WEBVTT 00:00:01.450 --> 00:00:05.002 In this video, we're going to be looking at the double angle 00:00:05.002 --> 00:00:11.130 formula. But to start with, we're going to start from the 00:00:11.130 --> 00:00:16.696 addition formula. Not all of them, just the ones that deal 00:00:16.696 --> 00:00:22.888 with A+B. So let's just write those down to begin with sign of 00:00:22.888 --> 00:00:25.396 A+B, we know is sign a. 00:00:25.950 --> 00:00:32.838 Cause B. Post cause a sign 00:00:32.838 --> 00:00:40.310 be. Next, we want the cause of A+B, which 00:00:40.310 --> 00:00:44.260 will be cause a calls 00:00:44.260 --> 00:00:51.387 B. Minus sign, a sign be and finally 00:00:51.387 --> 00:00:57.795 the tan one tan of A+B, which will 00:00:57.795 --> 00:01:00.999 be 10 A plus 00:01:00.999 --> 00:01:06.629 10B. Over 1 - 00:01:06.629 --> 00:01:10.688 10 a 10B. 00:01:12.290 --> 00:01:15.394 So those are three of our addition formula. 00:01:16.680 --> 00:01:20.088 And each one is to do with A+B. 00:01:20.990 --> 00:01:27.843 So what happens if we let a be equal to be? 00:01:28.480 --> 00:01:34.161 In other words, instead of having a plus B, we have a plus 00:01:34.161 --> 00:01:41.760 a. So that would be sign of A plus a would 00:01:41.760 --> 00:01:44.252 be signed to A. 00:01:45.720 --> 00:01:52.970 What does that do to this right hand side? Well 00:01:52.970 --> 00:02:00.220 gives us sign a cause, A plus cause a sign 00:02:00.220 --> 00:02:08.157 a. In other words, these two at the same, so we can just add 00:02:08.157 --> 00:02:15.642 them together. So sign of 2A is 2 sign a cause A and that's our 00:02:15.642 --> 00:02:20.133 first double angle formula double angle because it's 2A 00:02:20.133 --> 00:02:22.129 where doubling the angle. 00:02:22.670 --> 00:02:29.703 So what is it sign and so on. Let's do the same with 00:02:29.703 --> 00:02:36.736 cause. Let's put a equal to be. So will have cars to a 00:02:36.736 --> 00:02:42.687 is equal to what it was cause a Cosby. It's now 00:02:42.687 --> 00:02:48.638 going to be cause A cause a witches caused squared A. 00:02:49.720 --> 00:02:56.902 Sign a sign be when it's now going to be sign a sign a 00:02:56.902 --> 00:02:59.980 which is sine squared minus sign 00:02:59.980 --> 00:03:03.428 squared A. And that's how 00:03:03.428 --> 00:03:07.250 a second. Double angle formula. 00:03:07.930 --> 00:03:15.290 Doing the same with Tan Tan 2A is equal to. 00:03:15.940 --> 00:03:23.738 10A plus 10 B this is now angle a so it's 10A Plus 10A 00:03:23.738 --> 00:03:26.523 which is 2 Tab A. 00:03:26.540 --> 00:03:33.356 All over 1 - 10, eight and be. But this is now a instead of B, 00:03:33.356 --> 00:03:34.634 so it's tanae. 00:03:35.290 --> 00:03:41.167 10 eight times by 10 A is 10 squared. 00:03:41.167 --> 00:03:44.432 1 - 10 squared A. 00:03:46.150 --> 00:03:51.217 And here are our three double angle formula again 00:03:51.217 --> 00:03:56.847 to be learned to be recognized and to be used. 00:03:58.680 --> 00:04:04.477 Let's just have a look at this one cause to A. 00:04:05.750 --> 00:04:10.310 White pick out this one. Well this right hand side which is 00:04:10.310 --> 00:04:14.490 the bit that interests because it's got cost squared and sign 00:04:14.490 --> 00:04:17.150 squared in it and there is an 00:04:17.150 --> 00:04:21.130 identity. That's to do with cost squared. Plus sign squared 00:04:21.130 --> 00:04:26.800 equals 1. What that means is we can replace the sine squared. 00:04:27.340 --> 00:04:32.016 And get everything in terms of Cos squared. Or we can do it the 00:04:32.016 --> 00:04:33.018 other way round. 