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[0/9.44 Points]DETAILSPREVIOUS ANSWERSLARTRIG1O 2.3.060.Solve the equation. (Enter your answers as comma-separated list: Use n as an integer constant: Enter your re...

Question

[0/9.44 Points]DETAILSPREVIOUS ANSWERSLARTRIG1O 2.3.060.Solve the equation. (Enter your answers as comma-separated list: Use n as an integer constant: Enter your response in radians:) tan2(x) 5 tan(x) 6 = 0

[0/9.44 Points] DETAILS PREVIOUS ANSWERS LARTRIG1O 2.3.060. Solve the equation. (Enter your answers as comma-separated list: Use n as an integer constant: Enter your response in radians:) tan2(x) 5 tan(x) 6 = 0



Answers

Solve each equation ( $x$ in radians and $\theta$ in degrees) for all exact solutions where appropriate. Round approximate answers in radians to four decimal places and approximate answers in degrees to the nearest tenth. Write answers using the least possible non-negative angle measures. $$5+5 \tan ^{2} \theta=6 \sec \theta$$

Okay, so we have by plus five tangents. Squared of data is equal to six times Sequent data. Now, one thing to note is that we can actually make this five times one plus tangent squared of data. And if you remember the driven a metric identity, you can notice that this is just five seeking squared of data. So five seeking squared of data is equal to six seeking of data. And if we fact arise, this we get either seeking that data is equal to zero or Secret of data is equal to 6/5 now can secret if they ever equal to zero. No, because seeking with data is if you remember the definition, it's one over co sign of data, and there is no valley of data that can make this equal to zero. So we eliminate this, and focus on the Sikh imitator is equal to use six ever five, which is equivalent to saying again because seeking is one of the co sign that coast on a data. Is he with the 5/6? So since that's not one of the easily solvable valleys, we plug this into a calculator and gets that Breda is approximately either 33.6 degrees or 326.4 degrees. So we have our solution. Set up data is equal to 33.6 degrees plus and times 360 degrees three under 60. Because we're using co sign as our graph and coast under the period of three under 60 degrees and 326 0.4 degrees close and times 360 degrees where n is with then the set of integers and we're done.

Okay, So, uh, we know sine squared plus co sine squared is equal to one eso in the problem because we have sign of X by itself. We can't mess with that with the problem, But when we have co sine squared, that's right here. We can replace that with its equivalent, which would be solving for coastlines where it would be one minus sine squared of data. Um, whoops. It's X. I thought was state of first again, Uh, there's just variables. Please don't be discouraged by that. But there's still a six in front of that. It said six coast and squared is equal to five. So what we need to do is distribute this six in here, and I like to have a positive a term. So I'm gonna add it to the right side. So six sine squared of X minus sign of Exeter right side. And then this is six. When I subtracted over five. Minus six is negative one so we can factor this. Um, how do we get? Six is either 6 to 1 or three and two, and the only way to get negative one is one times one and one of these has to be negative. Well, if you do, your outside term is negative. Three and you're inside terms plus two. You do get negative one here in the middle. There's a lot of other ways that you can get the correct answer. You can use the quadratic formula you can. It depends on your teacher. Really? Eso you're supposed to say each of these equal to zero each and used zero product property. So that's a sign of X. Should equal negative one third or sign of X equals one half, you know, setting them equal to zero. You add one over and divide by two. Now, this is a perfect answer on the unit circle. Um, that's right here at Pi over six. You get the y coordinate to be one half. So this answer is perfect that you can say, 00 wait, where in degrees, aren't we? I'd prefer radiance. Just have it. It's at 30 degrees. Um, but it's the same sign is positive in the second quadrant has had the same reference angle 30 degrees, so 118 minus 30 gives you 100 50 degrees. All right, However, this answer is not on the unit circle. So you have to go to your calculator and hit second. Sign it right that out X equals second sign of negative one third. And what that tells you is you get an answer of negative 19.47 But what that means is on the unit circle that's going clockwise this far, which makes sense of Sinus negative in the fourth quarter. Well, how do you get the correct answer is adding 3 60 To that answer, you have to do a full revolution, and you get this answer of 340.53 degrees. Now. That's not the only correct answer, because Sinus also negative in the third quadrant, and it has the same reference angle. So how do you get that answer is do a Semicircle 180 and add to it this 19 points 47 And as I do that, I get my last answer of 199.47 degrees. So here your four answers boxed in now

All right, # 58. We've got tangent of x times tangent of X -2 equals five. So first thing we need to do is to distribute this tangent of X. So we get tangent squared of x minus two tangent of X equals five. Now we move our five over. So we have tangent squared of x minus two tangent of x minus five equals zero. Now this is not easily factored. So we're going to use the quadratic formula so it's negative B plus and minus the square root of B squared minus four A. C. All over two. A. So that's going to give us two plus and minus the square root of negative two squared -4 times one times negative five, All divided by two times 1. So now cleaning this up we get two plus and minus the square root of So that's gonna be four plus 20. So 24 over to Now this is not a special angle so we can go to our calculator. So we're gonna come over here we're gonna say second tangent And we've got two plus The square root of 24 divided by two. And we're in radiant mode. So our first solution is 1.2886. So and on this we're gonna have to break it apart into two plus route 24/2, which is what we just did. And then 2- Route 24 over to. So we just got 1.2 886 woman .2886. And then we're going to do plus pie To get our second solutions. That's 443 02 44302. And now we need to go through and do the same thing with two minus. So we're gonna say second tangent of two minus route 24 divided by two. And we get a negative solution so I'm gonna store that in X. And so we need to take an ad pie for one of our solutions. So 2.1747, 2.1747. And then we need to go back to what our value was originally and add two pi And that's five 3163 5.3163. So now we can go through and write out our solutions set. We have 12886 Plus two Pi n. So we've done that one and now this one's the next largest or 2.1747 Plus two Pi n. And then 44302 Plus two pi n. and finally 53163 plus two pi N. And all of these are where and is any into juror?

All right. So we have Tangela next times changing a X minus two is equal to five. This is not easily Fact arise. Also, we're going to expand this dangerous squared up X minus two times tangible X is he with a five? We're gonna add a one to both science, which makes the left hand side tangent of X minus one squared and the right hand side is going to become six because all we're doing is out of one to both sides. So we take the square root and we get tangent of X minus. One is equal to plus or minus square bit of six, which Jenkins says Tangent of X is equal to one plus or minus square root of six. Now, keeping in mind that attention has a period of 180 degrees, or in this case, hi Radiance says we're supposed to be doing this radiance we gets X is equal to well, approximately. I should say X is approximately 1.2886 and also approximately 2.17 47 So we get the answer. Set X is equal to 1.28 86 plus two n pi comma 2.1747 plus two N Pi where, of course, and is an integer and that's it, we're done.


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