‫Welcome back.
‫Let's now compute the Phi Beta and then the two variables, let's derive formulas for them.
‫The hints that I gave you were the following.
‫You should first put the equations in this form and in this form.
‫So if you're Z double that equals all that, then you're a Z, double that plus G equals cosine five
‫times cosine theta times you one over the mass of the drone.
‫So X double dot over a Z double that plus G equals this equation here, which is the X double date equation
‫here divided by this equation here.
‫So you can cancel out this and this, you can then separate this equation like this and then you can
‫cancel out this cosine phi with this cosine phi here.
‫Now from trigonometric identities, you know that sine X divided by cosine X equals tangent X, that
‫means that this part here sine theta over cosine Seeta.
‫That will be your tangent theta like this and then also sine phi over cosine phi.
‫This will be your tangent phi like this.
‫So I can rewrite this entire equation like this.
‫Similarly, why doubled that over a double Dods plus GS would be this equation here divided by this
‫equation here and you can see it here and now you can also cancel out you one over M and you one over
‫M.
‫You can then separate this equation in the same way and then you can cancel out counseling fi here and
‫counseling FI here again here, you will have signed Seeta over Cosine Theta and that will be your Tenjin
‫Seeta.
‫And then here you will have signed Phi Overconsume Phi and that will be your Tenjin Phi.
‫And so in the end this equation looks like this.
‫That's your end result.
‫And so in the end, you have these two equations here, so you have X, double dot over Z, double that,
‫plus it equals all this and then Y, double dot over Z, double that, plus G, it equals all this.
‫And now let's make a couple of abbreviations to write less.
‫Let's say that this thing here is a.
‫Then let's say that this thing here is B, so X double date over the plus G equals A and Y, double
‫date over Z, double that plus G equals B, then let's say that cosine Sysop R equals C, and therefore
‫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
‫one here is also D.
‫In that case, the first equation can be rewritten like this, a equals tangent theta times C like you
‫can see here plus.
‫Tangent five times D divided by cosine Seeta.
‫That's this first equation here, and then the second equation would be B equals tangent three times
‫D minus tangent five times C divided by cosine theta like this.
‫That would be your second equation here.
‫And so if I take these equations here, then I can rearrange them like this, I can take this term here
‫and then I can put it on the other side of the equation sine like this.
‫And then here I can take this term and then I can put it on the other side of the equation sine and
‫then I can take this B and then I can put it here on the other side of this equation.
‫Sine if I do that then here I will have tangent five times D over cosine Seeta which is this term here
‫equals a minus tangent Theta times C and then here I will have tangent five times C or Cosine Seeta,
‫which is this one here, and that equals tangent theta times D minus B.
‫I now want to express this tension phi in terms of everything else.
‫So Tangent Phi equals this term here and then I'm going to multiply it by cosine theta.
‫So this cosine theta will come here now and then I'm going to take this D and then I'm going to divide
‫this entire thing by D and I can do the same thing here.
‫I want to express this tension phi in terms of everything else.
‫So I'm going to do it like this tangent phi equals tangent theta times D minus B, that's this term
‫here.
‫And I'm going to multiply by this cosine Seeta and then I'm going to divide all that by C like this.
‫And now, look, you've got tension fire here and you've got tension fire here they are the same thing
‫like tangent five equals tangent fire like this.
‫Therefore, these two expressions here, they are equal to each other.
‫They are written in a different form, but they are the same thing.
‫And since they are equal to each other, then I can get rid of this cosine sheeter and this cosine Seeta.
‫And now I'm going to separate this equation like this, a over deed, so this divided by this minus
‫and then you will have Danjean Theta time C over D, and that equals tangent times D over C, so it's
‫here tension theta times the over C minus B over C is here, minus B oversee.
‫I can rearrange the terms and then I will have A over D plus B over C equals.
‫And then on the other side I will have tangent theta and then since I have to tension setas here and
‫this one will go on the other side of the equation sine then I can factor out tension theta and I will
‫have here D over C plus C over D, so I multiply my tension theta by this expression here and now I
‫can express tension theta in terms of everything else.
‫Tangent theta equals.
‫A over, D plus B over, see, that's this one here.
‫Divided this thing here, D.R.C., plus C over D like this.
‫We can now take this A over D plus B, oversee this numerator and we can see that.
‫This numerator here equals A time C plus B times D, divided by D, time C, so you just take the common
‫denominator and this is what you will get.
‫And then you can take this other expression, D over C plus C over D, and if you take the common denominator,
‫then you will have D squared plus C squared divided by D times C.
‫That means that this expression here can be rewritten like this.
‫So this numerator in this form will be divided by this denominator in this form here and of course,
‫you can cancel out the disks here and here and you're left with a time C plus B times D divided by.
‫D squared plus C squared.
‫And now what is D squared plus C squared?
‫Well, D squared plus C squared equals sine PSI sub R squared, plus cosine sysop r and that is also
‫squared because your deal was signed.
‫Sysop are in your C was cosine sysop r and this is also a trigonometric identity.
‫This trigonometric identities will equal one and that means that this denominator here will be one.
‫And that means that your tangent theta equals eight times C plus B times D.
‫In other words, your theta will be the arc tangent or the inverse tangent of eight times C plus B times
‫D.
‫And so if you replace all these letters here, then this is what you will have in the end, your theta
‫equals your inverse tangent and then all this.
‫So you have your theta.