00:04:33.600 --> 00:04:36.903 So I just have a look at that one. 00:04:38.280 --> 00:04:45.222 Cause to a cost squared, A 00:04:45.222 --> 00:04:48.693 minus sign squared 00:04:48.693 --> 00:04:55.745 a butt. Cost squared A plus sign, 00:04:55.745 --> 00:04:59.517 squared A equals 1. 00:05:00.200 --> 00:05:07.337 In other words, sign squared a is 1 minus Cos squared a so 00:05:07.337 --> 00:05:13.925 we can replace the sine squared here in our double angle formula 00:05:13.925 --> 00:05:19.964 by one minus Cos squared, so will have cause to A. 00:05:20.560 --> 00:05:28.140 Is cost squared A minus one minus Cos squared A? 00:05:29.150 --> 00:05:34.160 Using the brackets, notice to show I'm taking away all 00:05:34.160 --> 00:05:38.168 of it and now let's remove the brackets. 00:05:39.370 --> 00:05:46.630 Minus one. Minus minus gives me a plus 00:05:46.630 --> 00:05:53.330 cause squared a, so I now have two cost squared, 00:05:53.330 --> 00:06:00.030 A minus one, so that's another double angle formula for 00:06:00.030 --> 00:06:02.040 cost to a. 00:06:02.770 --> 00:06:09.346 Now because I replaced the sine squared here by one minus Cos 00:06:09.346 --> 00:06:16.470 squared, I can do the same again and replace the cost squared by 00:06:16.470 --> 00:06:23.594 one minus sign squared and what that will give main is cause 2A 00:06:23.594 --> 00:06:27.430 is 1 - 2 sine squared A. 00:06:27.460 --> 00:06:35.100 So lot of formally there. Let's just write them all 00:06:35.100 --> 00:06:42.306 down again. Sign to a 00:06:42.306 --> 00:06:46.210 IS2. Find a 00:06:46.210 --> 00:06:50.163 Kohl's A. Calls 00:06:50.163 --> 00:06:56.884 to a. Is cost squared A minus sign, 00:06:56.884 --> 00:07:03.674 squared A and we can rewrite that as two cost 00:07:03.674 --> 00:07:10.464 square day minus one or as 1 - 2 sine 00:07:10.464 --> 00:07:18.212 squared A. And then turn to a is 00:07:18.212 --> 00:07:21.128 equal to 2. 00:07:21.130 --> 00:07:27.538 Tam a over 1 - 10 00:07:27.538 --> 00:07:33.264 squared A. So there are our double angle formula 00:07:33.264 --> 00:07:37.360 formula to be learned formally to be remembered 00:07:37.360 --> 00:07:40.944 and most importantly recognized and used when 00:07:40.944 --> 00:07:42.480 we need them. 00:07:43.840 --> 00:07:49.566 So let's have a look at how we can make use of these double 00:07:49.566 --> 00:07:57.260 angle formula. So sign of three X. Is it possible to write 00:07:57.260 --> 00:08:02.957 sign of 3X all in terms of sine X? 00:08:04.480 --> 00:08:07.805 Well. Let's try and break this 00:08:07.805 --> 00:08:15.319 3X up. 3X is 2X Plus X, so we can write this a sign of 00:08:15.319 --> 00:08:16.738 2X Plus X. 00:08:18.230 --> 00:08:24.970 OK, this means we can use our addition formula sign 00:08:24.970 --> 00:08:28.340 of two X cause X. 00:08:28.930 --> 00:08:35.566 Plus cause of two X sign X. 00:08:36.550 --> 00:08:44.002 Now I can use my double angle formula here sign of two 00:08:44.002 --> 00:08:51.454 X is 2 sign X Cos X still to be multiplied by 00:08:51.454 --> 00:08:53.317 Cos X Plus. 00:08:54.040 --> 00:08:56.730 Now I have a choice. 