Find the exact value for each trigonometric function: Use reference angles wnen Dlnecdons: necessarx 8. COS 225 tn 330" 7 . sin 60"11. sec 135"12. ta...

Find the exact value for each trigonometric expression. $$\cot \left(\frac{\pi}{12}\right)$$

So in this problem, we are asked to find tangent of negative five pi over six, minus co Tangent of seven pi over two. So let's begin by finding 10 of negative five pi overseas. Yeah, using the unit circle, I can start at my or where the angle is zero and then go clockwise because it is negative. Now we'll go clockwise. Five pi over six radiance. Uh, these red lines and my accesses are, in essence, going high over six. So this makes negative one pi over six. Negative. Two pi over six. Oh, negative. Three pi over six. Oh, Uh huh. Negative four pi over six. And right here I will get negative. Five pi over six. We see. Yeah. Mhm that for negative five pi because it is tan. I'm going to need both the X and the Y flights now. Okay, I know that these will both be negative. What I'm gonna do is I'm gonna go up here to shoot the point. So because this is enough third quadrant, these will both be negative. Okay, I Furthermore, know that the numerical values will be the square root of 3/2 and a half Now, if we look at this, we can see the X value is larger than the y value. So I know my X value must be this crowed of 3/2 and the Y value must be a half my tan. We'll take the sign, which is negative. A half the y value over the coastline. Just the X value negative. Screwed of three over to okay. Mm. Yeah, we'll work on simplifying. Not in a bit. But first, let's find Coach Ancient of seven pi over two. So let's begin by finding out where this angle is. So I again started my where I have a zero angle and I'm going going to go counterclockwise because my angle is positive Once makes two pi or let's go by half. No. Yeah, this is one pi over two two pi over 23 pi over two four pi over two five pi over two six pi over two. And here I get seven pi over two. Yeah. Okay. Now I have the associated point. One thing to remember is that co tangent of seven pi over two is really just one over 10 of seven pi over two. Okay, my tangent. Or maybe a better way to do this as this will be the co sign of seven pi over two. Divided by the sine I have seven pi over two. My co sign zero. And my sign is negative one. Mm, yeah. Okay. So let's focus on simplifying the blue. Now my negatives will cancel to get me a positive and one half divided by the square root of 3/2 is really one half times to all over the square root of three. And my tools will cancel. Thus I'll be left with one all over the scroll of three minus zero, which is really just yeah, one of one over the square root of three. Since we do not like to leave radicals and our denominator, I'm going to multiply both the bottom and the top by the square root of three to get my final answer as the square root of three all over three

We'll talk about that. You can use a unit circle to evaluate the negative co tangent of high over four plus 17 pie. Co tangent is reciprocal to tangent. If tangent sine over casa and the co tangent, this would be the negative co sign over sign of this angle pi over four plus 17 pie. So let's find this angle. We've got um even pies because that's two pi zero and before pie, so on and so forth. This would be on pies including 17 pie and then we go another pi over four units from there to find the associated ordered pair. This would be equal to the negative cosine. Is the exporting it. And sign is the Y coordinate. Also negative spirit to over two. So these are the same thing when you do anything by itself, you're equal to one. So this is negative one. Yeah.

We can use the unit circle to evaluate the coast seeking of nine pi over four. That's going to be the same thing as one over the sign of nine pi over four. So let's find nine power over four in fourth. We've got four pi over 48 pi over four. We have to go one more pi over four. Find the associated ordered pair and then sign Is the why coordinate which is the square to to over two. So it's one over the square to to over two, which is a reciprocal To over square root to which rationalized is two square root 2/2. Which is to consuming the square root of two. Mhm.

Way. Want to exactly find the coast seeking of high over to plus the Coty engine Tau pi Over. Two up high over to we are at the 0.1 Co seeking is the reciprocal to sign sign is one. So, uh, the reciprocal toe one is one. Therefore, the cozying up I already was one co tangent. Um co tangent is their reciprocal two tangent tangent to sign over co sign then co tangent is co sign over sign, which is 0/1, which is zero in one plus year. I was one.