91Ó°ÊÓ

Let's denote the vertices of the regular polygon \(A_1, A_2, \ldots, A_n\). Since they are equally spaced points on the unit circle, the angle between the positive x-axis and the line connecting vertex \(A_i\) to the center, \(O\), is \(\alpha = \frac{2\pi}{n} \cdot (i - 1)\) for \(i = 1, 2, \ldots, n\). We can find the coordinates of the vertices using the unit circle definition: $$ A_i = (\cos\alpha,\sin\alpha)=\left(\cos\left(\frac{2\pi}{n}(i-1)\right), \sin\left(\frac{2\pi}{n}(i-1)\right)\right) $$

The distance between two points \((x_1, y_1)\) and \((x_2, y_2)\) in the coordinate plane is given by the distance formula: $$ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} $$ We will use this formula to find the length of the edge connecting vertices \(A_i\) and \(A_{i+1}\).

Let's consider the distance between vertices \(A_i\) and \(A_{i+1}\). Their coordinates will be as follows: $$ A_i= \left(\cos\left(\frac{2\pi}{n}(i-1)\right), \sin\left(\frac{2\pi}{n}(i-1)\right)\right) $$ $$ A_{i+1}= \left(\cos\left(\frac{2\pi}{n}(i)\right), \sin\left(\frac{2\pi}{n}(i)\right)\right) $$ Using the distance formula, the edge length will be: $$ \begin{aligned} d &= \sqrt{\left(\cos\left(\frac{2\pi}{n}(i)\right) - \cos\left(\frac{2\pi}{n}(i-1)\right)\right)^2 + \left(\sin\left(\frac{2\pi}{n}(i)\right) - \sin\left(\frac{2\pi}{n}(i-1)\right)\right)^2} \\ &= \sqrt{2 - 2\left(\cos\left(\frac{2\pi}{n}(i)\right) \cos\left(\frac{2\pi}{n}(i-1)\right) + \sin\left(\frac{2\pi}{n}(i)\right) \sin\left(\frac{2\pi}{n}(i-1)\right)\right)} \end{aligned} $$

Recall the cosine angle addition formula: $$ \cos(\alpha - \beta) = \cos\alpha\cos\beta + \sin\alpha\sin\beta $$ Here, \(\alpha = \frac{2\pi}{n}(i)\) and \(\beta = \frac{2\pi}{n}(i-1)\), so \(\alpha - \beta = \frac{2\pi}{n}\). Applying the formula: $$ \cos\left(\frac{2\pi}{n}\right) = \cos\left(\frac{2\pi}{n}(i)\right) \cos\left(\frac{2\pi}{n}(i-1)\right) + \sin\left(\frac{2\pi}{n}(i)\right) \sin\left(\frac{2\pi}{n}(i-1)\right) $$ Replacing this expression in the equation for the distance \(d\), we get:

The edge length of the regular polygon with n sides and vertices on the unit circle is: $$ d = \sqrt{2-2\cos\left(\frac{2\pi}{n}\right)} $$