00:08:57.380 --> 00:09:02.418 There are three double angle formula for cause 2X, so my 00:09:02.418 --> 00:09:09.288 choice is got to be governed by what it is I'm trying to do and 00:09:09.288 --> 00:09:14.784 we're trying to write sign 3X all in terms of sine X. 00:09:15.400 --> 00:09:22.218 That means the choice I have to make here is the one that's got 00:09:22.218 --> 00:09:28.062 signs in it, not cosines, but the one that's got signs and 00:09:28.062 --> 00:09:35.367 only signs, and the one that has that is 1 - 2 sine squared X 00:09:35.367 --> 00:09:38.776 still to be times by sign X. 00:09:39.580 --> 00:09:46.830 So this front term is going to be 2 sign 00:09:46.830 --> 00:09:49.730 X cause squared X. 00:09:50.300 --> 00:09:57.830 One times by Cynex is plus sign X minus and 00:09:57.830 --> 00:10:05.360 two sine squared X times. Biosynex is sine cubed X. 00:10:06.320 --> 00:10:11.204 Well, we're getting there. We've got sign here sign here. Sign 00:10:11.204 --> 00:10:13.424 cubed here. Cost squared here. 00:10:14.340 --> 00:10:19.940 But Cost Square can be rewritten using one of the fundamental 00:10:19.940 --> 00:10:24.637 identity's cost square plus sign squared is one so cost square 00:10:24.637 --> 00:10:26.772 can be replaced by Wang. 00:10:27.630 --> 00:10:32.850 Minus sign squared. 00:10:34.940 --> 00:10:40.256 And so we can see here. Everything is now in terms of 00:10:40.256 --> 00:10:44.243 sine X and all we need to do is 00:10:44.243 --> 00:10:51.580 tidied up. So we multiply out this bracket 2 sign X for 00:10:51.580 --> 00:10:58.540 the first term, 2 sign X times by one. Then we have 00:10:58.540 --> 00:11:04.920 two sign X times Y minus sign squared minus two sine 00:11:04.920 --> 00:11:10.720 cubed X plus sign X minus two sine cubed X. 00:11:11.730 --> 00:11:18.920 2 sign X plus sign X that's three sign X. 00:11:19.770 --> 00:11:27.366 Minus two sine cubed minus two sine cubed is minus 4 sign 00:11:27.366 --> 00:11:34.329 cubed X and that everything is in terms of sine X. 00:11:35.340 --> 00:11:41.598 You can do the same with cause as well cause 3X can be turned 00:11:41.598 --> 00:11:46.068 into an expression that's entirely in terms of cause X. 00:11:46.830 --> 00:11:52.386 That's an example of using our double angle formula in order to 00:11:52.386 --> 00:11:57.479 reduce if we like to use that expression and multiple angle 00:11:57.479 --> 00:12:03.961 sign 3X is a multiple angle down to a single angle in terms of 00:12:03.961 --> 00:12:09.980 the sign of that angle. Let's have a look now at solving an 00:12:09.980 --> 00:12:17.063 equation. Let's take cause 2X is equal to 00:12:17.063 --> 00:12:23.535 sign X and let's take a range of 00:12:23.535 --> 00:12:25.962 values for X. 00:12:26.540 --> 00:12:31.980 Which puts X between plus and minus pie. 00:12:33.590 --> 00:12:36.622 Again, I've deliberately chosen 00:12:36.622 --> 00:12:42.806 caused 2X. Be cause we have a choice, we have three 00:12:42.806 --> 00:12:45.076 possibilities. Which one do we 00:12:45.076 --> 00:12:50.640 choose? Well, if I want to solve an equation like this, I really 00:12:50.640 --> 00:12:53.680 need it all in terms of one trig 00:12:53.680 --> 00:12:56.740 function. Not two, but one. 00:12:58.050 --> 00:13:03.650 And here I've got sine X. Therefore makes sense here. 00:13:04.240 --> 00:13:10.696 To replace this by 1 - 2 sine 00:13:10.696 --> 00:13:13.924 squared, X equals sign 00:13:13.924 --> 00:13:19.866 X. Now we have a quadratic equation where the 00:13:19.866 --> 00:13:22.134 variable is sign X. 00:13:23.100 --> 00:13:26.898 Let's rearrange that so that it 00:13:26.898 --> 00:13:32.260 equals 0. Add the two sine squared to each side. 00:13:33.680 --> 00:13:36.875 Plus the sign 00:13:36.875 --> 00:13:41.799 X. And take one away from each side. 00:13:43.050 --> 00:13:47.200 This is now a quadratic equation. Can I factorize it? 00:13:47.200 --> 00:13:48.860 Let's have a look. 00:13:48.860 --> 00:13:54.283 Two brackets, 2 sign X and sign X when multiplied together, 00:13:54.283 --> 00:14:00.692 these two will give me the two sine squared I need minus one, 00:14:00.692 --> 00:14:07.101 so let's pop a one into each bracket, and one of them's got 00:14:07.101 --> 00:14:10.059 to be plus and one minus. 