1 00:00:00,970 --> 00:00:05,568 In this video, we're going to be solving whole collection of 2 00:00:05,568 --> 00:00:09,330 trigonometric equations now be cause it's the technique of 3 00:00:09,330 --> 00:00:13,510 solving the equation and in ensuring that we get enough 4 00:00:13,510 --> 00:00:17,272 solutions, that's important and not actually looking up the 5 00:00:17,272 --> 00:00:21,452 angle. All of these are designed around certain special angles, 6 00:00:21,452 --> 00:00:26,886 so I'm just going to list at the very beginning here the special 7 00:00:26,886 --> 00:00:31,484 angles and their sines, cosines, and tangents that are going to 8 00:00:31,484 --> 00:00:33,468 form. The basis of what we're doing. 9 00:00:37,830 --> 00:00:41,951 So the special angles that we're going to have a look at our 10 00:00:41,951 --> 00:00:49,444 zero. 30 4560 and 90 there in degrees. 11 00:00:49,444 --> 00:00:55,236 If we're thinking about radians, then there's zero. 12 00:00:55,940 --> 00:00:59,348 Pie by 6. 13 00:01:00,130 --> 00:01:03,718 Pie by 4. 14 00:01:04,670 --> 00:01:05,738 Pie by three. 15 00:01:06,370 --> 00:01:09,090 And Π by 2. 16 00:01:10,220 --> 00:01:15,786 Trig ratios we're going to be looking at are the sign. 17 00:01:15,790 --> 00:01:21,104 The cosine. On the tangent of each of these. 18 00:01:22,460 --> 00:01:25,850 Sign of 0 is 0. 19 00:01:26,690 --> 00:01:29,666 The sign of 30 is 1/2. 20 00:01:30,370 --> 00:01:34,338 Sign of 45 is one over Route 2. 21 00:01:34,850 --> 00:01:41,714 The sign of 60 is Route 3 over 2 and the sign of 90 is one. 22 00:01:43,440 --> 00:01:45,260 Cosine of 0 is one. 23 00:01:45,990 --> 00:01:52,665 Cosine of 30 is Route 3 over 2 cosine of 45 is one over Route 24 00:01:52,665 --> 00:01:58,895 2, the cosine of 60 is 1/2, and the cosine of 90 is 0. 25 00:01:59,570 --> 00:02:06,096 The town of 0 is 0 the town of 30 is one over 26 00:02:06,096 --> 00:02:12,622 Route 3 that Anna 45 is 110 of 60 is Route 3 and 27 00:02:12,622 --> 00:02:17,140 the town of 90 degrees is infinite, it's undefined. 28 00:02:18,630 --> 00:02:21,738 It's these that we're going to be looking at and working with. 29 00:02:22,560 --> 00:02:25,900 Let's look at our first 30 00:02:25,900 --> 00:02:30,546 equation then. We're going to begin with some very simple 31 00:02:30,546 --> 00:02:37,844 ones. So we take sign of X is equal to nought .5. Now 32 00:02:37,844 --> 00:02:43,751 invariably when we get an equation we get a range of 33 00:02:43,751 --> 00:02:45,899 values along with it. 34 00:02:46,610 --> 00:02:52,714 So in this case will take X is between North and 360. So what 35 00:02:52,714 --> 00:02:56,638 we're looking for is all the values of X. 36 00:02:57,200 --> 00:03:01,460 Husain gives us N .5. 37 00:03:03,970 --> 00:03:10,830 Let's sketch a graph of sine X over this range. 38 00:03:13,610 --> 00:03:16,165 And sign looks like that with 90. 39 00:03:17,310 --> 00:03:18,500 180 40 00:03:20,120 --> 00:03:27,152 270 and 360 and ranging between one 4 sign 90 and minus 41 00:03:27,152 --> 00:03:30,668 one for the sign of 270. 42 00:03:31,360 --> 00:03:36,610 Sign of X is nought .5. So we go there. 43 00:03:37,890 --> 00:03:38,720 And there. 44 00:03:39,780 --> 00:03:44,410 So there's our first angle, and there's our second angle. 45 00:03:45,900 --> 00:03:52,517 We know the first one is 30 degrees because sign of 30 is 46 00:03:52,517 --> 00:03:58,625 1/2, so our first angle is 30 degrees. This curve is symmetric 47 00:03:58,625 --> 00:04:05,242 and so because were 30 degrees in from there, this one's got to 48 00:04:05,242 --> 00:04:08,296 be 30 degrees back from there. 49 00:04:08,810 --> 00:04:12,610 That would make it 50 00:04:12,610 --> 00:04:17,080 150. There are no more answers because within this range as we 51 00:04:17,080 --> 00:04:18,360 go along this line. 52 00:04:18,890 --> 00:04:23,453 It doesn't cross the curve at any other points. 53 00:04:23,460 --> 00:04:30,004 Let's have a look at a cosine cause 54 00:04:30,004 --> 00:04:36,548 of X is minus nought .5 and the 55 00:04:36,548 --> 00:04:43,092 range for this X between North and 360. 56 00:04:43,820 --> 00:04:49,112 So again, let's have a look at a graph of the function. 57 00:04:50,070 --> 00:04:53,899 Involved in the equation, the cosine graph. 58 00:04:56,510 --> 00:05:00,094 Looks like that. One and 59 00:05:00,094 --> 00:05:04,819 minus one. This is 90. 60 00:05:06,300 --> 00:05:08,160 180 61 00:05:09,310 --> 00:05:15,838 270 and then here at the end, 360. 62 00:05:17,020 --> 00:05:19,770 Minus 9.5. 63 00:05:20,810 --> 00:05:26,387 Gain across there at minus 9.5 and down to their and 64 00:05:26,387 --> 00:05:27,908 down to their. 