let-pn-be-the-perimeter-of-an-n-gon-inscribed-in-a-unit-circle-figure-8-a-explain-intuitively-why-pn

Question

Let $P(n)$ be the perimeter of an $n$ -gon inscribed in a unit circle (Figure 8).
(a) Explain, intuitively, why $P(n)$ approaches 2$\pi$ as $n ightarrow \infty$
(b) Show that $P(n)=2 n \sin \left(\frac{\pi}{n}ight)$
(c) Combine ( a) and (b) to conclude that 
$$\lim _{n ightarrow \infty} \frac{n}{\pi} \sin \left(\frac{\pi}{n}ight)=1$$
(d)Use this to give another argument that
$$\lim _{	heta ightarrow 0} \frac{\sin 	heta}{	heta}=1$$

Bahar Tehranipoor · Accepted Answer

Okay, so this problem is a bit of a loved one. So we&#x27;re going to go through it part by far a through D and solve each part into the Julian with each part building off. So this question size let Pete and be the perimeter oven n gon inscribed in the US circle and the figure is talking about is right here. So if you can look at it, we have hex a gone and Nana gone, and it goes up. So you have all of these shapes inscribed inside us circle. So basically, all saying is that the radius is one to make it so a exploit intuitively wide PM and approaches to pie as end of purchasing. So let&#x27;s think about that for a second as an approaches infinity. That just means that there&#x27;s an infinite number of sides to this. Holly are. So when we have the shape and the sides are becoming an infinite amount, it starts to resemble just a circle because, as you can see as the sun size growth from 6 to 9 to 12 it starts to resemble more of a circle. So intuitively, when the sides approach infinity, the protocol approach to pie because it&#x27;s approaching the perimeter of the unis circle, and that is just too Hi are. And because our is just one here, our perimeter approaches 25 and that&#x27;s always asking you to do so we go to be show y. P even equals to end, sign high over. And now this is a more tricky problem to solve, and I think the best way to solve it if we just draw out a polygon to use as an example, and I&#x27;m just going to draw out, draw out. And often this is a very poorly drawn octagon, but it&#x27;s really just to get the idea. So we have a center point right here and we have the radius. You draw to radio I just to show, but it&#x27;s one and we&#x27;re gonna manage. This is inscribed in a circle, and really, that&#x27;s all we got. Way really need is the information that is described. A circle is that the radius is or one here, and they reached a corner points, of course, as he should. So we need to show the perimeter equals two inside high over, and I think the easiest way to do This is to use some trig that we could recall from pre calculus. So if we haven&#x27;t angle right here, call it speed up. No data. We have to know its data Peoples to pie over end Now, Why is that? Because the angles inside of a circle. Of course, if you cut off the circle, always angle. They&#x27;re going back to two pie. So we know that if we have these radius the radio and if we cut up this shape into all the angles, all the angles are going out of the two pi. And of course, in this situation, because we do have eight sides that there that would be too kind Ray or Hyo four. But we don&#x27;t need that information because, really, all we&#x27;re using this after going as just a shape to fewer problems. So we know that they are depending on how many sides are to this polygon is to pile around Now we can use a trick to figure out what this silent right here is the easiest way I think to do this this this wouldn&#x27;t have perpendicular to do it by sector, meaning that we&#x27;re going to split this angle in half and it&#x27;s gonna come straight down and we&#x27;re gonna call both sides right here. X now, using our trade identities. We know that sign well, from Kolkata, as we memorize a while ago, sine is the opposite over hype Linus and hear the opposite of the angle data over to now because we&#x27;re just using half this angle. His part is X, and we know that our high partners right here is just one. So we can say that sign of data over to which is just high over end equals X, which is our opposite, or our partners, which is just one. Now we know that X ignoring the one equals sign of pie over and and we know that two X is our silence to our polygon. So two exes, just two sign high over. And now when we have one side length and we have insides to find the perimeter, all we had to do is multiply our side length with the amount of size that we have. And if we multiply this five end, we get it to end sign hi over it. And that&#x27;s just be no. Our next part, see, is asking for us to combine what we&#x27;ve learned from a B to conclude thus and approaches infinity. The limit becomes one. So how do we do this? It&#x27;s crazy. We basically just have to look at the fact that when we concluded that p of n equals two and signed pipe end, we were able to prove that this is correct, that this is this is the perimeter. Now if we divide both sides of this by a certain something, we get what is stated right here and over Pi to get this over pie right here. All we had to do is divine both sides off. This of this equals to pie as and riches infinity as we as we did before because we use their intuition and a and then we really calculated out. And b we combine that together because we know as an approaches infinity, the perimeter becomes two eyes we can set that equals to buy and to get to this and over pie on the left side, Right here. All we have to do is divide both sides by two pipe and if we divide both sides by two pi, we get the born on this side and we get and over pi sign of piranhas because want but works out perfectly because as we divided well size by two pi, we get exactly what we need to prove right here. Now the last part of this question says to use this to give another argument that as data approaches, zero signed data over theta equals one. This will require some thinking as well. This&#x27;s pretty complex problem. What we can think about is this right here. You could rewrite it just like that. How? By saying that Seita here, peoples Hi over end. Now, how is it that we&#x27;re allowed to say that? Because as n approaches infinity, Saito will approach zero. Think about it as we drew this picture right here, even though we said data equals to power and there was just an arbitrary value we could think about. It just is this angle as an approaches infinity. This angle is going to grow smaller, smaller and smaller, and it&#x27;s going eventually. It&#x27;s going to start approaches. Zero, as in gets bigger and bigger so we can say as an approaches infinity. They&#x27;d approaches zero if they don&#x27;t people&#x27;s high over end Now all we have to do is take what we have right here and replace everything with the end of the pies with fatal. So if you know, data equals power and we have the limit data for a cheesy room and replace everything and over pie. If we have St Michael&#x27;s pyre end, it&#x27;s just one over data and we have signs of pie over and and that&#x27;s just sign Ophelia in that is exactly. But this identity says right here, and that&#x27;s really all this problem is asking you to solve. It&#x27;s just to show how, as a limited approaches infinity. We could think about it in our heads, especially in this terms of polygon. As it&#x27;s it&#x27;s like the circle is becoming a circle because as the size of Russian unity, it starts to look like this when it&#x27;s really all that I was asking for you to look at. Yeah, that&#x27;s it

