangent function right up to[br]540. So let's put in some more. 0:10:21.810,0:10:26.610 That's 360. On[br]there. 0:10:27.920,0:10:30.900 That's 450. 0:10:34.820,0:10:42.470 This one here will be[br]540 and that's as near 0:10:42.470,0:10:50.120 or as far as we[br]need to go. Tanner 3X 0:10:50.120,0:10:53.945 is minus one, so here's 0:10:53.945,0:10:59.124 minus one. And we go across here[br]picking off all the ones that we 0:10:59.124,0:11:00.936 need. So we've got one there. 0:11:01.690,0:11:08.862 There there. These are our[br]values, so 3X is equal 12. 0:11:08.862,0:11:16.686 Now we know that the angle[br]whose tangent is one is 45, 0:11:16.686,0:11:24.510 which is there. So again this[br]and this are symmetric bits of 0:11:24.510,0:11:32.334 curve, so this must be 45[br]further on. In other words 130. 0:11:32.340,0:11:32.980 5. 0:11:34.170,0:11:41.954 This one here has got to be[br]45 further on, so that will be 0:11:41.954,0:11:49.670 315. This one here has got[br]to be 45 further on, so that 0:11:49.670,0:11:57.440 will be 495, but it's X that[br]we want not 3X, so let's divide 0:11:57.440,0:12:04.100 throughout by three, so freezing[br]to that is 45 threes into that 0:12:04.100,0:12:11.315 is 105 and threes into that is[br]165. Those are our three answers 0:12:11.315,0:12:14.090 for that one 45 degrees. 0:12:14.100,0:12:17.106 105 degrees under 0:12:17.106,0:12:24.680 165. Let's take cause of[br]X over 2 this time. So 0:12:24.680,0:12:31.964 instead of multiplying by two or[br]by three, were now dividing by 0:12:31.964,0:12:38.641 two. Let's see what difference[br]this might make equals minus 1/2 0:12:38.641,0:12:45.925 and will take X to be[br]between North and 360. So let's 0:12:45.925,0:12:47.746 draw the graph. 0:12:48.830,0:12:55.970 All calls X between North[br]and 360, so there we've 0:12:55.970,0:12:58.826 got it 360 there. 0:12:59.560,0:13:06.346 180 there, we've got 90 and 270[br]there in their minus. 1/2 now 0:13:06.346,0:13:08.434 that's going to be. 0:13:09.990,0:13:17.466 Their cross and then these are[br]the ones that we are after. 0:13:19.060,0:13:25.682 So let's work with that. X over[br]2 is equal tool. Now where are 0:13:25.682,0:13:32.304 we? Well, we know that the angle[br]whose cosine is 1/2 is in fact 0:13:32.304,0:13:38.926 60 degrees, which is here 30 in[br]from there. So that must be 30 0:13:38.926,0:13:45.548 further on. In other words, 120[br]and this one must be 30 back. In 0:13:45.548,0:13:49.805 other words, 240. So now we[br]multiply it by. 0:13:49.850,0:13:57.078 Two, we get 240 and 480, but[br]of course this one is outside 0:13:57.078,0:14:04.306 the given range. The range is[br]not to 360, so we do not 0:14:04.306,0:14:07.642 need that answer, just want the 0:14:07.642,0:14:14.695 240. Now we've been working with[br]a range of North 360, or in one 0:14:14.695,0:14:20.480 case not to 180, so let's change[br]the range now so it's a 0:14:20.480,0:14:26.710 symmetric range in the Y axis,[br]so the range is now going to run 0:14:26.710,0:14:29.380 from minus 180 to plus 180 0:14:29.380,0:14:36.472 degrees. So we'll begin with[br]sign of X equals 1X is to 0:14:36.472,0:14:41.892 be between 180 degrees but[br]greater than minus 180 degrees. 