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Express tan(0) in terms of sec(0) for 8 in Quadrant II_tan(e)...

Question

Express tan(0) in terms of sec(0) for 8 in Quadrant II_tan(e)

Express tan(0) in terms of sec(0) for 8 in Quadrant II_ tan(e)



Answers

Use the half-angle identities to evaluate the given expression exactly. $$\cot \frac{\pi}{8}$$

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Hello. We have problem number 73. Uh, cost data is given as five x 8 Constituted five x 8. What? It is one we need to find the value of sake theater. So in first quarter and everything is positive. So we know that caused the days one by Saint Peter or we can say sec data is one by cost it from the reciprocal identity. So let us play in the value five x 8. So sick that I will be equal to eight buyer fight. Thank you so much.

Okay, so, um, we're gonna be used 1/2 angle formula. I'm silly. Right down with the formulas first. I don't have these memorized. I doubt your teachers making you memorize them as well. It's positive. Negative square root of one man. Isco, Sativex Arteta. Over, uh, to. Okay, um, and the other piece that they tell you is that you have three pi over eight. Really? At sign of three pi over eight. Well, think about where that is and what kind of answer you'll get. So, three power eight is smaller than pi over two. Pi over two is equal to four pi over eight before that makes sense. So think about if we have three pi over eight were in the first quadrant. Um, so we want the positive answer here. So, um, I'm kind of getting ahead of myself. Ah, Three power over eight is the same as three pi over four, divided by two. And ah, thing about you divide by a whole number is the same as sticking that in the denominator. So what does that mean? Is three piper for is the angle that's put into fatal here. So as I continue on my unit circle. I know 35 or four is over here, and that's the order pair native route to over to route to over to. So in my formula that I have right here, I want the positive answer of one minus. The co sign is negative route to over to all over to just following the formula. Now let's clean this up a bit and you want to get the same denominator. So one as a denominator choose to over to that becomes plus route to over to all divided by two. Well, let's make that better. Let's change that to have the denominator of to and you can just believe me, or you can actually work this out, that all of these twos will cancel out. And so how do you get to the answer that the book has, which is two plus route to all over to is you have to square root both the top and the bottom in the square to four is to This is the answer we're looking for, um, for this answer

Um So you're going to use 1/2 angle formula in this one? Um, so look in your form. The sheet for half angle formula for co sign of something over to and thats plus or minus square root of one plus co sign of data all over it, too. Just double check and I copied it down correctly. Okay, so how do you know it's 1/2 angle? Formidable in the problem, they tell you it's co sign of three pi over eight, which doesn't necessarily look like it's 1/2 angle formula. Um, except if you were to write it as co sign of three pi over four, cause we know that on the unit circle divided by two Well, if you divide by a whole number, that's just get stuck in the denominator. So these are equal eso in our problem. Everywhere there's a fada. You're going to replace it with three pie before so this thing is replaced with three Piper for where is that on the unit circle? Right here. Um, so we have the square root of one plus well at 35 or four, it said co signers, negatives with negative route to over to all over to now. Here's the question. Is this answer positive or negative? Because there's only one answer. Well, three pi over eight is in the first quadrant. Lies in the first quadrant is straight up. Is pi over two? Well, if you change that to be a denominator of eight, it's four. Pi over 83 is smaller than four, so it's in the first quadrant and co sign is positive in the first quarter. So when you do some simplification, I would get older, the dominators to be the same. Just how I like doing this problem. So one is to or two I'll just change it to minus this one already. Has it been on nerve to? And this, too? I carry right. It's four over. To that way I can cancel out all of these twos. I can't really do anything with the numerator. Just leave it as a square root of two minus route to just kind of ugly. But we allow it in the square to four in the denominators, too. That's what we want


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