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The area of a regular polygon that has been circumscribed about a circle of radius $r(\text { see figure })$ is given by the formula $A=n r^{2} \frac{\sin \left(\frac{\pi}{n}\right)}{\cos \left(\frac{\pi}{n}\right)}$,CAN'T COPY THE GRAPHwhere $n$ 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 area of a dodecagon ( 12 sides) circumscribed about the same circle.

a. $A=n r^{2} \tan \left(\frac{\pi}{n}\right)$b. $A=4 \cdot 4^{2} \tan \left(\frac{\pi}{4}\right)=64 \mathrm{m}^{2}$ c. $A=51.45 \mathrm{m}^{2}$

Precalculus

Algebra

Chapter 6

Trigonometric Identities, Inverses, and Equations

Section 1

Fundamental Identities and Families of Identities

Trigonometry

Functions

Campbell University

McMaster University

Idaho State University

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all right, This problem, They're talking about a polygon that's has a circle inscribed in it. And I just want the only thing I want to add is talking about what n and r. R n is. The number of sides and are is the radius of the inscribe circle. All right, so the first part asks us to rewrite it so that we have only one trig function, and that's kind of actually easy. I'm gonna rewrite this and recognize that sign of something over coastline of that same something is actually a fancy way of saying tangent of that something. So now we've simplified that. The second part says we'll do it with the square with that has a radius of four meters. So circles being inscribed into a square. The radius of that circle is four meters. And, um so let's just talk about us. Um, the area of this square, we know the area of squares. The area of a square is side squared, right? So if the this case, it tells us that the radius is four. If R equals four, then that means the side is the entire diameter, right? So it's gonna be eight. So in this case, hate squared is going to be 64. So that's the answer that we're expecting to get. Now, let's verify this because the square is a polygon. Let's verify it using this new formula, which is a little extra. But, you know, it works for, uh, polygons with more than four sites. So here we have foresighted polygon. The radius is four that's going to be squared and then we're gonna being tangent of pi over four. All right, all ends become four. And the cool thing here is that tangent of pyre before is just one, and then this is actually four cubed or 16 times for which is 64. So that kind of agrees with what we expected to see with a square What they're asking in The last part is they want us to use a 12 sided object. Um, that's uses the same circle of radius four. Okay, so a 12 sided object will be area of, uh, FIS. We will have 12 sides. Um, the theater is the number of side. I'm sorry. The radius is going to be four, and that will be squared. And then this is the interesting part. It's going to tangent of Pi over 12. Okay, so this this number's gonna not be a nice, neat number. That's okay. Um, looking at this, we have 12 times. 16 So 12 times 16 that's four square. That's gonna be 1 92 times Tangent of pi over 12 and we're gonna use our calculator for this. But please, please make sure pre calculus is that one course where you're gonna be switching between degrees and radiance a lot. So make sure you are in radiance when you type it in this way. So I'm doing that now. Have tangent a pi over 12 and I'm just gonna use a decimal approximation. It's It's about, um, 0.268 rounding to three decimal places. So I have 1 92 times that number, and I get that. The area is about 51.446 square meters. All right? And that's part C of this problem.

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