0:14:41.892,0:14:48.938 Let's sketch the graph of sign[br]in that range. So we want to 0:14:48.938,0:14:54.358 complete copy of it. It's going[br]to look like that. 0:14:55.200,0:15:00.192 Now we know that the angle[br]who sign is one is 90 0:15:00.192,0:15:04.352 degrees and so we know[br]that's one there and that's 0:15:04.352,0:15:09.344 90 there and we can see that[br]there is only the one 0:15:09.344,0:15:13.504 solution it meets the curve[br]once and once only, so 0:15:13.504,0:15:14.752 that's 90 degrees. 0:15:15.920,0:15:22.333 Once and once only, that is[br]within the defined range. Let's 0:15:22.333,0:15:24.082 take another one. 0:15:24.230,0:15:31.634 So now we use a multiple[br]angle cause 2 X equals 1/2 0:15:31.634,0:15:38.421 and will take X to be[br]between minus 180 degrees and 0:15:38.421,0:15:44.591 plus 180 degrees. So let's[br]sketch the graph. Let's remember 0:15:44.591,0:15:51.995 that if X is between minus[br]180 and plus one 80, then 0:15:51.995,0:15:55.697 2X will be between minus 360. 0:15:55.740,0:15:57.930 And plus 360. 0:16:02.830,0:16:07.835 So what we need to do is use the[br]periodicity of the cosine 0:16:07.835,0:16:09.375 function to sketch it. 0:16:09.980,0:16:14.630 In the range. So there's[br]the knocked 360 bit and 0:16:14.630,0:16:16.025 then we want. 0:16:19.550,0:16:25.972 To minus 360. So I just label up[br]the points. Here is 90. 0:16:26.940,0:16:28.090 180 0:16:29.240,0:16:36.540 Two 7360 and then back[br]this way minus 90 - 0:16:36.540,0:16:40.344 180. Minus 270 and 0:16:40.344,0:16:47.634 minus 360. Now cause[br]2X is 1/2, so here's a half. 0:16:48.260,0:16:52.745 Membrane that this goes between[br]plus one and minus one and if we 0:16:52.745,0:16:56.540 draw a line across to see where[br]it meets the curve. 0:16:58.530,0:17:04.900 Then we can see it meets it in[br]four places. There, there there 0:17:04.900,0:17:11.270 and there we know that the angle[br]where it meets here is 60 0:17:11.270,0:17:16.660 degrees. So our first value is 2[br]X equals 60 degrees. 0:17:17.600,0:17:23.768 By symmetry, this one back here[br]has got to be minus 60. 0:17:24.350,0:17:30.494 What about this one here? Well,[br]again, symmetry says that we are 0:17:30.494,0:17:37.662 60 from here, so we've got to[br]be 60 back from there, so this 0:17:37.662,0:17:44.318 must be 300 and our symmetry of[br]the curve says that this one 0:17:44.318,0:17:51.486 must be minus 300, and so we[br]have X is 30 degrees minus 30 0:17:51.486,0:17:54.046 degrees, 150 degrees and minus 0:17:54.046,0:18:00.824 150 degrees. Working with[br]the tangent function tan, two 0:18:00.824,0:18:07.744 X equals Route 3 and[br]again will place X between 0:18:07.744,0:18:14.664 180 degrees and minus 180[br]degrees. We want to sketch 0:18:14.664,0:18:21.584 the function for tangent and[br]we want to be aware 0:18:21.584,0:18:24.352 that we've got 2X. 0:18:24.970,0:18:32.245 So since X is between minus 118[br]+ 182, X is got to be between 0:18:32.245,0:18:34.670 minus 360 and plus 360. 0:18:38.670,0:18:40.756 So if we take the bit between. 0:18:45.860,0:18:48.630 North And 360. 0:18:49.530,0:18:55.130 Which is that bit of the curve[br]we need a copy of that between 0:18:55.130,0:18:59.130 minus 360 and 0 because again[br]the tangent function is 0:18:59.130,0:19:01.530 periodic, so we need this bit. 0:19:07.580,0:19:08.510 That 0:19:14.620,0:19:18.680 And we need that and it's Mark[br]off this axis so we know where 0:19:18.680,0:19:20.130 we are. This is 90. 