Question Solve the following trig equations:V3cscx _ 2 = 0tan 3r(tan € _ 1) sin? _ x = 3c0s2 € 2 sec? € + tan? _- 3 = 0sec € + tan € =...

Solving Trigonometric Equations Find all solutions of the given equation. $$ 3 \tan ^{2} \theta-1=0 $$

So in this question we have to solve Uh four legs late. So this discussion will be learning the concept of investing automatic function and its properties. to simplify four legs On the letters, we can use the formula of 2 10 universities. Right? So that would be equal to dine in West two Cossacks Divided by 1- Costs Credits. Mm hmm. And on the right hand side we have 10 university to core sex about psychics. We can compare literacy and knowledge is so we get to Cossacks one minus called squared X. We can write this sinus squared X. And on the right hand side we have to close X. Which can be written as one upon cynics. So when we simplify this, we get Cossacks equals to sine X. Old panics. Basic was 2. 1. Therefore the value of X would be by before like mhm.

In this question, we have to solve the questions. So our question is racket sign two times sine square eggs, Negative one, then squat eggs. Negative. Three equal to shoe. Now you can see that to sign Squire X. Negative one equal to zero. Oh Hmm. Square negative. Three equal. Does you know waken right like that? No. Uh huh. We ate one to the both side. Then we can to sign square eggs equal to one. Now what do I both sides by two. Then we get Sinus choir acts equal to one upon to after taking square root from the both sides. Then we get same eggs equal do flows minus one. A bomb described to here you can see that sign Eggs equal toe plus one of our own describe to And second value is negative one upon described to So we do that thing for this signed while the two in the 1st and 2nd quarante So same gonna sign X equal toe one upon squared too Well is by a bone four So you can write like that Eggs equal to by a born fool plus two times aimed by and for the second quarter and we know that fi Negative X So we know that execute toe by phone four. So we get three by 44 for the second current. So here but here, three time thigh upon four plus two times in five. Now we find the value off X for 10 square Xnegative three called us you So you can see that when we ate three to the both sides, Then we get then square eggs. We could do three. Now they take square toe the both sides. Then we get 10 eggs, equal toe squared tree and here sign will be positive and negative. So then, for the two value is in the first quarante and the third Cordant and father to So he had positive 10. You positive. 10 negative will be 2nd and 4th, so you can see that eggs equal toe by phone tree. So it will be in the first current plus And by no, you can find for the negative one, so you can see that here in the second quarter. And so in the second world and by negative by upon trees. So you get to buy bond three and I will put here X equal to to buy a 0.3 loans aimed pie. Now here we get for the positive one. So now he has toe find the negative one. So if we know that I do here sign will be negative in the third Cordant So by close eggs. So actually spy Pawn four. So you can see that by four. Four by phone. Pull. Sorry. 55.4. So now we put here eggs equal. Do five fine upon four plus to and by and you can see that here is positive on. So eggs 54 plus two time aimed by it is a final answer.

To solve this equation with data we're going to start by setting each factor equal to zero. So we've tangent Square data minus four equals zero, and we have to co sign fate of plus one equals zero, and I'm just spreading them apart. So I have plenty of room to work. Okay, so now we have two equations to solve. And for the 1st 1 we're going to add four to both sides, so tangent square data equals for and then square root. So tangent data equals plus or minus two and then take the inverse tangent. So Theda is inverse tangent of plus or minus two. Okay, so now we grab a calculator because this is not one of the exact values were familiar with. And we find the inverse tangent of positive, too. And that gives us approximately 1.11 radiance. Okay, so that gives us a quadrant. One answer tangent is also positive in quadrant three. And that answer would be pai Radiance away from the answer we just got. And then we're also going to be looking at negative two, and that's going to give us a quadrant to answer, and 1/4 of four. Answer. Okay, so are for a quadrant. One answer. We have our 1.11 radiance, and then we'll add pi Times K to represent all of the Quadrant one and all of the quadrant. Three answers. And then we find the inverse tangent of negative, too. And that is negative. 1.11 radiance. That's not a surprise. And so, for that answer and all the other quadrant, four answers and all the other quadrant two answers we just need to add pi Times K. And that means we're going to go half a circle around to get to another answer. OK, now it's time to go back to the second equation, so we have to co sign data Plus one equals zero. Let's at Let's subtract one from both sides and then divide by two. Okay, this one we can do exactly so let's see. Co sign is negative. In Quadrant two and quadrant three, we would have a one on the adjacent and the two on the high pot news, so that means we have a 30 60 90 triangle and we would have a square three on the opposite, and our reference angle would be 60 degrees, so we can see that one angle here. One answer would be 1 20 degrees, and the other answer would be 240 degrees. And if we write those in radiance, we have for 1 20 degrees. We have two pi over three radiance and 4 to 40 degrees. We have four pi over three radiance. Now we also need all the other answer center co terminal with these. So we're gonna right two pi or three plus two pi ke that would represent every answer that is co terminal with two pi over three and we're going to write four pi over three plus two pi ke that represents every answer That's co terminal with four pi over three.

This question asks us to solve the given equation by factory. What we know is we can take three tan squirt of data as a common factor as well as tan theta minus one as a common factor. We end up with 10 if it has one on 3 10 squared of theta is one. Therefore we have pie is however for is data. And given the fact that 10 if they does one, this means that tan of power before is one. Therefore, we have our first equation as 1/4 plus que times pi. Okay, now moving on to the next part. Remember, we had 3 10 squared If there is one, which means we hade 10 of data is plus or minus squirt of 1/3 squirt of one is just one. So it's one divide by squirt of three. This gives us two options pi over six and five pi over sex meaning we can write data is one sex plus que remember pulling the pilots We can read it in factored form and they does 56 was K times pi