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24. (a) Let $A_{n}$ be the area of a polygon with $n$ equal sides
inscribed in a circle with radius $r .$ By dividing the
polygon into $n$ congruent triangles with central angle
$2 \pi / n,$ show that
$$\begin{array}{c}{A_{\kappa}=\frac{1}{2} n r^{2} \sin \left(\frac{2 \pi}{n}\right)} \\ {\text { (b) Show that } \lim _{x \rightarrow \infty} A_{n}=\pi r^{2} \cdot[H i n t : \text { Use Equation } 2.4 .6} \\ {\text { on page } 133 .}\end{array}$$

No. In this question we're going to we have a circle here and we want to divide it into an equal triangles. We're going to divide this polynomial this circle into n. Congruent triangles or something like this. And we know that the angle in general is two pi over N. And we want to show that we actually get in itself a specific area. So we know that the area of a triangle Is equal to 1/2. And in general times B times C times the sign of a of a side of some angle alpha. And since the side lengths are just are R. And R. We can replace these within the own formula. So you get that the area of the of one triangle is equal to r squared divided by two times the sine of two pi over at all. Right. Mhm. All right. And because we have any of these, Yeah, because we have en triangles, the total area of this circle is going to be End times are squared divided by two times the sine of two pi over Ed. Right, okay, that's cool. And we want to prove that as the limit of as the limit goes to infinity we get pi times R squared. So how do we take care of this? Well, let's first begin with taking our limit. So we have the limit as angers to infinity. You've got 10 times r squared over two Times The sine of two pi over at. We want to eventually use the fact that's the limit As we want to. We want to show that the limit as T goes to zero Yeah. Of scient E over T equals one. We want to use this eventually. So how can we, how can we get it so that we have what we want? Well, I can reciprocate be right, right. Yeah. Within this limit, I can first bring out the R squared. I can bring out the R squared. Yeah. And then simply put this and over two down downstairs downstairs. So what do I mean by that? Well, I bring the R squared. I have the limit as N goes to infinity. We're going to have the sign of two pi over and divided by Yeah. One divided by N squared divided by N over two. Which we know becomes R squared times the limit S. N. Goes to infinity of the sign of two pi over. End over to over N. Yeah. Now you'll notice that we need to have a pie. Uh huh. That's okay. I can just multiply by pi as long as I divide by it right after. Finally, if I let t equal to pi over N. We get We know that as and goes to infinity will have that tea goes to zero. So this will become, in fact, pi R squared times the limit. Actually, let's write this in blue. We'll have the limit as T. Goes to infinity, Goes to goes to zero will have signed T over T. Which means that we know that this here is just one. So we get that the area that we're looking for is pi R squared. That's the answer to this question.

All right. We have? Yeah, we're pizza by. We're not divided into in slices, so these Peter has a review. So far, you're inviting. Dyinto displaces people angle so well, there is. This here is our on that angle here sequel for all those so divided in tow and his license prizes that he's that give up in spite, they're gonna get the whole the whole turn. So you get it's about, you know, equal toe by good. What is the area singles? Legs. What is that? We have a following. That is life. She's long rectangle. We have something like this. So here this is This is supposed to be the same. These areas that so you have you have here this angle This angle here is just their house more or here. So that will be you have these really is our So the area off this whole travel will be just the area off these rectangle because well, if you hold it, you get that same area, right? Also, the symbol here is their house on these very use. These are so this area will be people toe the line here calling X from the length times the length there. Why so that? No, he said basically to extent. Why, no. 26. What is what So X is gonna equal toe? Are them school? Sino bigger hubs? No sign of it, Hobbs. Why she go to our arms. Sorry. Oh, Hawks. So that the total area will be photo extent. White photo R squared. No sign of fair hubs, hubs times signed up their hopes. But, uh, well, the double angle identity Those tells us peace. It's peace. Because I am. Perhaps some sign of perhaps, is just because one, huh? Oh, sign off, for example. So sign off through times if it helps. So magic back to those who were you So that this area can be read to us R squared hopes. Well, sign there until these area of a single square a single, single triangle trying what total little aerial vehicles. So very will be ableto and them's that so in times R squared hopes. That's a sign. Well, you see gold too, right? Okay. Buddies. So sign off to buy over. So that well was that expression. Have a lead. So this is the area of dividing it into yeah, tangles in Frankel's. What happened in that area will be computed. Need us and those people. It seems to be bounded toe. Stay there. We need the circle of radius. Are so but it should converts this hoping true. So we can take the lead here for that expression. So you would have something off the form. Well, that goes to infinity on the side of that ghost zero. So sign up zero course zero. So it would be of the farm if you're zeroed so well, has tried to apply something of the formal lovey. Told to see you probably can bring there in down bitey bitey bite wondering. So it's you call toe Are square house? Yeah, that was his name because that are square house. We were outside. So you play the need we have here. Zero Somebody else is here so we can apply. Logical, Because Hello, love. It'll form. So you differentiate. It's up. You're gonna get out. Vehicle toe. Same goes for unity. It's been together. God are square house. Be different sheets trying to get a call sign go sign to buy over in They are being everything they devote to pi over and go to minors to buy over in, squirt over in square. Then we differentiate. This time you get minus one over and squirt wanted. So you want to compute this limit? You can cancel these with my ankles and squirt. So, Dad limit, He's goes to zero. I was gonna go through go sign of zero because I know 01 so good It does come first to something they leave it The same goes to infinity is gonna go to R squared hopes. But I'm still buy so that well, Amar cancellation. Wonderful. They too. So but these converges to the area of a circle, right? I'm thinking you're your pea sized physical. I wonder.

