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Use Ihe unit circle shown here solve the trigonamalic oquallon Solve aver [0,27)2087=(#4solulian su1 i5 oxalchHint ut uslng nua neudod_ Uauinlunora (Type wnHracllcn...

Question

Use Ihe unit circle shown here solve the trigonamalic oquallon Solve aver [0,27)2087=(#4solulian su1 i5 oxalchHint ut uslng nua neudod_ Uauinlunora (Type wnHracllcne lor any #umbnn4tha axpres53ncommiuaararo answversnrdudJo "(}

Use Ihe unit circle shown here solve the trigonamalic oquallon Solve aver [0,27) 2087= (#4 solulian su1 i5 oxalchHint ut uslng nua neudod_ Uauinlunora (Type wn Hracllcne lor any #umbnn4 tha axpres53n commi uaararo answvers nrdud Jo "(}



Answers

Derive Eqs. (33) and (35) from Eqs. (32) and $(34),$ respectively \[ \begin{aligned} R_{\text {closed }} &=\frac{4 \pi G \rho_{0}}{3 k c^{2}}[1-\cos (x)] \\ &=\frac{1}{2} \frac{\Omega_{0}}{\Omega_{0}-1}[1-\cos (x)] \\ t_{\text {closed }} &=\frac{4 \pi G \rho_{0}}{3 k^{3 / 2} c^{3}}[x-\sin (x)] \\ &=\frac{1}{2 H_{0}} \frac{\Omega_{0}}{\left(\Omega_{0}-1\right)^{3 / 2}}[x-\sin (x)] \end{aligned} \]

Where has to solve the initial condition. Problem D already say they equals three co sign squared of pirate for minus data and were given the initial condition that are zero. It was library. So first we need to find our data which will be the integral of three co sign squared of pyre reform. I miss data. Defeat it Before we go further integrating, we can rewrite that co sign squared term as one half plus one half times co sign of pirate to minus two theta. So substituting this back into the integral we get that our data equals the integral of three over two plus times one plus co sign of pie over to minus two three entities. David, we can evaluate the Senate roll using the substitution U equals pirate to minus tooth data so that d u is negative to the data and so made of one have to u equals de Sade. Plugging this substitution into are integral. We're looking at our fate. A equals three has times a negative one half times the integral of one plus coastline of you. Do you integrating this? We get negative three fourths times you plus sign of you Plus See, now if we undo the substitution here and by that I mean we plug in Piper to minus tooth data everywhere we see you and when we simplify, we see that our data equals negative. Three eighths pie plus three have stada minus three fourths sign of Piper to minus two data plus C. Now we need to solve for this constant C But we were given the initial condition that are zero equals pi for eight. So the plug in zero everywhere we see data, we should get pie over eight equals negative. Three aides pie plus three half time. Zero minus three fourth time Sign of pie over two plus seat since I knw of pie over two equals one introduces down to pie I ever ate equals negative eight pi minus night of three over eight times Pi Lina's three fourth plus C solving for see you see that C equals Piper to clustering forts. So finally we could put this back into our original equation that we found for our of data and after simplifying, we see that our data equals pi. Over eight plus three have stada minus three year before time. Sign of pie over to minus two data plus three