0:19:21.610,0:19:23.050 180 0:19:24.280,0:19:31.790 270 and 360. So this[br]must be minus 90 - 0:19:31.790,0:19:36.296 180 - 270 and minus[br]360. 0:19:37.420,0:19:42.672 Now 2X is Route 3, the angle[br]whose tangent is Route 3. We 0:19:42.672,0:19:49.136 know is 60, so we go across here[br]at Route 3 and we meet the curve 0:19:49.136,0:19:50.348 there and there. 0:19:51.440,0:19:57.656 And we come back this way. We[br]meet it there and we meet there. 0:19:57.656,0:20:00.320 So our answers are down here. 0:20:01.090,0:20:07.586 Working with this one, first we[br]know that that is 60, so 2X is 0:20:07.586,0:20:14.082 equal to 60 and so that that one[br]is 60 degrees on from that 0:20:14.082,0:20:19.650 point. Symmetry says there for[br]this one is also 60 degrees on 0:20:19.650,0:20:22.434 from there. In other words, it's 0:20:22.434,0:20:28.690 240. Let's work our way[br]backwards. This one must be 60 0:20:28.690,0:20:36.054 degrees on from minus 180, so it[br]must be at minus 120. This one 0:20:36.054,0:20:38.158 is 60 degrees on. 0:20:39.220,0:20:46.425 From minus 360 and so therefore[br]it must be minus 300. 0:20:47.120,0:20:54.188 And so if we divide throughout[br]by two, we have 31120 - 0:20:54.188,0:21:01.256 60 and minus 150 degrees. We[br]want to put degree signs on 0:21:01.256,0:21:06.557 all of these, so there are[br]four solutions there. 0:21:07.470,0:21:12.860 Trick equations often come up as[br]a result of having expressions 0:21:12.860,0:21:17.270 or other equations which are[br]rather more complicated than 0:21:17.270,0:21:19.230 that and depends upon 0:21:19.230,0:21:26.360 identity's. So I'm going to[br]have a look at a couple 0:21:26.360,0:21:31.805 of equations. These equations[br]both dependa pawn two identity's 0:21:31.805,0:21:37.855 that is expressions involving[br]trig functions that are true for 0:21:37.855,0:21:40.275 all values of X. 0:21:40.840,0:21:45.812 So the first one is sine[br]squared of X plus cost 0:21:45.812,0:21:51.688 squared of X is one. This is[br]true for all values of X. 0:21:52.890,0:21:56.130 The second one we derive from 0:21:56.130,0:22:02.119 this one. How we derive it[br]doesn't matter at the moment, 0:22:02.119,0:22:09.175 but what it tells us is that sex[br]squared X is equal to 1 + 10 0:22:09.175,0:22:15.343 squared X. So these are the two[br]identity's that I'm going to be 0:22:15.343,0:22:21.966 using. Sine squared X plus cost[br]squared X is one and sex squared 0:22:21.966,0:22:25.550 of X is 1 + 10 squared of 0:22:25.550,0:22:33.070 X OK. So how do we go[br]about using one of those to do 0:22:33.070,0:22:35.770 an equation like this? Cos 0:22:35.770,0:22:42.462 squared X? Plus cause[br]of X is equal 0:22:42.462,0:22:49.078 to sine squared of[br]X&X is between 180 0:22:49.078,0:22:51.559 and 0 degrees. 0:22:52.690,0:22:53.310 Well. 0:22:54.750,0:22:57.863 We've got a cost squared, A[br]cause and a sine squared. 0:22:58.680,0:23:03.852 If we were to use our identity[br]sine squared plus cost squared 0:23:03.852,0:23:08.593 is one to replace the sine[br]squared. Here I'd have a 0:23:08.593,0:23:15.058 quadratic in terms of Cos X, and[br]if I got a quadratic then I know 0:23:15.058,0:23:19.799 I can solve it either by[br]Factorizing or by using the 0:23:19.799,0:23:24.540 formula. So let me write down[br]sign squared X plus cost 0:23:24.540,0:23:29.281 squared. X is equal to 1, from[br]which we can see. 0:23:29.310,0:23:36.522 Sine squared X is equal to[br]1 minus Cos squared of X, 0:23:36.522,0:23:43.734 so I can take this and[br]plug it into their. So my 0:23:43.734,0:23:49.744 equation now becomes cost[br]squared X Plus X is equal 0:23:49.744,0:23:53.350 to 1 minus Cos squared X. 0:23:54.090,0:24:00.723 I want to get this as[br]a quadratic square term linear 0:24:00.723,0:24:07.959 term. Constant term equals 0, so[br]I begin by adding cost squared 0:24:07.959,0:24:09.768 to both sides. 