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If $heta$ and $varphi$ are the roots of the equation $8 x^{2}+22 x+5=$ 0 , then(A) both $sin ^{-1} heta$ and $sin ^{-1} varphi$ are real(B) both $sec ^{-1} heta$ an...

Question

If $heta$ and $varphi$ are the roots of the equation $8 x^{2}+22 x+5=$ 0 , then(A) both $sin ^{-1} heta$ and $sin ^{-1} varphi$ are real(B) both $sec ^{-1} heta$ and $sec ^{-1} varphi$ are real(C) both $an ^{-1} heta$ and $an ^{-1} varphi$ are real(D) none of these

If $ heta$ and $varphi$ are the roots of the equation $8 x^{2}+22 x+5=$ 0 , then (A) both $sin ^{-1} heta$ and $sin ^{-1} varphi$ are real (B) both $sec ^{-1} heta$ and $sec ^{-1} varphi$ are real (C) both $ an ^{-1} heta$ and $ an ^{-1} varphi$ are real (D) none of these



Answers

If $\theta$ and $\varphi$ are the roots of the equation $8 x^{2}+22 x+5=$ 0 , then (A) both $\sin ^{-1} \theta$ and $\sin ^{-1} \varphi$ are real (B) both $\sec ^{-1} \theta$ and $\sec ^{-1} \varphi$ are real (C) both $\tan ^{-1} \theta$ and $\tan ^{-1} \varphi$ are real (D) none of these

What? Hello. So we have this problem here where we're given the Pythagorean identity Co sign squared data. Plus I sine squared data equals one. We're given this and we're supposed to use it to derive one plus 10 square data equal sequence where data and co tangent worth data plus one equals ko seeking square. So how can we use this to derive these to equation? Well, whenever you're dealing with things like 10 seeking co tension co seek and these are all, um uh, express a ble in terms of coastline inside. So if you break them all into the basic components which is co sign and sign, then you can start to see where you can simplify where you can change things, manipulate it. So that's what I tend to do. So let's start with the top one right here. So pageants with data, the tangent is sign over post nine and isn't squared issued B sine squared data over confident with their the equivalent and seeking. It's the reciprocal of co sign. So to express this, you can also read it's one over co signs with data. If you notice both of these have same denominator using that we can change The one have the same dinner because you know, one you can express it with any denominator Very flexible. So we can turn the one to have a co sign squared data denominator and just have co signs with me on top. If you look at the denominators are all the same If you look at the numerator he realizes actually already exactly the same as the formula we have meaning that, uh, all we have to do is give it is that the nominees which is easy? All you have to do is multiply co signs where they did everything and then you get back here. So you want to go the opposite way. You'd have to divide everything my coastlines where data. And then you would get one plus 10 for data plus C squared data. Now for the 2nd 1 second one, you do the same thing. Break it up. Co tension. It's the end of the reciprocal tangent. Since tangent sine of the coastline co attention must be coast over, son. Then one in the same flexible coast seeking co seeking is the reciprocal of sign. So it's one over signs worth it. And both of these have a side squared data denominator so they don't. They're so everything has the same denominator. And again, the numerator is exactly the same. And all we have to do to undo its most pirating by science worth data. If we wanna go the opposite way, we can divide. So it's very useful toe break it down. But eventually you wantto kind of figure it out quicker than writing it all out. One easy trick is ah, if you see it in this general form and you have a hit that it linked to the parent them, or maybe it's outright given to you, you can look at the sequins or coast. It might not be in this sort of, but you'll find a beacon or coat speaking. If you find a seeking her Cosi get, then you know that to express that is one over co sign or one oversight. So if you see one of these, if you want to revert it back to this, chances are you would have to multiply science where they are co signs were there, depending which one you have. Look out for this. This will help you figure out the rest? Um, the CIA. That's pretty important to know, but this is just the first part of the problem. Now there's a second book. Second part says that this equation is true for all really numbers, uh, any complex in any real number for data and this relationship will still work. Now we have to prove whether or not these two work as well. So here's the thing. Uh, let's go back to this form this fraction form. So we're starting with the 1st 1 in the case. So here, let's say both sighing data ends up equaling zero. If a equals zero, that's fine. Um, scientific will probably not be there, and you'll end up getting one. All right, so well, you have to see, Can I put zero? That's usually a helpful indicated, whether it's possible or not. So here I look, the numerator can be zero. That's no problem. But if the denominators becomes zero, co sign squared equals zero, then all of a sudden that doesn't make sense, because if you have zero, that the nominators undefined. So, actually, any data that will make host I'm squared. Um, Dana zero actually does not exist. And by a similar note here for the second equation, we have science squared data on the bottom. And if science course data equals zero, then it's a similar issue. So, actually, uh, these two aren't, um, aren't true these two action. These two aren't true in all cases there on Lee. True, when co sign for this one wouldn't co signed it square thing. It does not equal zero. Whatever data will give you a nonce. Your answer. And for this one, whenever will give sine squared data not eat now equals zero. So, yeah, the main difference between these two is that this could be expresses a fraction this one cannot. So you don't have to worry about the whole denominator business. But this what you do, so I hope that helps