Carlos Pinilla · Answer

to have the following problem. There&#x27;s Ah, there&#x27;s some pizza for you want to slice? I want to be fighting D&#x27;oh! And he calls license. Oh, so he&#x27;s not perfect drying, but you already have. It wasn&#x27;t that bad angle where the same for all of these pieces. So I go there way have some radius here. So that way we do the whole turn you get, Uh, Bye on then. Each of these lice is us. Mango feta on olders and of them and his laces. So you want to figure out you have those in his license and the radios? What is the area? My theory of that shape. You there? Well, uh, if you believe that his life was the same Oh, area. Uh, What&#x27;s gonna be in times? It&#x27;s a van. Where s event is the area of a single slice here? Not shaded area there. That&#x27;s one of those pieces. A seven. So for his event, you have been following setting. So they&#x27;re triangle. Do you have the sango? Huh? On the his distance, there is the same LaSalle. This knows there are all right. So so loose travel would Well, we can cut it in how unfolded to see that the dollar here, it&#x27;s gonna be ableto Well, if you&#x27;re full Did is gonna vehicle to the area. Piece tangle. Right. We&#x27;re going for here. You have this angle. That&#x27;s Hobbs. That is the same wood there hops. Well, if you do, we have these triangle one of those. Well, these your triangle is the same. Was that about over travel? So here we are. No. Perhaps you have are. But our here&#x27;s be sense. So well, the area of these rectangle I was gonna be these distance. These those picks legal Does the snow&#x27;s X here is that the snow is white. All right, extends. Why, that is the area office when eggs, That&#x27;s why. But what is what is sex exists? Are was signed off the house. Our cause. Books? No, that is Baron. That is why are called on. Uh, that is way on eggs. Yeah. My butt on X is equal to our sign. Perhaps sign of her house. So, uh, also that if he did a product, you get the delivery is gonna be our squirt. We&#x27;ll sign off their hooks, then sign off the rocks. Sign off in the house on the DL angle. Identities tell you that these gadgets, they&#x27;re very simple Toe. What? Huh? Off sign off the that I&#x27;m going in their place. So one how both sign will be Sango two times two attacks. So that was very nice thing happens here. As you can guess, the two goes away, but and so you have area will still have the arts. Quarter square little area of these single piece will be of a single&#x27;s life. It&#x27;s gonna be our square loves sandal, but, uh Well, uh, what is it? Well, we have in his laces. So if there should be such that and that was that I go gives you the whole turn Did you buy? So we can be fired, but and wonderful. And they get this cancels so later we got to find a way to buy him. Great. So this area is abandoned on a vehicle toe our square replacing for what it is. Sign off to buy over in. So as we if it is drawing, we see that you Syria is cause trying to leave. Oh, it&#x27;s not. It can be greater than here of the circle because he&#x27;s constrained there so he can ask yourselves little area there. This deal is that how easily the same goes for you. Katie is early. Need us saying go to infinity. Hey. Hey, Here is that little area. So sometimes it&#x27;s a band is gonna be ableto in terms. This so is their lead to that area, Which will? There should be vehicles. You got these constrained to stay within the circle. So what is the lead off? N times are square hubs square house. So, you know, bye over in. And if you just go ahead the Bailey quitting Cy Baby, will you get the beauty times are square house That was signed off. All the results to zero. This is zero, but this is zero. So serious times infinity. What is that? But I will when you lose, you can manipulate this and already does, uh, wanna burn so peace Can we read us? Her square huffs sino good bye over end. We can bring that What? Multiplying white and is the same musty by biting by Want over so that well you think you are I&#x27;m going Divinity. We&#x27;re gonna have a local form, so it goes to zero zero or zero. Yeah, we&#x27;re gonna blow people. So after a fine no pickle Hello? Get her in. Shape it up with respect to end. So you&#x27;re The sign is that is what we have been through. They are square house people Sign is oh, sign, but well, the sign of the same But we have your depression with respect And so you get a factor of minus two by over a squared over since word And then we differentiate the bottom with respect when minus one Over in squared I said I&#x27;d Well, something wonderful happens here. I wonder where it ends where it comes to that so that well, you clean equals angle into beauty. You&#x27;re gonna have to co sign off zero, because by wearing goes baby Mr Zero. So, cause I hope, Cyril, but he&#x27;s gone. Okay, so So these whole expression is gonna go to grows through. Uh, so are square Hobbs. Them school sign of 01 times one times to buy them. Still buy. I do. But until another nice thing happens here to to to get that the leader is the area. Oh, the circle are scored in spite, but he&#x27;s here, not circle. Cool