00:14:11.240 --> 00:14:17.645 I need plus sign X in the middle going to make this one plus one, 00:14:17.645 --> 00:14:24.477 so I get +2 sign X, make that one minus so I get minus sign X 00:14:24.477 --> 00:14:29.174 and when I combine those two terms plus sign X there. 00:14:30.470 --> 00:14:37.098 This says. A bracket, a lump of algebra times by 00:14:37.098 --> 00:14:41.608 another bracket. Another lump of algebra is equal to 0. 00:14:42.630 --> 00:14:44.238 And so one. 00:14:44.850 --> 00:14:51.738 Or both of these brackets must be equal 00:14:51.738 --> 00:14:58.108 to 0. And so we've reduced this fairly complicated 00:14:58.108 --> 00:15:05.412 looking equation down to two simple ones, and this one tells 00:15:05.412 --> 00:15:13.380 us here. Sign X is equal to add 1 to each side 00:15:13.380 --> 00:15:16.036 and divide by two. 00:15:16.130 --> 00:15:21.915 Sign X is 1/2 or this one here tells us that sign X 00:15:21.915 --> 00:15:24.140 is equal to minus one. 00:15:25.700 --> 00:15:29.935 We've now got to extract the values of X from this 00:15:29.935 --> 00:15:34.555 information and those values of X must be between plus and minus 00:15:34.555 --> 00:15:41.300 pie. So let's sketch the graph of cynex between 00:15:41.300 --> 00:15:43.980 plus and minus pie. 00:15:45.310 --> 00:15:48.150 There's the graph, there's pie. 00:15:48.950 --> 00:15:55.880 Pie by 2 - π by two and minus pie and it goes between 00:15:55.880 --> 00:16:02.810 one and minus one. So let's take this one. First sign X is minus 00:16:02.810 --> 00:16:09.740 one. Well that goes across there and down to their, so X is minus 00:16:09.740 --> 00:16:13.700 π by two is one answer that we 00:16:13.700 --> 00:16:20.171 get there. Sign X is 1/2, half goes across there and we should 00:16:20.171 --> 00:16:25.343 recognize that this is one of those nice numbers. Sign X is 00:16:25.343 --> 00:16:31.377 1/2 for which we've got an exact answer, and So what we do know 00:16:31.377 --> 00:16:36.980 is that the sign of 30 degrees is 1/2, but where working in 00:16:36.980 --> 00:16:43.014 radians. So in fact 30 degrees is the same angle as pie by 6. 00:16:43.520 --> 00:16:47.870 And this is symmetrical. Remember the curve for sign is 00:16:47.870 --> 00:16:53.960 symmetrical, so if that's pie by 6 in there, that's got to be pie 00:16:53.960 --> 00:17:01.355 by 6 in there. So this, In other words will be 5 pie by 6, and so 00:17:01.355 --> 00:17:08.315 we have our two answers for this one pie by 6 and five pie by 6. 00:17:08.350 --> 00:17:14.447 So that we see that we've been able to solve our equation using 00:17:14.447 --> 00:17:19.137 our double angle formula and by making the right choice, 00:17:19.137 --> 00:17:24.765 particularly here when with cost 2X, we know that we have three 00:17:24.765 --> 00:17:31.562 possibilities. So. That is, have a look at another 00:17:31.562 --> 00:17:37.568 equation again using our double angle formula. Sign 2 X equals 00:17:37.568 --> 00:17:44.666 sign X. And again, let's take our value of X to lie between 00:17:44.666 --> 00:17:46.850 plus and minus pie. 00:17:48.160 --> 00:17:53.128 We've only one choice for sign 2X, that's two. 00:17:53.730 --> 00:18:00.604 Sign X Cos X equals sign X. 00:18:01.500 --> 00:18:07.604 Now. It's very, very tempting to say our common factor on each 00:18:07.604 --> 00:18:09.180 side. Cancel it out. 00:18:10.500 --> 00:18:11.860 And then we've lost it. 00:18:12.710 --> 00:18:18.446 And because we lose it, we might lose solutions to the equation. 