So this question here is asking us from what values of data is this equation. What is it? What? Is it gonna be true? And so really, when you want to think about it, here is that I would need for this toe always be a positive value. But it will be always a positive value for all values of data. Because if you think about it, what will co sign? So this is, uh, sign date up is equal to And then if you have a one minus And remember, I'm just multiplying. Co sign date at times co signed data. And what are the values that coast on data will be? It always is between. Remember, it starts high and goes low and continues to oscillate between one and negative one. So no matter how high I go, so if it's one when it's one a one times one is 11 minus one is zero. The square root of zero is zero. That's a valid number, and the sign of one is zero. So that because sign starts down here, I'll do in a different color and goes up first, right? So that will be true then what's the smallest number that I could have? Well, I could get closer. I could always Of course, I could have zero. So if I have zero, But if I have a smallest negative number, it's gonna be exactly the same thing. Because when I multiply that number by itself, then it turns positive. And so this is not does not become a negative number, and so this will not become negative. So basically, this will be true for all values of data.

In this problem, we have the multiple choice answers of three pi over 47 power for two pi and five pi over too. And our question is, which of the following Its not a possible solution for R given equation. So the first thing, especially because it's a multiple choice question. I want to make sure that my equation will d be defined at all of these choices. So the choice I want you to look at is Choice D, where it says that our answer choice is five pi over too well, remember, that's a code terminal angle of pi over too. However, I want you to notice that one of our functions is tangent squared. Well, the tangent of pi over too is equal to undefined because you would have zero in the denominator up your faction. Therefore, because the tangent of high over to is undefined tangent square the pirate, you would also be undefined. Therefore, this also tells us that the tangent of five pirate too is undefined. Therefore, choice d cannot possibly be one of our answer choices. So d

This question asked us to consider a series of statements and finding which one is false. And I'm just gonna write those statements down. The first Damon is that co sign squared Data is equal to one minus signs for data. The second statement is that CO tangent square data is equal to co Siegen Square data minus one. The third statement is that sine squared data is equal to co sign squared data minus one. And the final statement is tangent squared. Data is seeking square theatre minus one. And essentially all of these are just rearrangements of the trigger metric identities that are known as the Pythagorean identity. So the poor thing really started with this one. Generally, science, great data plus co sign squared equals one. And then we also write one plus tangent. Squared data is seeking square data, and finally, one plus ca change in squared data is equal to co seeking squared data. So all we have to do is look at these and see if they check out his rearranges of this. So the 1st 1 is just bringing the sine squared over the other side of this one would say yes. That is a correct statement. The 2nd 1 is a rearrangement of the third identity, So it's a rearrangement of this guy. But it's just bringing that one over and it checks odes, we say, Yeah, that's right. The 3rd 1 we see science crazy with co sign squared Fate of Minus one and I say, Well, hold on a second. That one isn't right And the reason is not right is because Science Square Theatre, if I rearranged this identity, is actually one minus co sign squared data. So it's got the signs backward. So we say that one's not correct. So that's our answer, and the sense of that's the statement that is not right. And just to check this last one is a rearrangement of this guy here and it just check out because it's just bringing the one to the other side. So that's the statement that is incorrect


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