Frank Lin · Answer

So here we use untrue I and goes to approximate a polygon to approximate area of the circle. So here, so it should eyes try. Go on, look at this. This one has radius. Are this wise radius? Are this is one over the the horse Sir Ho sang Go So the whole circle goes to part over Wei have learned from the high school that the area of the triangle this one over two length of this one times Lance off this one time sigh of this and go Oh, so it&#x27;s not yet also distressed area of the triangle is one over two times dance of this side and lands of this side Time side off the angle to pi over and the alien is simply trying goes at it together Therefore he gives us up This formula and computer are limit on fraud on course the infinity here that is this so here we still have to take limit here We put on over to pie here that we put pie are square here and we have side you pie over this just below there How to brought shake the limit here on this term This part it is. Actually, we actually use a famous result from calculus. What unconsciously infinity? This argument this too high over and goes to zero. And this can be view us divided by two pi over. So we used the famous result that science over hacks the limit of ab support zero give us one So this part of the limit will goes to one and we&#x27;re left with pi r squared.

Ayman Sherif · Answer

Okay. Hi. And this problem, we have a circulates Inside the circle is appointment of equal size, which can be by this. This is a center of the circle. This parliament is divided into a number of equal triangles. For example, this is this is one of this is one thing This is the next is the second language. This is the same triangle and support string this is this site is or And this site is also which is a sent a radius of the second. This angle is the central anger which is off. If you want to get the area off this triangle, we&#x27;ll be acquitted. Video, I hope deploy. But Roy Oh, off deploy swing is equal to boy over. And where n is the total number of triangles in sizable in this case with before so But we want to get the area of the overall volume. So you way will not deploy this area by end, which is a number off trains it so in which, which is a total area of the Taliban with the equal or squared Uh, sign Do boy deploy, which is the first requirement, is the second requirement. So we we knew that the area of the bargain will be equal and it will deploy or square deployed and oy, sign to over in. We want to take limit to this equation so limit from then turns to infinity. Half are square is a constant eyes is very circus constant So it will be equal hope squared limit when tends to infinity Oh, look it sighing to over in We will take this equation and multiply it mainly by one. But in our case it will be too boy over do for So this limit will become I will show in exploit to will deploy and over to boy must deploy sewing Cool boy boy over in if we talk If we take this turn on and make it a diminished denominator every year it would become equals to guard no deploy toe over and over to boy over in Let system be seated which is are some thrilling So in the next flight limit when end girls 24 in will be equal Oh squared deploy limit when sita zero Because what we&#x27;re increasing the number and the sita or the central angle will decrease gradually until it nearly become zero Flemington. See? It tends to zero toe boy Want deploy saying see you seek mint. We will take the two boy outside so exploit limit and well in turns to infinity will be acquitted. Pool off, squared, deploying to boy Look, Chloe, limit tens zero So I&#x27;m so in Saint Overseas. The tour Big gun with the two. So it will be equal. Sister is one. This time would be equipped to one because we see the term is 20 signs. It was one through This time is one. So this term will be equal next light limit and then turns to infinity. Boy R squared one. She could Boy. Oh, squish. This is a quote on This is the overall area of the city. This is the way as they used toe proof as overall area of the second as shown, uh, isn&#x27;t it you? Thank you.

