1 00:00:00,570 --> 00:00:01,450 ‫Welcome back. 2 00:00:02,190 --> 00:00:11,400 ‫Let's now compute the Phi Beta and then the two variables, let's derive formulas for them. 3 00:00:12,270 --> 00:00:14,880 ‫The hints that I gave you were the following. 4 00:00:15,750 --> 00:00:21,330 ‫You should first put the equations in this form and in this form. 5 00:00:22,200 --> 00:00:33,120 ‫So if you're Z double that equals all that, then you're a Z, double that plus G equals cosine five 6 00:00:33,840 --> 00:00:40,230 ‫times cosine theta times you one over the mass of the drone. 7 00:00:41,070 --> 00:00:52,140 ‫So X double dot over a Z double that plus G equals this equation here, which is the X double date equation 8 00:00:52,140 --> 00:00:56,360 ‫here divided by this equation here. 9 00:00:57,180 --> 00:01:05,610 ‫So you can cancel out this and this, you can then separate this equation like this and then you can 10 00:01:05,610 --> 00:01:10,140 ‫cancel out this cosine phi with this cosine phi here. 11 00:01:10,950 --> 00:01:24,780 ‫Now from trigonometric identities, you know that sine X divided by cosine X equals tangent X, that 12 00:01:24,780 --> 00:01:31,290 ‫means that this part here sine theta over cosine Seeta. 13 00:01:31,590 --> 00:01:40,920 ‫That will be your tangent theta like this and then also sine phi over cosine phi. 14 00:01:41,460 --> 00:01:45,420 ‫This will be your tangent phi like this. 15 00:01:46,600 --> 00:01:50,320 ‫So I can rewrite this entire equation like this. 16 00:01:51,760 --> 00:02:02,290 ‫Similarly, why doubled that over a double Dods plus GS would be this equation here divided by this 17 00:02:02,290 --> 00:02:12,880 ‫equation here and you can see it here and now you can also cancel out you one over M and you one over 18 00:02:12,880 --> 00:02:13,270 ‫M. 19 00:02:14,350 --> 00:02:22,750 ‫You can then separate this equation in the same way and then you can cancel out counseling fi here and 20 00:02:22,750 --> 00:02:33,610 ‫counseling FI here again here, you will have signed Seeta over Cosine Theta and that will be your Tenjin 21 00:02:33,620 --> 00:02:34,260 ‫Seeta. 22 00:02:34,780 --> 00:02:42,670 ‫And then here you will have signed Phi Overconsume Phi and that will be your Tenjin Phi. 23 00:02:43,450 --> 00:02:47,650 ‫And so in the end this equation looks like this. 24 00:02:47,920 --> 00:02:49,960 ‫That's your end result. 25 00:02:50,920 --> 00:02:57,340 ‫And so in the end, you have these two equations here, so you have X, double dot over Z, double that, 26 00:02:57,340 --> 00:03:05,910 ‫plus it equals all this and then Y, double dot over Z, double that, plus G, it equals all this. 27 00:03:06,580 --> 00:03:10,880 ‫And now let's make a couple of abbreviations to write less. 28 00:03:11,740 --> 00:03:17,860 ‫Let's say that this thing here is a. 29 00:03:18,920 --> 00:03:32,090 ‫Then let's say that this thing here is B, so X double date over the plus G equals A and Y, double 30 00:03:32,090 --> 00:03:45,350 ‫date over Z, double that plus G equals B, then let's say that cosine Sysop R equals C, and therefore 31 00:03:45,860 --> 00:04:03,230 ‫this is C and this is also C and then sine Sysop R equals D, so this one here is D and then also this 32 00:04:03,230 --> 00:04:05,840 ‫one here is also D. 33 00:04:06,950 --> 00:04:17,870 ‫In that case, the first equation can be rewritten like this, a equals tangent theta times C like you 34 00:04:17,870 --> 00:04:19,790 ‫can see here plus. 35 00:04:20,710 --> 00:04:28,720 ‫Tangent five times D divided by cosine Seeta. 36 00:04:29,860 --> 00:04:40,840 ‫That's this first equation here, and then the second equation would be B equals tangent three times 37 00:04:41,080 --> 00:04:52,980 ‫D minus tangent five times C divided by cosine theta like this. 38 00:04:53,890 --> 00:04:56,050 ‫That would be your second equation here. 39 00:04:57,060 --> 00:05:06,900 ‫And so if I take these equations here, then I can rearrange them like this, I can take this term here 40 00:05:07,110 --> 00:05:12,030 ‫and then I can put it on the other side of the equation sine like this. 41 00:05:12,810 --> 00:05:20,100 ‫And then here I can take this term and then I can put it on the other side of the equation sine and 42 00:05:20,100 --> 00:05:26,910 ‫then I can take this B and then I can put it here on the other side of this equation. 43 00:05:26,910 --> 00:05:37,380 ‫Sine if I do that then here I will have tangent five times D over cosine Seeta which is this term here 44 00:05:38,250 --> 00:05:51,780 ‫equals a minus tangent Theta times C and then here I will have tangent five times C or Cosine Seeta, 45 00:05:52,320 --> 00:06:00,630 ‫which is this one here, and that equals tangent theta times D minus B. 46 00:06:01,530 --> 00:06:06,660 ‫I now want to express this tension phi in terms of everything else. 47 00:06:07,530 --> 00:06:19,160 ‫So Tangent Phi equals this term here and then I'm going to multiply it by cosine theta. 