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