Mike Gaerlan · Answer

Okay, So if we take a look at this shape here first of all, the total area here this is going to be equal to Well, if we have, the total area is equal to pi r squared. And then this is going to be data over two pi. So then we&#x27;re going to have that This total area is going to be Bluetooth data divided by two pi or just equal to data R squared over two. So it&#x27;s gonna be equal to 1/2 data R squared. So that&#x27;s the total area of this part here. Now, the area of just this part here now that&#x27;s gonna be equal to this height here, times this length. So this length is equal to our, so it&#x27;s gonna be able to our and then times 1/2 of the height, so 1/2 base times height. Now, the height this is going to be able to are signed data here, so we&#x27;re gonna have our squared sine data, so that&#x27;s gonna be the area of this region here now to find the area of this region here, this is our and this is co sign of our or sorry coastline of data. So let me write that data. This is going to be close enough data. So our co sign a feta, and then this is going to be now ar minus our co sign of data. So the area of this region here is going to be 1/2 times are. And then times ar minus are cosigned of feta or sorry are signed data, which is the height. This is the height are signed data, and then the length is this ar minus archos idea. So that&#x27;s the area of that region here. So first, our limit is going to be, as data goes to zero of now on the bottom part. Okay, I&#x27;m gonna go ahead and multiply this out. We&#x27;re gonna have a 1/2 and then an r squared signed beta, and then a minus R squared and then signed peanut Times co sign data. That&#x27;s the same as 1/2 of sign tooth data. So I&#x27;m gonna rewrite it as that. So 1/2 times 1/2 is gonna be 1/4 then r squared sign of two theta and the next. Now, this region here is going to be able to this minus this part here. Remember, this was the region of this. So then this minus, this is going to be just this part here. So on the top, we&#x27;re gonna have 1/2 data R squared and then tying or minus 1/2 of our squared sine data. So now let&#x27;s take this limit as stated goes zero. So if we plug in directly, zero will have 0000 so we&#x27;ll get 0/0. So let&#x27;s simplify some things first. The 1/2 scan all cancel out, and this will become a 1/2 here instead. So now we&#x27;re going to simplify. This becomes limit as data goes, zero of 1/2. Sorry, uh, down the Thomson have data or squared minus R squared sine of feta. Uh, and then also noticed. Actually, before I write that that every term hasn&#x27;t are square, so I&#x27;m gonna cancel out that are squared as well. So on the top, I just have data minus sign data divided by and then I have signed data minus 1/2 sign of tooth AEA. Now let&#x27;s apply low petals rule. So, actually, I&#x27;m going to the bottom part out doing green here. Okay? So we&#x27;re gonna have that. This is when people to the limit as state a ghost. Zero of the derivative, the top part is gonna be one minus co sign of feta and then derivative of the bottom part will have cosigned data minus and then design to data. So if we try plugging in, substitution here will get zero. So co signers heroes with one. So get one minus 1/1 minus one, which is going to be 0/0. So let&#x27;s try applying loopholes rule again. So get the limit. State that goes to you. Data goes to zero of Then we&#x27;ll have signed data divided by now on the bottom. Here, the derivative is going to be will have negative, uh, signed data minus two or sorry, plus to sign tooth data. And again, if you plug in zero, we&#x27;re gonna get 0/0 plus zero, so that&#x27;s gonna be zero. So let&#x27;s try applying loopholes. Rule again. Limit as, say, that goes to zero on the top, will have a co sign or negative cosigner feta and then on the bottom will have a negative co sign of data and then plus four and then cosigned of tooth data. So now if we try direct substitution on the top part. So I had a minus here when I should have had a plus. Now, this is going to be equal to And then if I plug in zero I have are sorry. Zero close item zero is one. And then I have one over minus one plus four. So that&#x27;s gonna be no minus one over. Sorry. Positive one over. Minus one plus four. Which is gonna be 1/3, which is gonna be our final answer.

