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