5

-2 52- Find the exact value of cot sin...

Question

-2 52- Find the exact value of cot sin

-2 5 2- Find the exact value of cot sin



Answers

$$\text {Find the exact value.}$$ $$\cot \frac{5 \pi}{12}$$

In this question was asked to find Cho tangent of 510 degrees and 510 degrees. I'm going to chop 360 and use the code terminal angle of 150 degrees, so I basically just remove an extra two high revolution. So now to draw this, that's 90 degrees. 1 50 is gonna be like that. 1 80 minus 1 50 is thirties or reference singles. Right here. It's 30 degrees, and now we can start to draw the triumph. So opposite the 30 is gonna be one positive one opposite this 60 degree angle. We have 30 60 90 triangle. This is gonna be negative. Route three taipan uses to coach engine is a Jason over opposites and negative Route 3/1. It's just gonna be negative. Three

Don't Enemy birth Europol number 12 in which we need to get caught minus five by over 12 Need to get the value card minus fight. But militant So this should Britain is one over, then minus played by what would go? No, we just need this value and then we'll make it emits to get the value of God minds by pirate with So our primary focus should be. Did the value tan minus fight by over No, Stan minus Tita which they call two minus dented. Hello. This would become minus Dan. I buy over 12. No, I never went broke and Putin is three by less. Do you buy? Well, What then? Not precipitating of values. Make it the fire withdrew. Blessed do by over toe. This is by our full. This is by over six. So we're left with minus then by a 14 list by over six we have That's some might indeed feel are 10 function any place be well then a list, then be with one minus. Then they okay? It's playing in here. It becomes Dan like beings. We just need a bracket here. And by by four Must Dan by my stick or with one minus And by a fool. 10 point I asked been bio 41. And by over 61 by three, they went minus one into by three. No, I mean no have. If it does, just edition, it's gonna be well, people. It's one. I was three minus one over guilty. Just traditional affections. Okay, mind stand should be here we have This will begin still out if we get it flipped. Oh, we'll creep bliss one away. You look to minus one if you multiply listening to no natural ticket, no people is one over one. My answer would be. But remember, this is the value off. 10 minus bye Bye. No, this is the valuable Dan minus five. Well, liked it. Well, toe hold three plus 1/1 minus wolf three. But we need going my dance fight. Biola will one over can mine despite but what we're really you have which would be cool to one minus three over one, pleasantly. So they should be there. Answer. Thanks so much.

We're going to do problem number 32. In this question, we have to find the value off court. 510 degrees. Okay. So, for first of all, what we will do. Well, just subject 360 to it. So little subject in order and 60 use. Obstruct these 300 sixties. So we're going to get caught. 100 on 50 degree, 150 degree. Okay, now what? We can write to this. That is called 150 degree. We can write this US caught 180 degree man has started degree as it is lying this 100 degrees lying in the second quarter rent. So we have to put a negative sign over there, and we can write it a score. 30 degrees. Okay, So the value, of course. 30 degree is under three. So the lady does manners and little three. So this will be the answer of discussion. They're so thank you.

Here we are going to be evaluating co tangent at negative five Pi Divide by 12. This looks like a tricky problem in the sense that our identities are about tensions, not co tension. However, we can get around that difficulty by knowing that tangent is a reciprocal of the co tangent function. In other words, we can write that co tangent at negative five pi Divide by 12 is the same as one over tangent at the same angle. Negative five Pi divide by 12 so we get across that difficulty. The next issue is that five pi divide by 12 needs to be decomposed into two different angles. Let's go off to the right hand side of this layout and evaluate tangent at negative five Pi Divide by 12 by expressing it as tangent of two different angles. Those singles are going to be three pie. Divide by 12 minus eight Pi Divide by 12 least some back to negative five. Pi divide by 12 and when we reduce these two fractions, we get tangent at Pi Divide by four minus two. Pi divide by three so we know we're on the right track with this decomposition. Since these two angles are easy to work with for sign co sine and tangent. Next, we have the difference of two Ingles in the tension function, so we will be using this identity our next step. Weaken right that this is tangent of some angle minus tangent of some other angle. As for the identity will go divided by one plus tangent times. Tangent of the same breakfast, respective angles. Those Ingles were pira four and two pi over three. So let's fill those in so pira for to pira three in exactly this order Pirate for two Pi Divide by three. For our next step, let's evaluate each of these tangent functions. We know that tension that PIRA for evaluates to exactly one since Sign is rude to over two, and so is co sign at Pyro for for tangent at to Pirate three. Let's carefully work through this one first, the reference Inglis Pi over three and two. Pirate three ends up in quadrant two. That means the numerator, which will be signed at two Pirate three, becomes squared of three. Divide by two. The denominator is co sign of two pi or three and since were in quadrant two, we obtain a negative 1/2 so this reduces altogether too negative square root three. I will be substituting in this value and this value on the next step in this fraction. Let's do those substitution is right away for tension pie. Before we obtain a one minus attention and to Piper three, we obtain a negative square to three. Then down below, it'll be one plus tension. PIRA for here gives us a one times attention to Piper three gives us that negative square to three. So all the tangent functions have been evaluated and now this is equal to one plus the square to three divide by wouldn't minus square to three. So it gives us tangent debt Negative five pi over 12. Notice. We do not necessarily want to rationalize this expression because we're going to substitute in that value right exactly here in the original problem. So going back to co Tanja and negative five pi over 12 we have in the denominator the substitution that tangent itself at negative five pi over 12 is one plus a square to three. Divide by one minus the square to three. So this whole problem can be converted into the reciprocal of that denominator, which is now one minus X squared of three. Divide by one plus a square to three. And finally, we're a to simplify this by rationalization. Well, bay multiplying by a new fraction with the same quantity and numerator denominator, which is the reciprocal of that denominator, which will be one minus the square to three in both the numerator and denominator. Let's multiply this out next. When we multiply in the numerator first, I'll distribute the one through this group on the right, which isn't purple that gives us one minus. Route three. Next. Distribute Route three here, so we have minus root three again and here. Negative route three times negative. Route three will be positive. Three for the denominator, the quickest trick to uses the identity that a plus B times a minus bi is always equal to the difference of squares. A squared minus B squared. Here is one. B is a square to three sort. Denominator turns into one R, one squared minus three, which is the root three squared. So now it's simplified by combining like terms. First we have in the numerator four minus two copies of Route three. Divide by naked of two in the denominator. So with one last division, we now know that the quantity negative too plus the square to three is the exact evaluation of co tangent. Negative five PI Divide by 12.


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