For problem 56. Uh, you're given an initial value problem and which we need to solve it and get the initial equation for VR and data. So we're gonna set this up cause we're gonna have to integrate, right? So d r equals three co sign squared times pi over for minus data data. Right. And now we can integrate this and that will give us our and an equation for our right. So in order to integrate this when I have to use U substitution, so you equals pi over four minus data. Do U equals named of death data? Right? So now we can go ahead and write this out So we get the integral of Diarra just r equals. We'll pull out that three of the negative so negative three times the integral of a sine squared you, do you? And now we can use the power reducing formula to actually interpret this because we can't do just that. Cosette squared so r equals negative three house right. The power using formula has a one half out. Fronts will pull that out times the integral of one plus co sign So one plus co sign to you, do you? And there we go. So this is a formula that I recommend you work on memorizing one half times one plus co sign to you that is equal to co sign squared of you really useful for situations like this when you need to integrate. So now we can actually do that. So are equals negative. Three halfs times the integral Actually, not the integral for your integrated down this stuff. Times you plus the integral of co sign is going to just be signed. So sign of to you, right? And then we need to divide that by two. Since there's a two right here, then plus C. So there we go. There's are integral. Now we need to go ahead and distribute this a negative three halfs. You plus ar minus. There's native there minus 3/4 times a sign of to you policy. So now we can go ahead and back, substitute in and get our our right or our Sorry. Arteta. Right. So, uh, negative. Three halfs times I over for minus data minus 3/4 times the sign of two times I over for by s data, plus e. There we go. That's a quite the equation that we've got going on. But now we can finally solve for C. Wright. We can use its initial value to solve for that. So if I can get this to school down here, there we go. So we can step. I over ate pi over eight equal to this when, uh, data zero. So let's go ahead and do that. So negative three halfs times power before, right, cause it is just zero minus 3/4 times sign of two times pile for right And there we go and then plus C. So now let's proceed. Soap Ivory equals negative three half. So this is going to become negative. Three pi over eight minus 3/4 times. So the sign of PIRA for which is the same as pi over two. So the sign of pi over two is just one. So it's just times one, which is just negative. Three force, all plus C. So there we go. Um, now we can go ahead and add this to the other side. So what does that give us? Well, that gives us four I over eight plus 3/4 equal. See which we can simplify. Two pi over two plus 3/4 equals C. So is this correct? Well, we hope so, huh? No, it is. But, um Well, you can now go ahead and piece together this entire equation. So we have our data, all right? And that is going to equal. See if I can get this on the same page. I don't think I quite can't. Okay, so our Athena is going to equal negative three. House, you have three house times, our four minus data minus data minus 3/4 times. The sign minus the fourth, was a sighing of two times prior or four minus. There, two times. Pi over four, minus data. Here we go. And then that is then plus C, which we just found out is pie over two plus 3/4. So we can clean this up a little bit, but this is very close to our final answer. So our data equals eight of three pie over eight. Plus three data over to minus 3/4 time signed of What is this? This is just pie or two minus tooth data. Plus by over two plus 3/4. So, uh, we almost have this we can just write it in its final form, which is three halfs data minus 3/4 Sign, uh, pi over two minus tooth data plus pi or eight plus 3/4. And there we go.

We are given the function nine X rays to four plus terrify Rex where minus four equals zero. What we do here first as let's take X rays to full as you square This will give us nine years for the stratified u minus four equals zero, not refined the roots for the secretion on we conf activates it as nine years were pressed her distance u minus You buy nice for on. Then we get the value off you as one A born nine on a minus full, which gives as the value off eggs as plus minus 1.3 on a plus minus. Do I now reduced the I 84 calculator to find the real root? The first thing we do is set up there as I asked. That is, our X has to be my mystery to three. And why has to be my listing 200. So we sent up the scale when Max for our craft window one start is done, we go back to y equals on end our function that is nine X trees to fall last modified X square. My guess fall on. We type by two equals zero to find the eggs and disip. And now what will you do if I let the section between these two light to find out include We use the district option on the Calgary Indo Force intersection point this 0.0.33 I was on your screen. That's what my tree on. We find that second point that this minus 0.3 tree That's pilots. What about three?

Okay. This question wants us to solve the following differential equation. So we see that it is separable. So let's just multiply by D data to get d R is equal to three co sine squared of fada or sorry coastline squared of. However, for minus later data comment to find are we just integrate both sides So are is equal to the integral of three co sine squared of pi over four minus data deflator. And as is typical of trick functions, let's make a U substitution inside the argument. So u is equal to pi over four might s data. Do you is equal to negative data. Watch out for that sign there. So negative D'You is equal to D data. So now, looking back in, we get the integral of three co sine squared of you times negative d'you or negative three times the integral of co sign square to you, do you? And then the integral of coastline squared is just one that you need to memorize. So it's a 1/2 you and then with co sign it's plus sign of to You divided by four plus C can't now we can substitute back in for are you? So we get negative three times on half times. However, for my test Aita and then plus 1/4 sign of distributing the two. Here we get hi over to minus two Fada distributing that suit each term there plus C or no If we distribute out this negative three everywhere we get negative three pi over a plus three theta minus 3/4 Sign power over to minus two theater plus c So now we can solve for integration Constant. So we know that our of Fada is equal to looking back in Here are of Fada is equal to three theta minus 3/4 Sign of pi over two minus two theta plus C two and again if we have Constance, it doesn't really matter if we absorb them into our integration constant here. So I got rid of the negative three pi over eight because it is a constant. So it will be absorbed in the sea to so are zero it said was equal to pi over a. So we can use that fact to say that pi over a is equal to zero minus 3/4 times the sign of pi over two minus zero plus C two or Harvard eight is equal to, well, the sign of pirates whose one so negative 3/4 plus c two or see to is equal to pi over a plus 3/4. So plugging that in we get are of Fada is equal to three fada minus 3/4 sign of pi over two minus two fado and then plus our integration constant, which he said was, however, eight minus pirate plus 3/4. And this is our final answer because again, if we solved it a different way and, like didn't get rid of that negative three pi over eight at the beginning, we would still get the same constants in the We just have a bit more algebra to do to get here. But as a final test, let's just make sure initial conditions still works. So if we plug it zero on both sides, we get our of zero is equal toe zero minus 3/4 plus 3/4 plus pi over eight. So we get pi over eight. Yeah,


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