65 00:05:28,970 --> 00:05:35,450 Now the one thing we do know is that the cause of 60 is plus N 66 00:05:35,450 --> 00:05:40,715 .5, and so that's there. So we know there is 60. Now again, 67 00:05:40,715 --> 00:05:45,980 this curve is symmetric, so if that one is 30 back that way 68 00:05:45,980 --> 00:05:51,650 this one must be 30 further on. So I'll first angle must be 120 69 00:05:51,650 --> 00:05:56,915 degrees. This one's got to be in a similar position as this bit 70 00:05:56,915 --> 00:05:59,345 of the curve is again symmetric. 71 00:05:59,380 --> 00:06:07,216 So that's 270 and we need to come back 30 degrees, so 72 00:06:07,216 --> 00:06:12,512 that's 240. Now we're going to have a look at an example where 73 00:06:12,512 --> 00:06:16,494 we've got what we call on multiple angle. So instead of 74 00:06:16,494 --> 00:06:21,562 just being cause of X or sign of X, it's going to be something 75 00:06:21,562 --> 00:06:26,268 like sign of 2X or cause of three X. So let's begin with 76 00:06:26,268 --> 00:06:33,076 sign of. 2X is equal to Route 3 over 2 77 00:06:33,076 --> 00:06:40,046 and again will take X to be between North and 78 00:06:40,046 --> 00:06:40,743 360. 79 00:06:41,810 --> 00:06:44,840 Now we've got 2X here. 80 00:06:45,520 --> 00:06:52,501 So if we've got 2X and X is between Norton 360, then the 81 00:06:52,501 --> 00:06:59,482 total range that we're going to be looking at is not to 722. 82 00:06:59,482 --> 00:07:06,463 X is going to come between 0 and 720, and the sign function 83 00:07:06,463 --> 00:07:12,370 is periodic. It repeats itself every 360 degrees, so I'm going 84 00:07:12,370 --> 00:07:16,129 to need 2 copies of the sine 85 00:07:16,129 --> 00:07:22,410 curve. As the first one going up to 360 and now I need a second 86 00:07:22,410 --> 00:07:24,460 copy there going on till. 87 00:07:25,240 --> 00:07:27,320 720 88 00:07:28,370 --> 00:07:34,535 OK, so sign 2 X equals root, 3 over 2, but we know that the 89 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 90 00:07:41,933 --> 00:07:46,865 there, then it's going to be these along here as well. So 91 00:07:46,865 --> 00:07:52,208 what have we got? Well, the first one here we know is 60. 92 00:07:52,208 --> 00:07:57,962 This point we know is 180 so that one's got to be the same 93 00:07:57,962 --> 00:08:04,184 distance. Back in due to the symmetry 120, so we do know that 94 00:08:04,184 --> 00:08:10,932 2X will be 60 or 120, but we also now we've got these other 95 00:08:10,932 --> 00:08:16,716 points on here, so let's just count on where we are. There's 96 00:08:16,716 --> 00:08:22,018 the 1st loop of the sign function, the first copy, its 97 00:08:22,018 --> 00:08:27,320 periodic and repeats itself again. So now we need to know 98 00:08:27,320 --> 00:08:29,248 where are these well. 99 00:08:29,270 --> 00:08:36,302 This is an exact copy of that, so this must be 60 100 00:08:36,302 --> 00:08:43,334 further on. In other words, at 420, and this must be another 101 00:08:43,334 --> 00:08:50,366 120 further on. In other words, at 480. So we've got two 102 00:08:50,366 --> 00:08:57,206 more answers. And it's X that we actually want, not 103 00:08:57,206 --> 00:09:00,641 2X. So this is 3060. 104 00:09:00,820 --> 00:09:04,441 210 and finally 105 00:09:04,441 --> 00:09:11,657 240. Let's have a look at that with a tangent 106 00:09:11,657 --> 00:09:17,587 function. This time tan or three X is equal to. 107 00:09:18,160 --> 00:09:24,864 Minus one and will take X to be 108 00:09:24,864 --> 00:09:28,216 between North and 180. 109 00:09:29,740 --> 00:09:36,040 So we draw a graph of the tangent function. 110 00:09:37,140 --> 00:09:38,380 So we go up. 111 00:09:40,990 --> 00:09:43,780 We've got that there. That's 90. 112 00:09:53,090 --> 00:09:59,837 This is 180 and this is 270 now. It's 3X. X is between 113 00:09:59,837 --> 00:10:07,103 Norton 180, so 3X can be between North and 3 * 180 which is 114 00:10:07,103 --> 00:10:13,850 540. So I need to get copies of this using the periodicity of 115 00:10:13,850 --> 00:10:20,597 the tangent function right up to 540. So let's put in some more. 116 00:10:21,810 --> 00:10:26,610 That's 360. On there. 117 00:10:27,920 --> 00:10:30,900 That's 450. 118 00:10:34,820 --> 00:10:42,470 This one here will be 540 and that's as near 119 00:10:42,470 --> 00:10:50,120 or as far as we need to go. Tanner 3X 120 00:10:50,120 --> 00:10:53,945 is minus one, so here's 121 00:10:53,945 --> 00:10:59,124 minus one. And we go across here picking off all the ones that we 122 00:10:59,124 --> 00:11:00,936 need. So we've got one there. 123 00:11:01,690 --> 00:11:08,862 There there. These are our values, so 3X is equal 12. 124 00:11:08,862 --> 00:11:16,686 Now we know that the angle whose tangent is one is 45, 125 00:11:16,686 --> 00:11:24,510 which is there. So again this and this are symmetric bits of 126 00:11:24,510 --> 00:11:32,334 curve, so this must be 45 further on. In other words 130. 