48 00:06:19,590 --> 00:06:27,180 ‫So this cosine theta will come here now and then I'm going to take this D and then I'm going to divide 49 00:06:27,210 --> 00:06:32,360 ‫this entire thing by D and I can do the same thing here. 50 00:06:32,460 --> 00:06:37,380 ‫I want to express this tension phi in terms of everything else. 51 00:06:38,370 --> 00:06:50,490 ‫So I'm going to do it like this tangent phi equals tangent theta times D minus B, that's this term 52 00:06:50,490 --> 00:06:51,060 ‫here. 53 00:06:51,540 --> 00:06:59,190 ‫And I'm going to multiply by this cosine Seeta and then I'm going to divide all that by C like this. 54 00:07:00,310 --> 00:07:07,450 ‫And now, look, you've got tension fire here and you've got tension fire here they are the same thing 55 00:07:08,350 --> 00:07:14,310 ‫like tangent five equals tangent fire like this. 56 00:07:15,130 --> 00:07:20,380 ‫Therefore, these two expressions here, they are equal to each other. 57 00:07:21,160 --> 00:07:25,090 ‫They are written in a different form, but they are the same thing. 58 00:07:25,750 --> 00:07:33,640 ‫And since they are equal to each other, then I can get rid of this cosine sheeter and this cosine Seeta. 59 00:07:34,590 --> 00:07:44,690 ‫And now I'm going to separate this equation like this, a over deed, so this divided by this minus 60 00:07:45,060 --> 00:07:58,830 ‫and then you will have Danjean Theta time C over D, and that equals tangent times D over C, so it's 61 00:07:58,830 --> 00:08:08,760 ‫here tension theta times the over C minus B over C is here, minus B oversee. 62 00:08:09,630 --> 00:08:19,590 ‫I can rearrange the terms and then I will have A over D plus B over C equals. 63 00:08:20,100 --> 00:08:29,520 ‫And then on the other side I will have tangent theta and then since I have to tension setas here and 64 00:08:29,550 --> 00:08:34,860 ‫this one will go on the other side of the equation sine then I can factor out tension theta and I will 65 00:08:34,860 --> 00:08:50,610 ‫have here D over C plus C over D, so I multiply my tension theta by this expression here and now I 66 00:08:50,610 --> 00:08:54,330 ‫can express tension theta in terms of everything else. 67 00:08:55,230 --> 00:08:57,180 ‫Tangent theta equals. 68 00:08:58,280 --> 00:09:03,200 ‫A over, D plus B over, see, that's this one here. 69 00:09:04,140 --> 00:09:13,380 ‫Divided this thing here, D.R.C., plus C over D like this. 70 00:09:14,400 --> 00:09:20,910 ‫We can now take this A over D plus B, oversee this numerator and we can see that. 71 00:09:21,950 --> 00:09:35,690 ‫This numerator here equals A time C plus B times D, divided by D, time C, so you just take the common 72 00:09:35,690 --> 00:09:38,540 ‫denominator and this is what you will get. 73 00:09:39,380 --> 00:09:50,060 ‫And then you can take this other expression, D over C plus C over D, and if you take the common denominator, 74 00:09:50,360 --> 00:09:59,060 ‫then you will have D squared plus C squared divided by D times C. 75 00:10:00,010 --> 00:10:04,090 ‫That means that this expression here can be rewritten like this. 76 00:10:05,130 --> 00:10:15,000 ‫So this numerator in this form will be divided by this denominator in this form here and of course, 77 00:10:15,000 --> 00:10:24,750 ‫you can cancel out the disks here and here and you're left with a time C plus B times D divided by. 78 00:10:25,770 --> 00:10:28,980 ‫D squared plus C squared. 79 00:10:29,850 --> 00:10:33,210 ‫And now what is D squared plus C squared? 80 00:10:34,110 --> 00:10:48,780 ‫Well, D squared plus C squared equals sine PSI sub R squared, plus cosine sysop r and that is also 81 00:10:48,780 --> 00:10:52,830 ‫squared because your deal was signed. 82 00:10:53,190 --> 00:11:01,060 ‫Sysop are in your C was cosine sysop r and this is also a trigonometric identity. 83 00:11:01,410 --> 00:11:11,640 ‫This trigonometric identities will equal one and that means that this denominator here will be one. 84 00:11:12,360 --> 00:11:21,530 ‫And that means that your tangent theta equals eight times C plus B times D. 85 00:11:22,590 --> 00:11:35,190 ‫In other words, your theta will be the arc tangent or the inverse tangent of eight times C plus B times 86 00:11:35,310 --> 00:11:35,840 ‫D. 87 00:11:36,750 --> 00:11:43,950 ‫And so if you replace all these letters here, then this is what you will have in the end, your theta 88 00:11:43,950 --> 00:11:47,910 ‫equals your inverse tangent and then all this. 89 00:11:48,840 --> 00:11:50,700 ‫So you have your theta.