0:24:09.870,0:24:16.331 So adding on a cost squared[br]there makes 2 Cos squared X plus 0:24:16.331,0:24:23.289 cause X equals 1. 'cause I added[br]cost square to get rid of that 0:24:23.289,0:24:30.247 one. Now I need to take one away[br]from both sides to cost squared 0:24:30.247,0:24:33.726 X Plus X minus one equals 0. 0:24:34.850,0:24:38.964 Now this is just a quadratic[br]equation, so the first question 0:24:38.964,0:24:43.826 I've got to ask is does it[br]factorize? So let's see if we 0:24:43.826,0:24:45.696 can get it to factorize. 0:24:46.550,0:24:51.295 I'll put two calls X in there[br]and cause X in there because 0:24:51.295,0:24:56.770 that 2 cause X times that cause[br]X gives Me 2 cost squared and I 0:24:56.770,0:25:01.880 put a one under one there 'cause[br]one times by one gives me one 0:25:01.880,0:25:07.355 and now I know to get a minus[br]sign. One's got to be minus and 0:25:07.355,0:25:13.195 one's got to be plus now I want[br]plus cause X so if I make this 0:25:13.195,0:25:16.845 one plus I'll have two cause X[br]times by one. 0:25:16.880,0:25:21.664 Is to cause X if I make this one[br]minus I'll have minus Cos X from 0:25:21.664,0:25:26.330 there. Taking those two[br]together, +2 cause X minus Cos X 0:25:26.330,0:25:31.296 is going to give me the plus[br]Kozaks in there, so that equals 0:25:31.296,0:25:36.290 0. Now, if not equal 0, I'm[br]multiplying 2 numbers together. 0:25:36.290,0:25:42.290 This one 2 cause X minus one and[br]this one cause X plus one, so 0:25:42.290,0:25:47.890 one of them or both of them have[br]got to be equal to 0. 0:25:48.770,0:25:54.714 So 2 calls X minus[br]one is 0. 0:25:55.560,0:26:02.952 All cause of X Plus One is[br]0, so this one tells me that 0:26:02.952,0:26:06.648 cause of X is equal to 1/2. 0:26:07.660,0:26:13.224 And this one tells Maine that[br]cause of X is equal to minus 0:26:13.224,0:26:17.932 one, and both of these are[br]possibilities. So I've got to 0:26:17.932,0:26:22.640 solve both equations to get the[br]total solution to the original 0:26:22.640,0:26:28.204 equation. So let's begin with[br]this cause of X is equal to 1/2. 0:26:28.830,0:26:35.526 And if you remember the range[br]of values was nought to 180 0:26:35.526,0:26:41.664 degrees, so let me sketch[br]cause of X between North and 0:26:41.664,0:26:46.686 180 degrees, and it looks[br]like that zero 9180. 0:26:47.740,0:26:53.096 We go across there at half and[br]come down there and there is 0:26:53.096,0:26:58.864 only one answer in the range, so[br]that's X is equal to 60 degrees. 0:27:00.220,0:27:06.892 But this one again let's sketch[br]cause of X between North and 0:27:06.892,0:27:13.560 180. There and there between[br]minus one and plus one and we 0:27:13.560,0:27:20.140 want cause of X equal to minus[br]one just at one point there and 0:27:20.140,0:27:26.250 so therefore X is equal to 180[br]degrees. So those are our two 0:27:26.250,0:27:30.010 answers to the full equation[br]that we had. 0:27:30.060,0:27:33.819 So it's now have a look at 0:27:33.819,0:27:41.220 three. 