127 00:11:32,340 --> 00:11:32,980 5. 128 00:11:34,170 --> 00:11:41,954 This one here has got to be 45 further on, so that will be 129 00:11:41,954 --> 00:11:49,670 315. This one here has got to be 45 further on, so that 130 00:11:49,670 --> 00:11:57,440 will be 495, but it's X that we want not 3X, so let's divide 131 00:11:57,440 --> 00:12:04,100 throughout by three, so freezing to that is 45 threes into that 132 00:12:04,100 --> 00:12:11,315 is 105 and threes into that is 165. Those are our three answers 133 00:12:11,315 --> 00:12:14,090 for that one 45 degrees. 134 00:12:14,100 --> 00:12:17,106 105 degrees under 135 00:12:17,106 --> 00:12:24,680 165. Let's take cause of X over 2 this time. So 136 00:12:24,680 --> 00:12:31,964 instead of multiplying by two or by three, were now dividing by 137 00:12:31,964 --> 00:12:38,641 two. Let's see what difference this might make equals minus 1/2 138 00:12:38,641 --> 00:12:45,925 and will take X to be between North and 360. So let's 139 00:12:45,925 --> 00:12:47,746 draw the graph. 140 00:12:48,830 --> 00:12:55,970 All calls X between North and 360, so there we've 141 00:12:55,970 --> 00:12:58,826 got it 360 there. 142 00:12:59,560 --> 00:13:06,346 180 there, we've got 90 and 270 there in their minus. 1/2 now 143 00:13:06,346 --> 00:13:08,434 that's going to be. 144 00:13:09,990 --> 00:13:17,466 Their cross and then these are the ones that we are after. 145 00:13:19,060 --> 00:13:25,682 So let's work with that. X over 2 is equal tool. Now where are 146 00:13:25,682 --> 00:13:32,304 we? Well, we know that the angle whose cosine is 1/2 is in fact 147 00:13:32,304 --> 00:13:38,926 60 degrees, which is here 30 in from there. So that must be 30 148 00:13:38,926 --> 00:13:45,548 further on. In other words, 120 and this one must be 30 back. In 149 00:13:45,548 --> 00:13:49,805 other words, 240. So now we multiply it by. 150 00:13:49,850 --> 00:13:57,078 Two, we get 240 and 480, but of course this one is outside 151 00:13:57,078 --> 00:14:04,306 the given range. The range is not to 360, so we do not 152 00:14:04,306 --> 00:14:07,642 need that answer, just want the 153 00:14:07,642 --> 00:14:14,695 240. Now we've been working with a range of North 360, or in one 154 00:14:14,695 --> 00:14:20,480 case not to 180, so let's change the range now so it's a 155 00:14:20,480 --> 00:14:26,710 symmetric range in the Y axis, so the range is now going to run 156 00:14:26,710 --> 00:14:29,380 from minus 180 to plus 180 157 00:14:29,380 --> 00:14:36,472 degrees. So we'll begin with sign of X equals 1X is to 158 00:14:36,472 --> 00:14:41,892 be between 180 degrees but greater than minus 180 degrees. 159 00:14:41,892 --> 00:14:48,938 Let's sketch the graph of sign in that range. So we want to 160 00:14:48,938 --> 00:14:54,358 complete copy of it. It's going to look like that. 161 00:14:55,200 --> 00:15:00,192 Now we know that the angle who sign is one is 90 162 00:15:00,192 --> 00:15:04,352 degrees and so we know that's one there and that's 163 00:15:04,352 --> 00:15:09,344 90 there and we can see that there is only the one 164 00:15:09,344 --> 00:15:13,504 solution it meets the curve once and once only, so 165 00:15:13,504 --> 00:15:14,752 that's 90 degrees. 166 00:15:15,920 --> 00:15:22,333 Once and once only, that is within the defined range. Let's 167 00:15:22,333 --> 00:15:24,082 take another one. 168 00:15:24,230 --> 00:15:31,634 So now we use a multiple angle cause 2 X equals 1/2 169 00:15:31,634 --> 00:15:38,421 and will take X to be between minus 180 degrees and 170 00:15:38,421 --> 00:15:44,591 plus 180 degrees. So let's sketch the graph. Let's remember 171 00:15:44,591 --> 00:15:51,995 that if X is between minus 180 and plus one 80, then 172 00:15:51,995 --> 00:15:55,697 2X will be between minus 360. 173 00:15:55,740 --> 00:15:57,930 And plus 360. 174 00:16:02,830 --> 00:16:07,835 So what we need to do is use the periodicity of the cosine 175 00:16:07,835 --> 00:16:09,375 function to sketch it. 176 00:16:09,980 --> 00:16:14,630 In the range. So there's the knocked 360 bit and 177 00:16:14,630 --> 00:16:16,025 then we want. 178 00:16:19,550 --> 00:16:25,972 To minus 360. So I just label up the points. Here is 90. 179 00:16:26,940 --> 00:16:28,090 180 180 00:16:29,240 --> 00:16:36,540 Two 7360 and then back this way minus 90 - 181 00:16:36,540 --> 00:16:40,344 180. Minus 270 and 182 00:16:40,344 --> 00:16:47,634 minus 360. Now cause 2X is 1/2, so here's a half. 183 00:16:48,260 --> 00:16:52,745 Membrane that this goes between plus one and minus one and if we 184 00:16:52,745 --> 00:16:56,540 draw a line across to see where it meets the curve. 