All right. So we have our our pizza by. I want to divide it into and, uh, you see the pieces, so assume that that is any form. So, Dave, is that you have some of these guns here are leading by he divided and cut it into slices. But how goes as some angle on, uh, same angle for everyone For each one. I'm the same radios. So, uh, a lot of those laces books. So feel like these Where you have here radios Here is some angle. Yeah, so that we have divided into end devices and his laces arena and all those angles she give you the whole The whole turn should leave you to buy. So any time said I should be able to do, but But All right, So what is the area off single? It's like both. We can explain it into half bus lines like these. Like the drawings. So album here. So very news. It starts there. I'm the here disarm new angle ways. The old one was divided by house their hubs. So that, uh, well, the home slice here would be piece like times outlet. You call this length Slink picks about why the area was gonna be very of these tangle. Because that can be seen us just for being for the puppies so that they have two of those. So you're there is gonna be nix Thames away. That egg's another way. So we're always but it's sex so acceptable to our call sign off its hubs. Why you should go to our sign with the house. So the exams, Why is b r? Squared? No sign of fair hubs comes sign off their lives. So, uh, so for D. C. We can use the double angle identity. That is a very nice identity that tells us that co sign of some mango times Sign off. The other angle is what? How off? Sign off boys that are so 1/2 off. Sign off two times, But I'm a little bit down But we failed our hubs so right, boo Better so he's gone. Gone. So the area of these triangle cover everything us r squared. You see this idea provided by 1/2 signed off Data house? No. So well from these weaken, you can simplify what he stayed on my divided by him. Bye bye and So how fair Isaac will do? Bye bye. Bye bye. N So the area fussing those lice was evil lies That teacher is equal to r squared. I'm simplifying for what it's there there's off There's event so are square but to sign called Flora Jesu to body divided by and very nice You know, the area while single so most evil triangle So what he's gonna be the total area total area for pizza is gonna be ableto so this would be area so bad a sink over time area We're equal to just adding And of those so many times square loves sign Bye. We're in so well much ourselves, huh? Is that area we call these total areas of an Is there a limit? Assemble, say affinity about area P ait Ben So well, you should be the case because Well, because you keep on making that larger and larger It just constrained this day within that circle. So most of the media is gonna be area over a single, right? So for looking at these limit way need to look at the limit Singles leave it for inviting fragility. Oh, in James R squared hubs So you know, Bye. Or and so you just go ahead and blowing and equals infinity, You get Get impunity there. Dams ball signed off zero Because a single thing for you to buy ivory angles is you. What kind of zeros able to steer So you get a far off. See you at times. So what is that? But what we can do is turn it right. These name ID us. Well, it's right. There are square house. Oh, they're so go right at us. Sign ups to buy over in invited. You bite it by one over So that you did the streak we're gonna get Same goes to infinity on hope it'll form because of zero You so it's gonna be up from 00 So if we apply in love, I love you we'll be done. So, uh, differentiate the duck. They have the votes Scientist called side. So go sign bye or so. Remember, here were different shaping with respect to end because that is survival that you're taking really needed respect. So theoretical science co signed but the relative ease off to off to buy or end this mind stood by over in squared Get down on top on their devotion. One of his minus one over ants, Word minus one over square. So you still have the our square house. And so you see here, Jim, the council, that from that Nice. So these limit legal the limit a single safe in any of these. I love it. I believe all of this is goes this year's It's what we call sign of zero there, which is one one. So, uh, so this limit will be ableto are square of I'm super, but I'm very nice to lose weight. So is the same. Nothing that is about to our square pie. The chair is the area off the tickle. So all that's a very nice result the meaning of the aerial view. Our pizza goes toe to the physical as you could pay. His life is very done. Very narrows license. You can have you, can you making little in radius license. So that is in


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