Adam Cornes · Answer

who question gives those of regular polygon with insides inscribed inside the circle with radius one. So the death rate to do this is just to draw the circle. This is going to help us imagine what we&#x27;re talking about. So this is just gonna be the science. So this is gonna be a side hair. This doesn&#x27;t So this is a rough sketch. So there&#x27;s gonna be however many sides and in total So these points had such in the circle are going to be points a I won&#x27;t be to These will make a triangle by touching oh, in the middle and we&#x27;ll do the dash lines. Help was with the equations later. Run with the midpoint. This line no donated his acts to me on the name. Why? This angle here is gonna ankle 180 ever end. The best way to do this is we can easily work out this angle here. So a be simply gonna three engine 60 breeze here in a circle divided by and number side saying a to be it&#x27;s going to be one side on the sides. Go enough in either direction which are going to make more triangles So however many sides gonna be holding many triangles, so the number of that want to speak to know it is Then So you know, X is exactly half as we conceive by that Dutch line to the right angle you be. That&#x27;s just over two. Who stood that top it in. So 360 degrees at the end. But I like to Well, um, in 100 90 degrees, everyone So that sand sets upon a So I just put a there. Thank Bay. Must have showed that an a X So the length of ex hair Zuko to sign 1 80 ever end. So gonna be easier We just draw the triangle out So just drawing on its side to make it easy It&#x27;s a show So this here is gonna be putting oh on And this would be the right angle This will be X. This will be a This is gonna be one. So the best way we can work out the length of a of axes We can simply use the sign rule. So we know that sign of X have around gonna he cool You pulled in the sign of a well That&#x27;s cool. Sign. Oh, uh, hey, thanks. So we can ignore this middle bit. Just focus on those. Last two was a role equivalent and we&#x27;ll rearrange for a of X ray. That&#x27;s I&#x27;m gonna eat whole sign. Although there this sign of X times Ah, and we know Oh, so this is the angle, your wreck. She&#x27;s also just the same as angle. That&#x27;s another no zero. So we completed that in and we know that our equals one from the diagram So weak local All those in you should get sign when you see never in So I have signed 90 is one cause exes Night Great degrees When r equals one that has come slow We left with you Fine 1 80 degrees and bring that&#x27;s your answer To be there in part C This is to show looks oh, of x equals And it says Overbeck should equal cause of 1 80 of Iran. And we need to show you that So again, using the diagram, it&#x27;s time we could just use their care saying, 00 was gonna equal that case and right there and use so in turns off the angular the Jason gonna be this angle here. Now you put high parting uses. Obviously are was one. So you put those in. That&#x27;s just people over there over there with a and painful over that one. And of course, we know that that Angulo is wanted to your urine. So therefore rearranged her bath and equal because, 018 number. That&#x27;s answer to see So I d. That&#x27;s was assured that the parents of the polygon P equals to end time sign 1 80 avery. So we know that the province over to alternative be an equal and number of science times the length of one side, which in this case, is a B. So goes the diagram. See that one length, one length is a B, and we know that Eddie Be is made up of 28 arrests because that&#x27;s exactly half vainly. No way to exit equals. So you can say that equals and times two time his app. So to say to any time to act Were you clicking eight x, which we know is sign when it&#x27;s ever ran So it&#x27;s gonna be too and sign It&#x27;s well down When 80 never. There you go. So he has to show that the area of the polygon is a equals ensign. When a tabor in times cause money saver in so the area of a public gun there is a equals. So this day, is there a problem? Guns up in there? That is a p of the afrisam times the perimeter. So the apathy is the line that goes perpendicular from the side of the triangle to the center of circles. So in this case, that&#x27;s overtax So And look, these numbers in So you know, it&#x27;s hard. Both ex times pay and we know all those numbers, so we put them in Good God. Oh, I was 18 c and ran times just without pay to in time Sign one again. See on. And these two have been accounts allow natural levers with and because 18 So you, your friend times 182 and room cancer, etc.

Problem

Every limit as $x \rightarrow \infty$ can be rewr…

Problem 44 Hard Difficulty

Answer

a) $2 \sin \left(\frac{\pi}{n}\right)$
b) $2 \pi$
c) 1

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a) $2 \sin \left(\frac{\pi}{n}\right)$b) $2 \pi$c) 1

Calculus for AP

Kayleah T.

Caleb E.

Kristen K.

Michael J.

Precalculus Review - Intro

Functions on the Real Line - Overview

Jon Rogawski & Ray CannonCalculus for AP

Kayleah T.

Caleb E.

Kristen K.

Michael J.

a) $2 \sin \left(\frac{\pi}{n}\right)$
b) $2 \pi$
c) 1

Jon Rogawski & Ray Cannon

Calculus for AP