Computing cos and sin

Notes

Here's a Python implementation of the algorithm we developed:

Consider a regular $n$ -gon inscribed in a circle of radius 1: Show that its perimeter is given by $2n \sin\left(\frac{360^\circ}{2n}\right).$
Hint Solution

Look at problem 4 from the trigonometry review.

Subdivide the $n$ -gon into $n$ triangles, as shown in the figure. Each such triangle shares a side with the $n$ -gon; we write $b$ for the length of this side and $\theta$ for the angle opposite to it. The perimeter of the $n$ -gon is thus $nb;$ note also that $\theta = \frac{360^\circ}{n}.$ In problem 4 from the trigonometry review, you showed that $b = 2\sin(\theta/2).$ Combining these, we get that the perimeter is $2n\sin\left(\frac{360^\circ}{2n}\right).$
$\theta$ $b$
Recall that $\Twopi$ is defined to be the circumference of a unit circle, and $\pi$ is defined to be half this quantity. Compute approximate values of $\Twopi$ and $\pi$ (to 4 decimal points, say).
Solution