185 00:16:58,530 --> 00:17:04,900 Then we can see it meets it in four places. There, there there 186 00:17:04,900 --> 00:17:11,270 and there we know that the angle where it meets here is 60 187 00:17:11,270 --> 00:17:16,660 degrees. So our first value is 2 X equals 60 degrees. 188 00:17:17,600 --> 00:17:23,768 By symmetry, this one back here has got to be minus 60. 189 00:17:24,350 --> 00:17:30,494 What about this one here? Well, again, symmetry says that we are 190 00:17:30,494 --> 00:17:37,662 60 from here, so we've got to be 60 back from there, so this 191 00:17:37,662 --> 00:17:44,318 must be 300 and our symmetry of the curve says that this one 192 00:17:44,318 --> 00:17:51,486 must be minus 300, and so we have X is 30 degrees minus 30 193 00:17:51,486 --> 00:17:54,046 degrees, 150 degrees and minus 194 00:17:54,046 --> 00:18:00,824 150 degrees. Working with the tangent function tan, two 195 00:18:00,824 --> 00:18:07,744 X equals Route 3 and again will place X between 196 00:18:07,744 --> 00:18:14,664 180 degrees and minus 180 degrees. We want to sketch 197 00:18:14,664 --> 00:18:21,584 the function for tangent and we want to be aware 198 00:18:21,584 --> 00:18:24,352 that we've got 2X. 199 00:18:24,970 --> 00:18:32,245 So since X is between minus 118 + 182, X is got to be between 200 00:18:32,245 --> 00:18:34,670 minus 360 and plus 360. 201 00:18:38,670 --> 00:18:40,756 So if we take the bit between. 202 00:18:45,860 --> 00:18:48,630 North And 360. 203 00:18:49,530 --> 00:18:55,130 Which is that bit of the curve we need a copy of that between 204 00:18:55,130 --> 00:18:59,130 minus 360 and 0 because again the tangent function is 205 00:18:59,130 --> 00:19:01,530 periodic, so we need this bit. 206 00:19:07,580 --> 00:19:08,510 That 207 00:19:14,620 --> 00:19:18,680 And we need that and it's Mark off this axis so we know where 208 00:19:18,680 --> 00:19:20,130 we are. This is 90. 209 00:19:21,610 --> 00:19:23,050 180 210 00:19:24,280 --> 00:19:31,790 270 and 360. So this must be minus 90 - 211 00:19:31,790 --> 00:19:36,296 180 - 270 and minus 360. 212 00:19:37,420 --> 00:19:42,672 Now 2X is Route 3, the angle whose tangent is Route 3. We 213 00:19:42,672 --> 00:19:49,136 know is 60, so we go across here at Route 3 and we meet the curve 214 00:19:49,136 --> 00:19:50,348 there and there. 215 00:19:51,440 --> 00:19:57,656 And we come back this way. We meet it there and we meet there. 216 00:19:57,656 --> 00:20:00,320 So our answers are down here. 217 00:20:01,090 --> 00:20:07,586 Working with this one, first we know that that is 60, so 2X is 218 00:20:07,586 --> 00:20:14,082 equal to 60 and so that that one is 60 degrees on from that 219 00:20:14,082 --> 00:20:19,650 point. Symmetry says there for this one is also 60 degrees on 220 00:20:19,650 --> 00:20:22,434 from there. In other words, it's 221 00:20:22,434 --> 00:20:28,690 240. Let's work our way backwards. This one must be 60 222 00:20:28,690 --> 00:20:36,054 degrees on from minus 180, so it must be at minus 120. This one 223 00:20:36,054 --> 00:20:38,158 is 60 degrees on. 224 00:20:39,220 --> 00:20:46,425 From minus 360 and so therefore it must be minus 300. 225 00:20:47,120 --> 00:20:54,188 And so if we divide throughout by two, we have 31120 - 226 00:20:54,188 --> 00:21:01,256 60 and minus 150 degrees. We want to put degree signs on 227 00:21:01,256 --> 00:21:06,557 all of these, so there are four solutions there. 228 00:21:07,470 --> 00:21:12,860 Trick equations often come up as a result of having expressions 229 00:21:12,860 --> 00:21:17,270 or other equations which are rather more complicated than 230 00:21:17,270 --> 00:21:19,230 that and depends upon 231 00:21:19,230 --> 00:21:26,360 identity's. So I'm going to have a look at a couple 232 00:21:26,360 --> 00:21:31,805 of equations. These equations both dependa pawn two identity's 233 00:21:31,805 --> 00:21:37,855 that is expressions involving trig functions that are true for 234 00:21:37,855 --> 00:21:40,275 all values of X. 235 00:21:40,840 --> 00:21:45,812 So the first one is sine squared of X plus cost 236 00:21:45,812 --> 00:21:51,688 squared of X is one. This is true for all values of X. 237 00:21:52,890 --> 00:21:56,130 The second one we derive from 238 00:21:56,130 --> 00:22:02,119 this one. How we derive it doesn't matter at the moment, 239 00:22:02,119 --> 00:22:09,175 but what it tells us is that sex squared X is equal to 1 + 10 240 00:22:09,175 --> 00:22:15,343 squared X. So these are the two identity's that I'm going to be 241 00:22:15,343 --> 00:22:21,966 using. Sine squared X plus cost squared X is one and sex squared 242 00:22:21,966 --> 00:22:25,550 of X is 1 + 10 squared of 243 00:22:25,550 --> 00:22:33,070 X OK. So how do we go about using one of those to do 244 00:22:33,070 --> 00:22:35,770 an equation like this? Cos 245 00:22:35,770 --> 00:22:42,462 squared X? Plus cause of X is equal 246 00:22:42,462 --> 00:22:49,078 to sine squared of X&X is between 180 247 00:22:49,078 --> 00:22:51,559 and 0 degrees. 