10 squared X is[br]equal to two sex squared X 0:27:41.220,0:27:44.904 Plus One and this time will 0:27:44.904,0:27:50.790 take X. To be between North and[br]180 degrees. Now, the identity 0:27:50.790,0:27:56.146 that we want is obviously the[br]one, the second one of the two 0:27:56.146,0:28:01.090 that we had before. In other[br]words, the one that tells us 0:28:01.090,0:28:08.094 that sex squared X is equal to 1[br]+ 10 squared X and we want to be 0:28:08.094,0:28:14.274 able to take this 1 + 10 squared[br]and put it into their. So we've 0:28:14.274,0:28:21.392 got 3. 10 squared[br]X is equal to 2 0:28:21.392,0:28:29.052 * 1 + 10 squared[br]X Plus one. Multiply out 0:28:29.052,0:28:36.712 this bracket. 310 squared X[br]is 2 + 210 squared 0:28:36.712,0:28:39.010 X plus one. 0:28:39.570,0:28:44.549 We can combine the two and the[br]one that will give us 3. 0:28:45.070,0:28:51.024 And we can take the 210 squared[br]X away from the three times 0:28:51.024,0:28:57.436 squared X there. That will give[br]us 10 squared X. Now we take the 0:28:57.436,0:29:03.848 square root of both sides so we[br]have 10X is equal to plus Route 0:29:03.848,0:29:06.138 3 or minus Route 3. 0:29:07.890,0:29:11.474 And we need to look at each of 0:29:11.474,0:29:15.180 these separately. So. 0:29:15.770,0:29:19.042 Time X equals Route 0:29:19.042,0:29:26.067 3. And Tan X[br]equals minus Route 3. 0:29:26.690,0:29:34.074 Access to be between North and[br]180, so let's have a sketch of 0:29:34.074,0:29:40.322 the graph of tan between those[br]values, so there is 90. 0:29:41.930,0:29:48.573 And there is 180 the angle whose[br]tangent is Route 3, we know. 0:29:49.820,0:29:56.780 Is there at 60 so we[br]know that X is equal to 0:29:56.780,0:30:02.167 60 degrees? Here we've[br]got minus Route 3, so 0:30:02.167,0:30:03.730 again, little sketch. 0:30:04.970,0:30:10.129 Between North and 180 range over[br]which were working here, we've 0:30:10.129,0:30:15.757 got minus Route 3 go across[br]there and down to their and 0:30:15.757,0:30:22.323 symmetry says it's got to be the[br]same as this one. Over here it's 0:30:22.323,0:30:29.358 got to be the same either side.[br]So in fact if that was 60 there 0:30:29.358,0:30:34.986 this must be 120 here, so X is[br]equal to 120 degrees. 0:30:35.160,0:30:39.670 So far we've been working in[br]degrees, but it makes little 0:30:39.670,0:30:43.360 difference if we're actually[br]working in radians and let's 0:30:43.360,0:30:49.100 just have a look at one or two[br]examples where in fact the range 0:30:49.100,0:30:55.250 of values that we've got is in[br]radians. So if we take Tan, X is 0:30:55.250,0:31:00.580 minus one and we take X to be[br]between plus or minus pie. 0:31:00.870,0:31:06.162 Another way of looking at[br]that would be if we were in 0:31:06.162,0:31:10.572 degrees. It will be between[br]plus and minus 180. Let's 0:31:10.572,0:31:14.100 sketch the graph of tangent[br]within that range. 0:31:15.780,0:31:17.178 Up to there. 0:31:18.060,0:31:20.728 That's π by 2. 0:31:22.950,0:31:25.668 Up to their which is π. 0:31:26.280,0:31:33.208 Minus Π[br]by 2. 