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2. (15 points) Find all solutions of the differential equation valid in the interval 2 t < 2 = = (tan t)r sin t_...

Question

2. (15 points) Find all solutions of the differential equation valid in the interval 2 t < 2 = = (tan t)r sin t_

2. (15 points) Find all solutions of the differential equation valid in the interval 2 t < 2 = = (tan t)r sin t_



Answers

Find all solutions on the interval $[0,2 \pi)$. $$ 2 \tan ^{2}(t)=3 \sec (t) $$

We are going to do problem number 55 In this question, what we need to do is we need to just find all possible solutions in between 0-2 pi. This is given to 10 T sick is choir thi this is equals to zero. So converting tenants second sign and cause form. So we're having to sign T by cost -1 by Costly Square T. This is equal to zero. So just dividing, We are just obstructing these two. So we are having courses 14 denominators, we'll be having to sci fi into cost Management equals to zero. So multiplying causes clear to both sides. We will be having only are two sci fi into costea -1 equals to zero. Now we know that sign to theater. Sign to theater this is equals to two science theater into cost theater. So into cost theaters on the basis of that we can just try to test so into data so you're right, this a sign Do K to- when he goes to zero. So here scientists signed to theatre this is he goes to one. Actually in place of theater we have to be so bright T Okay. What is given in the caution? That is the correct procedure. Yeah, that's perfect. Yeah. No, we'll just proceed now. Uh The value of sand function when it is going to be one When only when it is it's value would be by by two. Okay, so He will be constructive way to go to the risk by by four. This is power for No. What do you know what we could do next is And to solve this question, we can just take one more value here. Okay that at which the sine function is going to be The same function is going to be one. Okay, that is just check your talk here. If you're going to take this value, okay Then only at 5 to the value is one. So if you just try to general equation for this at this point at 20 plus, like this at point t The general question will be by by four plus. Buy in. So you can just add one and to get a solution in between the two pi. that will be by by four plus uh by So that will be equals to fight by by four. Okay, So this is under to buy. So we've got two solutions in from here that is by by four and fight. Bye bye. These are the answers in this case. That's all. Thank you.

First record that tan jing to square to T Plus one. He calls sick hand square teeth. Hence we substitute second square tea with tangent, square T plus one. It becomes miners tangent square T plus twice of tangent, T -1 equals zero. There is miners Tangent T -1 Squared Geico zero. Hence tangent E. He calls one so T equals hi or fall of five pi or fall.

Hello and welcome to this video. We're gonna solve this equation. Sine squared T. Is equal to co sign T. We're gonna find all the solutions from 0 to 2 pi now we're gonna do is we're also gonna use a graphing calculator to help us out with this a little bit. Now if we wanted to we could actually just type in sine square T. And co sign T. And we can find the intersections on the graphing calculator. But let's at least talk about some of the math that's going on behind here to kind of figure out what we're doing what we're doing. So notice that we have two different trig functions. Sine squared T and co sign T. Well I'm gonna use a basic trig identity sine squared plus co sine squared is equal to one. To kind of help me out with this. So if sine squared plus cosine squared is one that means that sine squared is equal to one minus co sine squared. So I'm gonna go one minus co sign squared, T. Is equal to co sign of T. Now the next step is to get all this on the same side of the equation. So I'm going to have zero equals co sign squared T. Plus co sign of T minus what. Now the reason why I do it like this is I want to make sure that this guy was positive. Now here's what we have we have a hidden and embedded quadratic function right here and to make life a little bit easier and trying to because like this thing won't factor nicely. Let's just go to a graph and how to help us solve it. So I'm gonna go over here, we go to my functions and I'm gonna type in co sign right here, I'll go squared and it doesn't matter what variable you go with. I'm gonna go with X. I should probably close my parentheses plus functions co signed X Mine or close the parenthesis again -1 and there we go. Now. What I'm curious about is where this is going to equal zero. And so I'm gonna do is I'm gonna go to my graph and I'm going to zoom in right here. What I want to find are those intercepts. And so let's see here, I have one at 0.9 05. I have another one at 5.379. And I want to go between zero and two pi And remember remember that two pi is roughly 6.28. So let's see if we have another 12 of this is The 7.11. So that's clearly not going to work. Therefore, these are my two solutions.

Yeah. In this problem we have given the equation Tangent to T -2 into society is equal to zero and we are asked to find the solutions of the equation that are in the interval zero comma to buy. So we have Attention to T -2. Into Society is equal to zero. By using the pragmatic function we can write engine to T as signed today upon. Co sign to t -2 into society is equals to zero. Now by using the techno metric formula of science duty we have scientology is equal to you. Went to see I intend to co sign thing Vi get to into society into a society upon because I into b minus two interco scientists. It was 20 by taking two interco scientific common. We get to enter society into sci fi upon Because I into T -1 is equal to zero by taking each factories equals to zero. We get because scientists equals to zero. All 70 Upon goes into T -1 is equals to zero. Further we get Society is equal to zero or 17 minus Consigned to be is equals to zero by simplifying it. We get do into science where the plus 70 -1 is equal to zero. It would be equals two To win 2 17 -1 In two. Scientific Plus one is equal to zero. No hell we have. So scientists equals to zero. All Science is equal to one upon to all. scientists equals 2 -1 because scientists equals to zero. It implies that They would be equals two by upon to Coma three by upon to or scientists equals to one a point to. It implies that They would be equals two by upon six comma by minus by upon six go on. Scientists equals 2 -1, implies that They would be equals 2 3 5.2. Therefore we get These equals two pi upon six comma by a point to coma 5 5.6 comma three by a Pontoon. These are the solutions


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