248 00:22:52,690 --> 00:22:53,310 Well. 249 00:22:54,750 --> 00:22:57,863 We've got a cost squared, A cause and a sine squared. 250 00:22:58,680 --> 00:23:03,852 If we were to use our identity sine squared plus cost squared 251 00:23:03,852 --> 00:23:08,593 is one to replace the sine squared. Here I'd have a 252 00:23:08,593 --> 00:23:15,058 quadratic in terms of Cos X, and if I got a quadratic then I know 253 00:23:15,058 --> 00:23:19,799 I can solve it either by Factorizing or by using the 254 00:23:19,799 --> 00:23:24,540 formula. So let me write down sign squared X plus cost 255 00:23:24,540 --> 00:23:29,281 squared. X is equal to 1, from which we can see. 256 00:23:29,310 --> 00:23:36,522 Sine squared X is equal to 1 minus Cos squared of X, 257 00:23:36,522 --> 00:23:43,734 so I can take this and plug it into their. So my 258 00:23:43,734 --> 00:23:49,744 equation now becomes cost squared X Plus X is equal 259 00:23:49,744 --> 00:23:53,350 to 1 minus Cos squared X. 260 00:23:54,090 --> 00:24:00,723 I want to get this as a quadratic square term linear 261 00:24:00,723 --> 00:24:07,959 term. Constant term equals 0, so I begin by adding cost squared 262 00:24:07,959 --> 00:24:09,768 to both sides. 263 00:24:09,870 --> 00:24:16,331 So adding on a cost squared there makes 2 Cos squared X plus 264 00:24:16,331 --> 00:24:23,289 cause X equals 1. 'cause I added cost square to get rid of that 265 00:24:23,289 --> 00:24:30,247 one. Now I need to take one away from both sides to cost squared 266 00:24:30,247 --> 00:24:33,726 X Plus X minus one equals 0. 267 00:24:34,850 --> 00:24:38,964 Now this is just a quadratic equation, so the first question 268 00:24:38,964 --> 00:24:43,826 I've got to ask is does it factorize? So let's see if we 269 00:24:43,826 --> 00:24:45,696 can get it to factorize. 270 00:24:46,550 --> 00:24:51,295 I'll put two calls X in there and cause X in there because 271 00:24:51,295 --> 00:24:56,770 that 2 cause X times that cause X gives Me 2 cost squared and I 272 00:24:56,770 --> 00:25:01,880 put a one under one there 'cause one times by one gives me one 273 00:25:01,880 --> 00:25:07,355 and now I know to get a minus sign. One's got to be minus and 274 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 275 00:25:13,195 --> 00:25:16,845 one plus I'll have two cause X times by one. 276 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 277 00:25:21,664 --> 00:25:26,330 there. Taking those two together, +2 cause X minus Cos X 278 00:25:26,330 --> 00:25:31,296 is going to give me the plus Kozaks in there, so that equals 279 00:25:31,296 --> 00:25:36,290 0. Now, if not equal 0, I'm multiplying 2 numbers together. 280 00:25:36,290 --> 00:25:42,290 This one 2 cause X minus one and this one cause X plus one, so 281 00:25:42,290 --> 00:25:47,890 one of them or both of them have got to be equal to 0. 282 00:25:48,770 --> 00:25:54,714 So 2 calls X minus one is 0. 283 00:25:55,560 --> 00:26:02,952 All cause of X Plus One is 0, so this one tells me that 284 00:26:02,952 --> 00:26:06,648 cause of X is equal to 1/2. 285 00:26:07,660 --> 00:26:13,224 And this one tells Maine that cause of X is equal to minus 286 00:26:13,224 --> 00:26:17,932 one, and both of these are possibilities. So I've got to 287 00:26:17,932 --> 00:26:22,640 solve both equations to get the total solution to the original 288 00:26:22,640 --> 00:26:28,204 equation. So let's begin with this cause of X is equal to 1/2. 289 00:26:28,830 --> 00:26:35,526 And if you remember the range of values was nought to 180 290 00:26:35,526 --> 00:26:41,664 degrees, so let me sketch cause of X between North and 291 00:26:41,664 --> 00:26:46,686 180 degrees, and it looks like that zero 9180. 292 00:26:47,740 --> 00:26:53,096 We go across there at half and come down there and there is 293 00:26:53,096 --> 00:26:58,864 only one answer in the range, so that's X is equal to 60 degrees. 294 00:27:00,220 --> 00:27:06,892 But this one again let's sketch cause of X between North and 295 00:27:06,892 --> 00:27:13,560 180. There and there between minus one and plus one and we 296 00:27:13,560 --> 00:27:20,140 want cause of X equal to minus one just at one point there and 297 00:27:20,140 --> 00:27:26,250 so therefore X is equal to 180 degrees. So those are our two 298 00:27:26,250 --> 00:27:30,010 answers to the full equation that we had. 299 00:27:30,060 --> 00:27:33,819 So it's now have a look at 300 00:27:33,819 --> 00:27:41,220 three. 