0:31:36.370,0:31:39.870 Their minus 0:31:39.870,0:31:45.266 pie. Ton of X is[br]minus one, so somewhere 0:31:45.266,0:31:49.193 across here it's going to[br]meet the curve and we can 0:31:49.193,0:31:50.978 see that means it here. 0:31:52.120,0:31:56.060 And here giving us these[br]solutions at these points. Well, 0:31:56.060,0:32:01.576 we know that the angle whose[br]tangent is plus one is π by 4. 0:32:02.330,0:32:08.960 So this must be pie by 4 further[br]on, and so we have X is equal to 0:32:08.960,0:32:16.370 pie by 2 + π by 4. That will be[br]3/4 of Π or three π by 4, and 0:32:16.370,0:32:18.320 this one here must be. 0:32:19.080,0:32:25.295 Minus Π by 4 back there, so[br]minus π by 4. 0:32:26.010,0:32:30.385 Let's take one with[br]a multiple angle. 0:32:32.240,0:32:39.104 So we'll have a look cause[br]of two X is equal to Route 0:32:39.104,0:32:40.688 3 over 2. 0:32:41.730,0:32:47.826 I will take[br]X between North 0:32:47.826,0:32:54.560 and 2π. Now if[br]X is between North and 2π, and 0:32:54.560,0:32:55.850 we've got 2X. 0:32:56.680,0:33:01.372 And that means that 2X can be[br]between North and four π. 0:33:02.270,0:33:07.270 So again, we've got to make use[br]of the periodicity. 0:33:08.080,0:33:12.484 Of the graph of cosine to get a[br]second copy of it. 0:33:14.380,0:33:20.704 So there's the first copy[br]between North and 2π, and now we 0:33:20.704,0:33:27.028 want a second copy that goes[br]from 2π up till four π. 0:33:28.380,0:33:32.880 We can mark these off that one[br]will be pie by two. 0:33:33.450,0:33:34.220 Pie. 0:33:35.290,0:33:37.310 Three π by 2. 0:33:38.150,0:33:46.018 This one will be 5 Pi by[br]two. This one three Pi and this 0:33:46.018,0:33:48.828 one Seven π by 2. 0:33:49.790,0:33:55.362 So where are we with this cost?[br]2 X equals. Well, in fact we 0:33:55.362,0:34:00.934 know cost to access Route 3 over[br]2. We know that the angle that 0:34:00.934,0:34:06.904 gives us the cosine that is[br]Route 3 over 2 is π by 6. So 0:34:06.904,0:34:12.874 I'll first one is π Phi six,[br]root 3 over 2. Up here we go 0:34:12.874,0:34:18.446 across we meet the curve we come[br]down. We know that this one here 0:34:18.446,0:34:20.038 is π by 6. 0:34:20.080,0:34:25.540 Let's keep going across the[br]curves and see where we come to, 0:34:25.540,0:34:32.365 what we come to one here which[br]is π by 6 short of 2π. So 0:34:32.365,0:34:39.645 let me write it down as 2π -[br]Π by 6, and then again we come 0:34:39.645,0:34:45.105 to one here. Symmetry suggests[br]it should be pie by 6 further 0:34:45.105,0:34:51.020 on, so that's 2π + π by 6,[br]and then this one here. 0:34:51.030,0:34:57.606 Is symmetry would suggest his[br]pie by 6 short of four Pi, 0:34:57.606,0:35:04.730 so four π - π by 6.[br]So let's do that arithmetic 2X 0:35:04.730,0:35:06.922 is π by 6. 0:35:07.530,0:35:12.954 Now, how many sixths are there[br]in two? Well, the answer. Is 0:35:12.954,0:35:19.282 there a 12 of them and we're[br]going to take one of them away, 0:35:19.282,0:35:26.062 so that's eleven π by 6. We're[br]going to now add a 6th on, so 0:35:26.062,0:35:28.322 that's 13 Pi by 6. 0:35:29.630,0:35:37.022 How many 6th are there in four[br]or there are 24 of them? We're 0:35:37.022,0:35:43.886 going to take one away, so[br]that's 23. Pi over 6. Now we 0:35:43.886,0:35:51.278 want X, so we divide each of[br]these by 2π by 1211 Pi by 0:35:51.278,0:35:58.670 12:13, pie by 12, and 20, three[br]π by 12, and there are our 0:35:58.670,0:36:05.900 four solutions. Let's have a[br]look at one where we've got the 0:36:05.900,0:36:12.572 X divided by two rather than[br]multiplied by two. So the sign 0:36:12.572,0:36:18.132 of X over 2 is minus Route[br]3 over 2. 