00:18:19.120 --> 00:18:25.511 So what's better than canceling out is to get everything to 00:18:25.511 --> 00:18:32.756 one side. By taking cynex away from each side. 00:18:33.310 --> 00:18:39.770 And then. Take out a common factor, and here there's 00:18:39.770 --> 00:18:43.160 a common factor of sign X. 00:18:43.730 --> 00:18:50.530 Which will leave us 2 cause AX minus 00:18:50.530 --> 00:18:58.037 one. Two expressions multiplied together give us 0. 00:18:58.960 --> 00:19:06.520 So either or both of these expressions is equal to 0, so 00:19:06.520 --> 00:19:12.190 either sign X equals 0 or two cause X. 00:19:12.830 --> 00:19:15.518 Minus one equals 0. 00:19:16.330 --> 00:19:22.358 Now we've managed to reduce this equation to two smaller, simpler 00:19:22.358 --> 00:19:27.290 equations once sign and the other ones for cause. 00:19:27.820 --> 00:19:33.115 But each is going to give us a value of X. Let's take this one 00:19:33.115 --> 00:19:35.233 first, sign of X is 0. 00:19:36.290 --> 00:19:43.670 And let's draw a sketch up here of sign of X. There 00:19:43.670 --> 00:19:51.050 it is and it's zero here, here and here on the X 00:19:51.050 --> 00:19:57.815 axis minus Π Zero and Pi straightaway. We've got X equals 00:19:57.815 --> 00:20:05.195 minus π and 0, not pie, because pie is excluded from the 00:20:05.195 --> 00:20:07.070 range that. We've got. 00:20:07.580 --> 00:20:14.255 But notice sign X equals 0 gave us 2 answers for X if we have 00:20:14.255 --> 00:20:20.930 cancelled sign X out up here and just got rid of it, we would not 00:20:20.930 --> 00:20:23.155 have got those two answers. 00:20:23.770 --> 00:20:28.582 Let's go to this equation now. 2 cause X minus one is 00:20:28.582 --> 00:20:33.394 0, so that tells us that cause X is equal to 1/2. 00:20:34.540 --> 00:20:39.448 And what we need to do is sketch the graph of cosine. 00:20:40.430 --> 00:20:42.490 And the graph of cosine. 00:20:43.500 --> 00:20:50.250 Looks. Like that between pie and minus pie? 00:20:51.130 --> 00:20:54.620 And here is the half. 00:20:56.170 --> 00:21:02.338 And cause X equals 1/2. This is another one of these nice 00:21:02.338 --> 00:21:07.992 relationships and we know that this one is 60 degrees or 00:21:07.992 --> 00:21:13.646 because we're working in radians Pi by three and because of 00:21:13.646 --> 00:21:20.328 symmetry this one here has got to be minus π by three, so 00:21:20.328 --> 00:21:25.468 X is minus π by 3 or π by 3. 00:21:25.590 --> 00:21:31.530 And so we've got our four solutions for this equation. 00:21:32.530 --> 00:21:37.359 In doing this work with double angles, in effect, we've been 00:21:37.359 --> 00:21:41.749 looking at what are called multiple angles, and here I've 00:21:41.749 --> 00:21:46.578 been drawing sketches of a single angle. If you like, just 00:21:46.578 --> 00:21:52.724 sign X Cos X. So the question is, what does the graph of sine 00:21:52.724 --> 00:21:54.041 2X look like? 00:21:54.670 --> 00:22:00.429 What does the graph of cause 3X look like? Or for that matter, 00:22:00.429 --> 00:22:03.087 what about sign of 1/2 X? 00:22:03.100 --> 00:22:08.824 Well, let's just explore that sign X. Let's just have a look 00:22:08.824 --> 00:22:10.255 at its graph. 00:22:11.240 --> 00:22:14.795 Between North And 00:22:14.795 --> 00:22:21.140 2π So that's sine 00:22:21.140 --> 00:22:26.280 X. What about sign 2X? 00:22:28.740 --> 00:22:32.108 Over the same range. 