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9 Solve each trigonometric equation exactly on 0 < 0 < Zt. a) secx 2sinx b) sin( Zx) + sinx = 0...

Question

9 Solve each trigonometric equation exactly on 0 < 0 < Zt. a) secx 2sinx b) sin( Zx) + sinx = 0

9 Solve each trigonometric equation exactly on 0 < 0 < Zt. a) secx 2sinx b) sin( Zx) + sinx = 0



Answers

Solve each equation for solutions over the interval $\left[0^{\circ}, 360^{\circ}\right) .$ Give solutions to the near. est tenth as appropriate. $$9 \sin ^{2} \theta-6 \sin \theta=1$$

Okay so we have yeah Toast of two x -3 x equals two. And we can use a trig identity that coasts of two X. Is going to be one minus two sine squared of X. So we get uh one minus two signs squared vacs mhm minus three sine of X. And since we're gonna be factoring here we may as well move the two. Uh actually we don't need to move the two ankles too. So you get -1, -2 squared two. Sine squared X minus three. Synnex equals two. And so we get so we get minus to sign squared X minus three sign. Yeah X Equals one. Because we move we subtract everything here by one And the two x 1. So then we get Yeah. Yeah. Uh huh. So then we if we move everything to the right side we can get two sine squared X minus three. Sign X plus one. Mhm. Mhm Yeah Equals zero. And then we can factor this out into to sign of X. Mhm plus one times. Uh Sine of X plus one. And if we fact and if we multiply all this together you can see that it's gonna create this sine of X plus one. So then we get Syn Ack so we get Synnex is equal to mhm -1 half. Because for here remove them though, We move the positive 1-0 and then divided by two so minus one half and equals Um it also equals just -1 for this sine of X. So we get X equals look five Hi over six and 11 Pi over six. Yeah mm and three pi over two. Because as we can see on this trigger as we can see on this trig circle right here. The only time we're going to get minus one half is when for the for the sign is what? Sorry I've actually made a mistake here. This is going to be seven pie or six because you can see that the only times Sine is going to be minus one half on this trick circle is seven pi over six year, one half and 11 pi over six year again minus one half. And then also the only time the one and only time is going to be -1 is when it's going to be three pi over two. And then obviously we can you can add multiple times and multiple times to piety each of these values and you'll get the same thing. Okay So this would all be plus two pie in. Yeah, so for a that's a that's A. And for B. We have yeah. Um to close I to co sign squared of x -3 Synnex -3 We can Eagle zero and then we can take out the coast squared by substituting in the substituting and sign and you know from the pathetic thank you and see him that coast squared is going to be one minus sine squared. So that gives us so I factor that all in and we get two minus two. Sorry I need to put the -3 in there. Sine squared of x minus three. Synnex -3 equals zero. So it's going to give us so we can just take out the two here. So it's going to be three minus three plus two is just going to minus one. And since everything here is going to be negative we can just moving over the other side. So we get two signs we can just times it all by -1. Get to sign squared X plus three sign of X plus one equals zero. And then again this is going to get and then we see again and this is actually the exact same formula as in a So we can just so we can then see that she's going to factor it again into two. Sine X plus one times Sine X plus one. So it's just gonna be X Is equal to seven pi over six. Yeah 11 is gonna be the exact same answer. 11 pi over six comma three pi over to. So now for see we have three co 2nd x. Mhm minus sine X equals two. So we have it's gonna be since co second is the universe of sign. We can just get three over Sine of X -1. Yeah yeah equals two. So we get so then if we want to put it all under over the first, the left hand side. All under sine of X. We can just times this part by sine of X is going to be three minus sine squared of X over Sine of X. Yeah, equals two. Yeah. Right. And then we get three minus sign squared of X. Mhm equals Yeah to sign Mhm Uh huh. X Mhm No. Mhm. And so then we can bring this all to once so then we can bring this all to one side. So we can have sign squared of X plus two. Sign have X -3 is going to be zero. And so we can then factor that out to be sine of X plus three. And then sign of x minus one And then all equal zero. All right. So now we can see that that's going to give us a sign of X is equal to Uh -3 and one. So then X is going to be some solution for this. And we get this by finding out in each of these factors when this is going to be equal to zero. So sine of X plus three equals zero. It's going to be sine of X equals minus three. Since they're since sine of X goes from -1-1, this is this has no solution. So we only look at when it's equal to one and that's going to be exclusively pie over to and then obviously plus two pi N for any indicator. And so yeah, that's the solution. That's that's that's the one and only solution here and now for d we have tangent squared of X Plus two equals 0. And we need to bring all of this and remember we need to bring all these into signs so we get Tangent squared is equal. We can start by saying tangent squared is equal to one minus coast squared minus coast to X. So it's going to be 1- Coast sign of Yeah two x over one plus co sign of two X. Yeah plus two equals zero. And then we change the coat wind. We used a co sign of two X. Before the identity of coastline of two X. Before so it's going to give us um 1 -2 sine squared x. So one minus one. Highness too. Sine squared of X. Yeah. Yeah. And then um one plus one line is two sine squared of X Plus two equals 0. Okay. Mhm And then two sine squared of X over 2 -2. Sine Squared of X. Yeah. Mhm Plus two equals 0. And then we need to bring all this together so we take everything out of the bottom. So we multiply everything by 2 -2 sine squared effects. That gives us to sign squared of X plus four because there's two times four times two minus two Sine squared of X. So then minus for sine squared of X. Yeah equals zero. So we can then say we can then look at, We can then take four. It's going to be consumption, we can to sine squared of x minus four. Sine squared is just going to minus two sine squared effects. We can then move that to the left side so it's going to be four equals two sine squared of x. Two. Yeah. Mhm. Sign squared of X. So we get to equal sine squared of X. Yeah. And then we get Sine of x equals root of two. Yeah But the root of two is bigger than one and we know that the sine of X only goes that this functionally goes from -1-1. So there is no solution here. Yeah. And that's it for three A to D.