10 squared X is equal to two sex squared X 301 00:27:41,220 --> 00:27:44,904 Plus One and this time will 302 00:27:44,904 --> 00:27:50,790 take X. To be between North and 180 degrees. Now, the identity 303 00:27:50,790 --> 00:27:56,146 that we want is obviously the one, the second one of the two 304 00:27:56,146 --> 00:28:01,090 that we had before. In other words, the one that tells us 305 00:28:01,090 --> 00:28:08,094 that sex squared X is equal to 1 + 10 squared X and we want to be 306 00:28:08,094 --> 00:28:14,274 able to take this 1 + 10 squared and put it into their. So we've 307 00:28:14,274 --> 00:28:21,392 got 3. 10 squared X is equal to 2 308 00:28:21,392 --> 00:28:29,052 * 1 + 10 squared X Plus one. Multiply out 309 00:28:29,052 --> 00:28:36,712 this bracket. 310 squared X is 2 + 210 squared 310 00:28:36,712 --> 00:28:39,010 X plus one. 311 00:28:39,570 --> 00:28:44,549 We can combine the two and the one that will give us 3. 312 00:28:45,070 --> 00:28:51,024 And we can take the 210 squared X away from the three times 313 00:28:51,024 --> 00:28:57,436 squared X there. That will give us 10 squared X. Now we take the 314 00:28:57,436 --> 00:29:03,848 square root of both sides so we have 10X is equal to plus Route 315 00:29:03,848 --> 00:29:06,138 3 or minus Route 3. 316 00:29:07,890 --> 00:29:11,474 And we need to look at each of 317 00:29:11,474 --> 00:29:15,180 these separately. So. 318 00:29:15,770 --> 00:29:19,042 Time X equals Route 319 00:29:19,042 --> 00:29:26,067 3. And Tan X equals minus Route 3. 320 00:29:26,690 --> 00:29:34,074 Access to be between North and 180, so let's have a sketch of 321 00:29:34,074 --> 00:29:40,322 the graph of tan between those values, so there is 90. 322 00:29:41,930 --> 00:29:48,573 And there is 180 the angle whose tangent is Route 3, we know. 323 00:29:49,820 --> 00:29:56,780 Is there at 60 so we know that X is equal to 324 00:29:56,780 --> 00:30:02,167 60 degrees? Here we've got minus Route 3, so 325 00:30:02,167 --> 00:30:03,730 again, little sketch. 326 00:30:04,970 --> 00:30:10,129 Between North and 180 range over which were working here, we've 327 00:30:10,129 --> 00:30:15,757 got minus Route 3 go across there and down to their and 328 00:30:15,757 --> 00:30:22,323 symmetry says it's got to be the same as this one. Over here it's 329 00:30:22,323 --> 00:30:29,358 got to be the same either side. So in fact if that was 60 there 330 00:30:29,358 --> 00:30:34,986 this must be 120 here, so X is equal to 120 degrees. 331 00:30:35,160 --> 00:30:39,670 So far we've been working in degrees, but it makes little 332 00:30:39,670 --> 00:30:43,360 difference if we're actually working in radians and let's 333 00:30:43,360 --> 00:30:49,100 just have a look at one or two examples where in fact the range 334 00:30:49,100 --> 00:30:55,250 of values that we've got is in radians. So if we take Tan, X is 335 00:30:55,250 --> 00:31:00,580 minus one and we take X to be between plus or minus pie. 336 00:31:00,870 --> 00:31:06,162 Another way of looking at that would be if we were in 337 00:31:06,162 --> 00:31:10,572 degrees. It will be between plus and minus 180. Let's 338 00:31:10,572 --> 00:31:14,100 sketch the graph of tangent within that range. 339 00:31:15,780 --> 00:31:17,178 Up to there. 340 00:31:18,060 --> 00:31:20,728 That's π by 2. 341 00:31:22,950 --> 00:31:25,668 Up to their which is π. 342 00:31:26,280 --> 00:31:33,208 Minus Π by 2. 343 00:31:36,370 --> 00:31:39,870 Their minus 344 00:31:39,870 --> 00:31:45,266 pie. Ton of X is minus one, so somewhere 345 00:31:45,266 --> 00:31:49,193 across here it's going to meet the curve and we can 346 00:31:49,193 --> 00:31:50,978 see that means it here. 347 00:31:52,120 --> 00:31:56,060 And here giving us these solutions at these points. Well, 348 00:31:56,060 --> 00:32:01,576 we know that the angle whose tangent is plus one is π by 4. 349 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 350 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 351 00:32:16,370 --> 00:32:18,320 this one here must be. 352 00:32:19,080 --> 00:32:25,295 Minus Π by 4 back there, so minus π by 4. 353 00:32:26,010 --> 00:32:30,385 Let's take one with a multiple angle. 354 00:32:32,240 --> 00:32:39,104 So we'll have a look cause of two X is equal to Route 355 00:32:39,104 --> 00:32:40,688 3 over 2. 356 00:32:41,730 --> 00:32:47,826 I will take X between North 357 00:32:47,826 --> 00:32:54,560 and 2π. Now if X is between North and 2π, and 358 00:32:54,560 --> 00:32:55,850 we've got 2X. 359 00:32:56,680 --> 00:33:01,372 And that means that 2X can be between North and four π. 360 00:33:02,270 --> 00:33:07,270 So again, we've got to make use of the periodicity. 361 00:33:08,080 --> 00:33:12,484 Of the graph of cosine to get a second copy of it. 362 00:33:14,380 --> 00:33:20,704 So there's the first copy between North and 2π, and now we 363 00:33:20,704 --> 00:33:27,028 want a second copy that goes from 2π up till four π. 364 00:33:28,380 --> 00:33:32,880 We can mark these off that one will be pie by two. 365 00:33:33,450 --> 00:33:34,220 Pie. 366 00:33:35,290 --> 00:33:37,310 Three π by 2. 