0:36:18.640,0:36:26.272 And let's take X to be[br]between pie and minus π. So 0:36:26.272,0:36:33.268 will sketch the graph of sign[br]between those limited, so it's 0:36:33.268,0:36:40.149 there. And their π[br]zero and minus pie. 0:36:40.750,0:36:46.535 Where looking for minus three[br]over 2. Now the one thing we do 0:36:46.535,0:36:53.210 know is that the angle who sign[br]is 3 over 2 is π by 3. 0:36:53.750,0:36:59.150 But we want minus Route 3 over[br]2, so that's down there. 0:36:59.740,0:37:01.510 We go across. 0:37:02.160,0:37:04.512 And we meet the curve these two 0:37:04.512,0:37:08.724 points. Now this curve is[br]symmetric with this one. 0:37:09.230,0:37:12.070 So if we know that. 0:37:12.710,0:37:14.900 Plus Route 3 over 2. 0:37:15.450,0:37:21.190 This one was Pi by three. Then[br]we know that this one must be 0:37:21.190,0:37:22.830 minus π by 3. 0:37:23.350,0:37:30.998 This one is π by three back, so[br]it's at 2π by three, so this one 0:37:30.998,0:37:38.168 must be minus 2π by three, and[br]so we have X over 2 is equal 0:37:38.168,0:37:45.338 to minus 2π by three and minus,[br]π by three, but it's X that we 0:37:45.338,0:37:52.030 want, so we multiply up X equals[br]minus four Pi by three and minus 0:37:52.030,0:37:53.464 2π by 3. 0:37:54.210,0:37:59.622 Let's just check on these[br]values. How do they fit with the 0:37:59.622,0:38:05.936 given range? Well, this 1 - 2π[br]by three is in that given range. 0:38:06.540,0:38:11.060 This one is outside, so we don't[br]want that one. 0:38:12.010,0:38:18.918 A final example here, working[br]with the idea again of using 0:38:18.918,0:38:24.570 those identities and will take 2[br]cost squared X. 0:38:25.490,0:38:31.167 Plus sign X is[br]equal to 1. 0:38:31.970,0:38:37.874 And we'll take X between[br]North and 2π. 0:38:38.780,0:38:43.060 We've got causes and signs,[br]so the identity that we're 0:38:43.060,0:38:47.768 going to want to help us[br]will be sine squared plus 0:38:47.768,0:38:49.908 cost. Squared X equals 1. 0:38:51.000,0:38:52.560 Cost squared here. 0:38:54.500,0:38:59.725 Cost squared here. Let's use[br]this identity to tell us that 0:38:59.725,0:39:05.900 cost squared X is equal to 1[br]minus sign squared X and make 0:39:05.900,0:39:08.750 the replacement up here for cost 0:39:08.750,0:39:14.460 squared. Because that as we will[br]see when we do it. 0:39:14.630,0:39:22.624 Leads to a quadratic in sign X,[br]so it's multiply this out 2 - 0:39:22.624,0:39:30.618 2 sine squared X plus sign X[br]is equal to 1 and I want 0:39:30.618,0:39:37.470 it as a quadratic, so I want[br]positive square term and then 0:39:37.470,0:39:44.893 the linear term and then the[br]constant term. So I need to add. 0:39:44.920,0:39:51.262 This to both sides of 0 equals 2[br]sine squared X. Adding it to 0:39:51.262,0:39:57.604 both sides. Now I need to take[br]this away minus sign X from both 0:39:57.604,0:40:03.946 sides and I need to take the two[br]away from both sides. So one 0:40:03.946,0:40:06.211 takeaway two is minus one. 0:40:07.040,0:40:10.930 And now does this factorize?