According to question in cost heretics plus three signings minus off nine equals zero, where X belongs to zero to fight, right? No, What we're going to do is they're going to replace cost critics with one minor science critics. Now you might ask why you were doing it. That is because so that this equation becomes a quadratic in Sinus. And then we can easily find the value of signings. So it's a become in Koesterich can. Britain is one minus. Science projects Science predicts. Let's three signings minus off. Nine equals zero go. This is akin, minus off pain Science predicts. Plus three signings minus off nine equals It'll it will be minus off pain science projects. Let's resigning. Plus one equals +03 Freeman to play both sides with minus one, then the signs off all the terms really change 13 cents per day minus of three Cy Knicks minors off one equals zero. Open for the simplification. This will give a This will become who sign X minus one multiplied by quite signings. 1st 1 equals zero. So from here we're going to get to different values of X one is going to sign X minus one equals zero and another one in five sine X plus one equals +072 sine X minus one equals zero. Then sign off. Fix V equals 1.2 Howard when five sine x plus one equals zero, then sign off. X equals minus one on one fight. Therefore, for Sinus equals 1.2 scientists is closed. We win. It is in post Cordant and second Corden so valuable X V equals five on six by upon six and fight by upon six. And when sine X equals minus one upon five, that means, he decided, present in court corden or four quartering. So the value corresponding values of X will be called three points be four 30 when it is present in third quarter and six point video 818 20 This present in four quadrants. So in total were getting four different values off X, which satisfies the initially question. So this will be declared cancer

39. So this one c'est nine signs were Peter Marina. Six. ST Peter is equal to one. So let's leader this equation as nine stains were treated minus six, ST Tita minus one is a good zero. Well, it's subsitute ST Tita as let's invite. So this equation becomes nine y squared minus 65 minus one is close to zero. Not this is a quadratic equation, so we can solve it using the quadratic formula. So we have by as minus off minus six. Let's minus square root off minus six whole square minus 49 minus one were you? Pains name. So this becomes six plus minus square off 36 plus 36 who were 18. So it has become six plus maintenance. Over here we have six Grew to over 18. So this can't afford a decent defied as Mr Great the new military denominator by six. So we have one bust minus one. From here we get two equations. One s y is equal to one plus two or three. And why isn't in what scientists would be a scientist as one plus plus who treat so from here? If we take if If I know value. We get one of the value s sine inverse one plus group three, which comes as 53.6 degrees. We have to use the calculators. This is 50 treatment 60. These one value on the other value and a range of 03 60 will be 1 80 minus despite so when a d minus 53.6, which which is 1 26 painful. So these other two solutions from the left side Let's talk about the right one. The right we have. Why, as one witness who do a tree, why is nothing but scientific tells we have scientific ties. People do one minus two or tree. So we have treated a sign and worse, one minus three. No, this is my new actually will come out is negative but we're not interested in negative values. On another value from here will be 1 80 minus this values when 80 minus saying in verse one minus. So this will actually give a positive value, which is 1 87.9 degrees Onda. If you talk about this now, here is sending a devalue. But another value, which we can get from here will be 3 60 degree Prestes trees exchange, agree to Davalos. A week or two trees extremely replace slain and worse. One minus route to over treat. And this consoles comes out of 3 52.1 degrees. So from left, we have two solutions and from right as well we have these deceptions in total for solutions.

They said. We call that sine squared plus coastline squared Physical 21 This is our protectorate and intensity, so we want to hear it sine squared in terms of coaching, so we can only have so we have Onley. One trick function. So let's not radioing that we have to one minus co sites. Quarterbacks for three and co sign of that you could sell. So let's want to point out that still get to minus two coastlines Where Rex stringing clothes. Next to know. Now let's multiply this by negative one. So we get that are hash power has a positive, um, coefficient. So we get to close science quarterback when his three co center back. No, it is 200 And now let's use our degree or are quadratic formula so they get that close Interfax is equal to They gotta be used on us three plus minus square roots. Um um B squared. So it's nine and then plus 16 over four, but it gives us great post minus five over, for that's equal to We get, um, to a negative one over to the when is co sign of X equal to to. But That's not possible because goes on of exes between negative one one. So I only have one cold sign of X is equal to negative 1/2 and coastline is negative in quadrant two and quadrant three. So this gives us, um, X is equal to ah quite into this. Would be, um it's dry out I triangle. This isn't a good one. To sit is the square root of three plus minus. Still, in quadrant two, this is a variation of part three. So this is hi, I minus pirc agree today she pie number three and then pi plus part of three. That's four pi over three. So I stations for X R two pi over three and four pi over three.


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