367 00:33:38,150 --> 00:33:46,018 This one will be 5 Pi by two. This one three Pi and this 368 00:33:46,018 --> 00:33:48,828 one Seven π by 2. 369 00:33:49,790 --> 00:33:55,362 So where are we with this cost? 2 X equals. Well, in fact we 370 00:33:55,362 --> 00:34:00,934 know cost to access Route 3 over 2. We know that the angle that 371 00:34:00,934 --> 00:34:06,904 gives us the cosine that is Route 3 over 2 is π by 6. So 372 00:34:06,904 --> 00:34:12,874 I'll first one is π Phi six, root 3 over 2. Up here we go 373 00:34:12,874 --> 00:34:18,446 across we meet the curve we come down. We know that this one here 374 00:34:18,446 --> 00:34:20,038 is π by 6. 375 00:34:20,080 --> 00:34:25,540 Let's keep going across the curves and see where we come to, 376 00:34:25,540 --> 00:34:32,365 what we come to one here which is π by 6 short of 2π. So 377 00:34:32,365 --> 00:34:39,645 let me write it down as 2π - Π by 6, and then again we come 378 00:34:39,645 --> 00:34:45,105 to one here. Symmetry suggests it should be pie by 6 further 379 00:34:45,105 --> 00:34:51,020 on, so that's 2π + π by 6, and then this one here. 380 00:34:51,030 --> 00:34:57,606 Is symmetry would suggest his pie by 6 short of four Pi, 381 00:34:57,606 --> 00:35:04,730 so four π - π by 6. So let's do that arithmetic 2X 382 00:35:04,730 --> 00:35:06,922 is π by 6. 383 00:35:07,530 --> 00:35:12,954 Now, how many sixths are there in two? Well, the answer. Is 384 00:35:12,954 --> 00:35:19,282 there a 12 of them and we're going to take one of them away, 385 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 386 00:35:26,062 --> 00:35:28,322 that's 13 Pi by 6. 387 00:35:29,630 --> 00:35:37,022 How many 6th are there in four or there are 24 of them? We're 388 00:35:37,022 --> 00:35:43,886 going to take one away, so that's 23. Pi over 6. Now we 389 00:35:43,886 --> 00:35:51,278 want X, so we divide each of these by 2π by 1211 Pi by 390 00:35:51,278 --> 00:35:58,670 12:13, pie by 12, and 20, three π by 12, and there are our 391 00:35:58,670 --> 00:36:05,900 four solutions. Let's have a look at one where we've got the 392 00:36:05,900 --> 00:36:12,572 X divided by two rather than multiplied by two. So the sign 393 00:36:12,572 --> 00:36:18,132 of X over 2 is minus Route 3 over 2. 394 00:36:18,640 --> 00:36:26,272 And let's take X to be between pie and minus π. So 395 00:36:26,272 --> 00:36:33,268 will sketch the graph of sign between those limited, so it's 396 00:36:33,268 --> 00:36:40,149 there. And their π zero and minus pie. 397 00:36:40,750 --> 00:36:46,535 Where looking for minus three over 2. Now the one thing we do 398 00:36:46,535 --> 00:36:53,210 know is that the angle who sign is 3 over 2 is π by 3. 399 00:36:53,750 --> 00:36:59,150 But we want minus Route 3 over 2, so that's down there. 400 00:36:59,740 --> 00:37:01,510 We go across. 401 00:37:02,160 --> 00:37:04,512 And we meet the curve these two 402 00:37:04,512 --> 00:37:08,724 points. Now this curve is symmetric with this one. 403 00:37:09,230 --> 00:37:12,070 So if we know that. 404 00:37:12,710 --> 00:37:14,900 Plus Route 3 over 2. 405 00:37:15,450 --> 00:37:21,190 This one was Pi by three. Then we know that this one must be 406 00:37:21,190 --> 00:37:22,830 minus π by 3. 407 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 408 00:37:30,998 --> 00:37:38,168 must be minus 2π by three, and so we have X over 2 is equal 409 00:37:38,168 --> 00:37:45,338 to minus 2π by three and minus, π by three, but it's X that we 410 00:37:45,338 --> 00:37:52,030 want, so we multiply up X equals minus four Pi by three and minus 411 00:37:52,030 --> 00:37:53,464 2π by 3. 412 00:37:54,210 --> 00:37:59,622 Let's just check on these values. How do they fit with the 413 00:37:59,622 --> 00:38:05,936 given range? Well, this 1 - 2π by three is in that given range. 414 00:38:06,540 --> 00:38:11,060 This one is outside, so we don't want that one. 415 00:38:12,010 --> 00:38:18,918 A final example here, working with the idea again of using 416 00:38:18,918 --> 00:38:24,570 those identities and will take 2 cost squared X. 417 00:38:25,490 --> 00:38:31,167 Plus sign X is equal to 1. 418 00:38:31,970 --> 00:38:37,874 And we'll take X between North and 2π. 419 00:38:38,780 --> 00:38:43,060 We've got causes and signs, so the identity that we're 420 00:38:43,060 --> 00:38:47,768 going to want to help us will be sine squared plus 421 00:38:47,768 --> 00:38:49,908 cost. Squared X equals 1. 