110. Sum of Cosines. Calculate the sum

$$
\cos 5^{\circ}+\cos 77^{\circ}+\cos 149^{\circ}+\cos 221^{\circ}+\cos 293^{\circ}
$$
110. Project the sides of an arbitrary polygon onto a line lying in the plane of this polygon. Then the sum of such projections, taken with the appropriate sign, is zero. Now let's take a regular pentagon with a unit side and note that its exterior angle is $72^{\circ}$. The terms

![](https://cdn.mathpix.com/cropped/2024_05_21_2b40d984baf9da275c68g-109.jpg?height=426&width=610&top_left_y=2026&top_left_x=723)
of our sum represent the projections of the sides of this pentagon onto a line that forms an angle of $5^{\circ}$ with one of these sides. Therefore,

$$
\cos 5^{\circ}+\cos 77^{\circ}+\cos 149^{\circ}+\cos 221^{\circ}+\cos 293^{\circ}=0
$$

[M. S. Klamkin, M. M., 28, 293 (May 1955).]