00:22:33.330 --> 00:22:39.750 I just think about it. What are we doing when we're multiplying 00:22:39.750 --> 00:22:45.807 by two? Well, we're doubling yes, but what that means is 00:22:45.807 --> 00:22:48.122 everything that happens on this 00:22:48.122 --> 00:22:52.567 graph. Happened twice as fast on this graph. 00:22:53.920 --> 00:22:59.560 So in the time it takes this to go from North to 2π, it's done 00:22:59.560 --> 00:23:01.816 all of that in this space. 00:23:02.970 --> 00:23:10.374 So that bit of graph appears in this space so that we 00:23:10.374 --> 00:23:17.161 have up down there. Then of course it's periodic, so it 00:23:17.161 --> 00:23:19.012 does it again. 00:23:20.700 --> 00:23:24.360 So sign of 2 X. 00:23:26.220 --> 00:23:31.164 Gets through everything twice as quickly as sign X and sign of 00:23:31.164 --> 00:23:36.932 three. X will get through it 3 times as quickly, so I'll have 3 00:23:36.932 --> 00:23:41.464 copies of this graph in the same space, not to pie. 00:23:42.060 --> 00:23:48.720 And the same with cause 2X and cause 3X and cost 4X. 00:23:49.290 --> 00:23:54.519 What if I take sign of 1/2 of X? 00:23:54.520 --> 00:24:01.430 Do that. Sign of 1/2 of X. 00:24:02.210 --> 00:24:09.425 I'll do sign of X first just to give us the picture again. 00:24:09.940 --> 00:24:13.760 That's our graph between North 00:24:13.760 --> 00:24:17.269 and 2π. So what about? 00:24:17.840 --> 00:24:23.198 Sign. Of half of X on the same scale. 00:24:24.160 --> 00:24:27.385 Well, things are happening half 00:24:27.385 --> 00:24:34.310 as quickly. 1/2 of two pies just pie, so we will only 00:24:34.310 --> 00:24:40.834 have got through that bit of the curve by the time we've got to 00:24:40.834 --> 00:24:47.358 2π, so that in effect the graph of sign a half pie looks like 00:24:47.358 --> 00:24:49.222 that and continues on. 00:24:49.890 --> 00:24:52.450 In double the space. 00:24:53.900 --> 00:24:58.509 So graphs of multiple angles look a little bit different to 00:24:58.509 --> 00:24:59.766 the ordinary angle. 00:25:00.480 --> 00:25:05.492 But the thing that we have to remember is that if we have a 00:25:05.492 --> 00:25:06.924 multiple here, that's greater 00:25:06.924 --> 00:25:11.780 than one. Then it's going to get through it much quicker and 00:25:11.780 --> 00:25:13.730 we're going to see the graph 00:25:13.730 --> 00:25:18.262 repeated. If we've got a multiple here, that's less than 00:25:18.262 --> 00:25:21.934 one, it's going to take longer to get through. 00:25:22.810 --> 00:25:28.282 The graph and we're going to see the graph extended and drawn 00:25:28.282 --> 00:25:34.452 out. Let's have a look at the 00:25:34.452 --> 00:25:40.066 graph of cause X and cause 2X. 00:25:42.760 --> 00:25:45.448 Again between North. 00:25:45.950 --> 00:25:48.150 And. 2π 00:25:49.430 --> 00:25:51.430 So there's our graph. 00:25:53.980 --> 00:26:00.217 What about? Mark the points in the same positions. This is 00:26:00.217 --> 00:26:06.013 going to go through twice as quickly, so we're going to see 00:26:06.013 --> 00:26:11.326 that shape repeated in this space here, so we're going to 00:26:11.326 --> 00:26:18.088 see that come down and go up, and then we're going to see it 00:26:18.088 --> 00:26:21.895 repeated again. I thought so. Sketching these graphs of 00:26:21.895 --> 00:26:25.990 multiple angles is quite easy. All you need to do is look at 00:26:25.990 --> 00:26:29.455 the original graph and judge the number of times that it's 00:26:29.455 --> 00:26:31.975 going to be repeated over the given range.