Problem

At their point of intersection, the angle $\theta…

01:52

Problem 74 Hard Difficulty

The perimeter of a regular polygon circumscribed about a circle of radius $r$ is given by the formula $P=2 n r \frac{\sin \left(\frac{\pi}{n}\right)}{\cos \left(\frac{\pi}{n}\right)}$.
where $n$ represents the number of sides. (a) Rewrite the formula in terms of a single trig function;
(b) verify the formula for a square circumscribed about a circle with radius $4 \mathrm{m} ;$ and (c) Find the perimeter of a dodecagon (12 sides) circumscribed about the same circle.

Answer

a. $P=2 n r \tan \left(\frac{\pi}{n}\right)$
b. $P=32 \mathrm{m}$
c. $P \approx 25.72 \mathrm{m}$

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Precalculus

Chapter 6

Trigonometric Identities, Inverses, and Equations

Section 1

Fundamental Identities and Families of Identities

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Video Transcript

all right in this problem. We're talking about the perimeter of a polygon, and they've given us this equation here too. And our times of this quantity here and I have highlighted because the first test they want us to do is to re write it as a as a trig function. And that's only one trick function. Right now we have signing coastline, but we only want one. So sign over co sign of the same angle we can rewrite as tangent. So this is too and our times tangent of pi over and the next task requires us to make sure that we are verifying this with a with a known value. So this, in this case, this is a primitive, a polygon that's circumscribed around a circle of radius R. So that's where our comes in as the radius of that circle and then n is the number of sides for that polygon. So in this case, they want us to do a square, and the square has four sides. And then we also know that if r is the radius of the circumscribe, um um, figure, um, I have something that looks like this little bad drawing, but it's okay. We have our here. That means that decide length of this square is going to be to our Okay, so too are is the side link. And we know that the perimeter of a square we know that that's going to be side Plus I plus I plus side or four s in this case. So the perimeter of this is going to be to times are but are we know is four. So I'm gonna take this for out, are out and make it for so it's eight times four, or in this case, it's going to be 32. So that's the answer that I'm going to expect from B by using this equation in a. So the perimeter is going to be two times the number of signs which work dealing with this square. The radius is going to be four. And then we're gonna multiply this by tangents of pi over four. And the cool thing about tangent of Higher before is that's just gonna be one. So I'm left with two times four times four. Either way, you look at it, it's going to be 32 I get 32 as my answer, and it matches what I expect it. So I've kind of demonstrated that this equation works foursquare, so it should work for polygons that have more sides. And that's exactly what it asks us to do in part C in part C. They want us to do a a doe Decca Gone, which has 12 sides, and they want the same radius to be four. Okay, so in this case, we have our perimeter of this 12 sided polygon is going to be two times and which is 12 times four times tangent of high over 12. All right, that's not on the unit circle, so we will need a calculator for this. But the cool thing about this is, um, two times 12 is 24 multiply by four. That should be 96. I have 96 times Tangent of pi over 12. And when I multiply pi over 12 or Tanja Pirate 12 the one thing you want to make sure is that you are in radiance. If you are in this class, you you need Teoh be switching between degrees and radiance quite often, probably. But this is one where when you have a pie in it. Usually you're gonna be dealing with radiance. There's no degree symbol. Your you have to assume radiance. So 96 is going to be multiplied by this, a decimal approximation of 268 And our final answer is gonna be the perimeter. And it's gonna be not an on an exact number. Nice whole number for us, but it will be 25.7 to 3, uh, meters.

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