[br]It's clearly a quadratic. Let's 0:40:10.930,0:40:16.765 look to see if we can make it[br]factorize 2 sign X and sign X. 0:40:16.765,0:40:20.655 Because multiplied together,[br]these two will give Me 2 sine 0:40:20.655,0:40:24.156 squared one and one because[br]multiplied together, these two 0:40:24.156,0:40:29.602 will give me one, but one of[br]them needs to be minus. To make 0:40:29.602,0:40:34.659 this a minus sign here. So I[br]think I'll have minus there and 0:40:34.659,0:40:39.327 plus there because two sign X[br]times by minus one gives me. 0:40:39.390,0:40:45.598 Minus 2 sign X one times by sign[br]X gives me sign X and if I 0:40:45.598,0:40:50.254 combine sign X with minus two[br]sign XI get minus sign X. 0:40:50.770,0:40:55.291 I have two numbers multiplied[br]together. This number 2 sign X 0:40:55.291,0:40:59.812 Plus One and this number sign X[br]minus one. They multiply 0:40:59.812,0:41:05.977 together to give me 0, so one or[br]both of them must be 0. Let's 0:41:05.977,0:41:07.210 write that down. 0:41:07.940,0:41:15.604 2 sign X Plus One is equal to[br]0 and sign X minus one is equal 0:41:15.604,0:41:23.268 to 0, so this tells me that sign[br]of X is equal. To take one away 0:41:23.268,0:41:29.974 from both sides and divide by[br]two. So sign X is minus 1/2 and 0:41:29.974,0:41:35.243 this one tells me that sign X is[br]equal to 1. 0:41:35.810,0:41:40.386 I'm now in a position to solve[br]these two separate equations. 0:41:40.910,0:41:43.360 So let me take this one first. 0:41:43.980,0:41:51.123 Now. We were working between[br]North and 2π, so we'll have a 0:41:51.123,0:41:53.488 sketch between North and 2π. 0:41:53.990,0:41:59.528 Of the sine curve and we want[br]sign X equals one. Well, there's 0:41:59.528,0:42:05.492 one and there's where it meets,[br]and that's pie by two, so we can 0:42:05.492,0:42:08.900 see that X is equal to pie by 0:42:08.900,0:42:15.744 two. Sign X equals minus 1/2.[br]Again, the range that we've been 0:42:15.744,0:42:21.618 given is between North and 2π.[br]So let's sketch between Norton 0:42:21.618,0:42:23.220 2π There's 2π. 0:42:25.450,0:42:27.090 Three π by 2. 0:42:27.810,0:42:33.966 Pie pie by two 0 - 1/2,[br]so that's coming along between 0:42:33.966,0:42:39.609 minus one and plus one that's[br]going to come along there. 0:42:40.890,0:42:45.869 And meet the curve there and[br]there. Now the one thing that we 0:42:45.869,0:42:48.933 do know is the angle who sign is 0:42:48.933,0:42:55.576 plus 1/2. Is π by 6, so we're[br]looking at plus 1/2. It will be 0:42:55.576,0:42:58.792 there and it would be pie by 6. 0:42:59.870,0:43:06.520 So it's π by 6 in from there,[br]so symmetry tells us that this 0:43:06.520,0:43:14.120 must be pie by 6 in from there,[br]so we've got X is equal to π 0:43:14.120,0:43:21.720 + π by 6, and symmetry tells us[br]it's pie by 6 in. From there, 2π 0:43:21.720,0:43:23.620 - Π by 6. 0:43:25.340,0:43:32.634 There are six sixths in pie, so[br]that's Seven π by 6. There is 0:43:32.634,0:43:39.407 1216, two Pi. We're taking one[br]of them away, so it will be 0:43:39.407,0:43:41.491 11 Pi over 6. 0:43:41.840,0:43:46.910 So we've shown there how to[br]solve some trig equations. 0:43:46.910,0:43:51.980 The important thing is the[br]sketch the graph. Find the 0:43:51.980,0:43:56.543 initial value and then[br]workout where the others are 0:43:56.543,0:44:01.106 from the graphs. Remember,[br]the graphs are all symmetric 0:44:01.106,0:44:05.669 and they're all periodic, so[br]they repeat themselves every 0:44:05.669,0:44:08.204 2π or every 360 degrees.