422 00:38:51,000 --> 00:38:52,560 Cost squared here. 423 00:38:54,500 --> 00:38:59,725 Cost squared here. Let's use this identity to tell us that 424 00:38:59,725 --> 00:39:05,900 cost squared X is equal to 1 minus sign squared X and make 425 00:39:05,900 --> 00:39:08,750 the replacement up here for cost 426 00:39:08,750 --> 00:39:14,460 squared. Because that as we will see when we do it. 427 00:39:14,630 --> 00:39:22,624 Leads to a quadratic in sign X, so it's multiply this out 2 - 428 00:39:22,624 --> 00:39:30,618 2 sine squared X plus sign X is equal to 1 and I want 429 00:39:30,618 --> 00:39:37,470 it as a quadratic, so I want positive square term and then 430 00:39:37,470 --> 00:39:44,893 the linear term and then the constant term. So I need to add. 431 00:39:44,920 --> 00:39:51,262 This to both sides of 0 equals 2 sine squared X. Adding it to 432 00:39:51,262 --> 00:39:57,604 both sides. Now I need to take this away minus sign X from both 433 00:39:57,604 --> 00:40:03,946 sides and I need to take the two away from both sides. So one 434 00:40:03,946 --> 00:40:06,211 takeaway two is minus one. 435 00:40:07,040 --> 00:40:10,930 And now does this factorize? It's clearly a quadratic. Let's 436 00:40:10,930 --> 00:40:16,765 look to see if we can make it factorize 2 sign X and sign X. 437 00:40:16,765 --> 00:40:20,655 Because multiplied together, these two will give Me 2 sine 438 00:40:20,655 --> 00:40:24,156 squared one and one because multiplied together, these two 439 00:40:24,156 --> 00:40:29,602 will give me one, but one of them needs to be minus. To make 440 00:40:29,602 --> 00:40:34,659 this a minus sign here. So I think I'll have minus there and 441 00:40:34,659 --> 00:40:39,327 plus there because two sign X times by minus one gives me. 442 00:40:39,390 --> 00:40:45,598 Minus 2 sign X one times by sign X gives me sign X and if I 443 00:40:45,598 --> 00:40:50,254 combine sign X with minus two sign XI get minus sign X. 444 00:40:50,770 --> 00:40:55,291 I have two numbers multiplied together. This number 2 sign X 445 00:40:55,291 --> 00:40:59,812 Plus One and this number sign X minus one. They multiply 446 00:40:59,812 --> 00:41:05,977 together to give me 0, so one or both of them must be 0. Let's 447 00:41:05,977 --> 00:41:07,210 write that down. 448 00:41:07,940 --> 00:41:15,604 2 sign X Plus One is equal to 0 and sign X minus one is equal 449 00:41:15,604 --> 00:41:23,268 to 0, so this tells me that sign of X is equal. To take one away 450 00:41:23,268 --> 00:41:29,974 from both sides and divide by two. So sign X is minus 1/2 and 451 00:41:29,974 --> 00:41:35,243 this one tells me that sign X is equal to 1. 452 00:41:35,810 --> 00:41:40,386 I'm now in a position to solve these two separate equations. 453 00:41:40,910 --> 00:41:43,360 So let me take this one first. 454 00:41:43,980 --> 00:41:51,123 Now. We were working between North and 2π, so we'll have a 455 00:41:51,123 --> 00:41:53,488 sketch between North and 2π. 456 00:41:53,990 --> 00:41:59,528 Of the sine curve and we want sign X equals one. Well, there's 457 00:41:59,528 --> 00:42:05,492 one and there's where it meets, and that's pie by two, so we can 458 00:42:05,492 --> 00:42:08,900 see that X is equal to pie by 459 00:42:08,900 --> 00:42:15,744 two. Sign X equals minus 1/2. Again, the range that we've been 460 00:42:15,744 --> 00:42:21,618 given is between North and 2π. So let's sketch between Norton 461 00:42:21,618 --> 00:42:23,220 2π There's 2π. 462 00:42:25,450 --> 00:42:27,090 Three π by 2. 463 00:42:27,810 --> 00:42:33,966 Pie pie by two 0 - 1/2, so that's coming along between 464 00:42:33,966 --> 00:42:39,609 minus one and plus one that's going to come along there. 465 00:42:40,890 --> 00:42:45,869 And meet the curve there and there. Now the one thing that we 466 00:42:45,869 --> 00:42:48,933 do know is the angle who sign is 467 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 468 00:42:55,576 --> 00:42:58,792 there and it would be pie by 6. 469 00:42:59,870 --> 00:43:06,520 So it's π by 6 in from there, so symmetry tells us that this 470 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 471 00:43:14,120 --> 00:43:21,720 + π by 6, and symmetry tells us it's pie by 6 in. From there, 2π 472 00:43:21,720 --> 00:43:23,620 - Π by 6. 473 00:43:25,340 --> 00:43:32,634 There are six sixths in pie, so that's Seven π by 6. There is 474 00:43:32,634 --> 00:43:39,407 1216, two Pi. We're taking one of them away, so it will be 475 00:43:39,407 --> 00:43:41,491 11 Pi over 6. 476 00:43:41,840 --> 00:43:46,910 So we've shown there how to solve some trig equations. 477 00:43:46,910 --> 00:43:51,980 The important thing is the sketch the graph. Find the 478 00:43:51,980 --> 00:43:56,543 initial value and then workout where the others are 479 00:43:56,543 --> 00:44:01,106 from the graphs. Remember, the graphs are all symmetric 480 00:44:01,106 --> 00:44:05,669 and they're all periodic, so they repeat themselves every 481 00:44:05,669 --> 00:44:08,204 2π or every 360 degrees.