Chapter 9.3, Problem 51E

Solution

A.

To identify: The perimeter of each polygon in the sequence is given by $a_n=2n\sin \left(\raisebox{1ex}{$\pi $}\!\left/ \!\raisebox{-1ex}{$n$}\right.\right)$ where $n$ is the number of sides in the polygon.

In the following sequence of diagrams, regular polygons are inscribed in unit circles with at least one side of each polygon perpendicular to the positive x-axis.

  

The perimeter is $nAB=2n\sin \frac{{180}^{\circ }}{n}$ .

Given information:

The given term is $a_n=2n\sin \left(\raisebox{1ex}{$\pi $}\!\left/ \!\raisebox{-1ex}{$n$}\right.\right)$

  

Explanation:

Let’s consider a regular polygon with n sides and one of the n congruent triangles. $AM\perp AB$ .

Determine the angle $MOB$

  $\angle MOB=\frac{\angle MOB}{2}=\frac{\frac{{360}^{\circ }}{n}}{2}=\frac{{180}^{\circ }}{n}$

Determine MB:

  $OB\sin MOB=MB=1\cdot \sin \frac{{180}^{\circ }}{n}=\sin \frac{{180}^{\circ }}{n}$

Determine the polygon’s side:

  $AB=2MB=2\sin \frac{{180}^{\circ }}{n}$

So, perimeter is $nAB=2n\sin \frac{{180}^{\circ }}{n}$ .

B.

To identify: The conclusion $a_n\to 2\pi asn\to \infty$ investigate the value $a_n$ for $n=10,100,1000$ .

That as $n$ increase, $a_n$ gets closer and closer to $2\pi$ the unit circle’s length $a_n\to 2\pi asn\to \infty$

Given information:

The value of $n=10,100,1000$ ,

Explanation:

Consider the given information.

Use the formula for the perimeter of the regular polygon.

  $a_n=2n\sin \frac{{180}^{\circ }}{n}$

Put the value of $n=10,100,1000$ into $a_n=2n\sin \frac{{180}^{\circ }}{n}$

  $\begin{array}{l}{a}_{10}=2\left(10\right)\sin \frac{{180}^{\circ }}{10}\\ a_{10}=20\sin {18}^{\circ }\\ a_{10}\approx 6.18034\\ a_{100}=2\left(100\right)\sin \frac{{180}^{\circ }}{100}\\ a_{100}=200\sin {1.8}^{\circ }\\ a_{100}\approx 6.282152\\ a_{1000}=2\left(1000\right)\sin \frac{{180}^{\circ }}{1000}\\ a_{1000}=2000\sin {0.18}^{\circ }\\ a_{1000}\approx 6.283175\\ a_{10000}=2\left(10000\right)\sin \frac{{180}^{\circ }}{10000}\\ a_{10000}=20000\sin {0.018}^{\circ }\\ a_{10000}\approx 6.283185\end{array}$

Notice that as $n$ increase, $a_n$ gets closer and closer to $2\pi$

So, $a_n\to 2\pi asn\to \infty$