A regular dodecagon is inscribed in a circle with radius 1. Compute

(The sum includes all terms of the form where

Let denote the vector starting at the origin and ending at the vertex of the 12-gon. There is an angle of (360/12)º = 30º between consecutive vectors.

Recall that for any two vectors , we have

with the angle between the two vectors. Also recall that

For , is the length of the vector . So

The 12-gon is inscribed in a circle of radius 1, which means each vector has length 1, and from this we have

where is the angle between vectors and with , and these angles are multiples of 30º.

There are terms in the sum (from 12 total vertices, you take 2 at a time).

So we have

d

step-by-step explanation: