Let A1 A2 A3 A4 A5 A6 A1 be a regular hexagon. Write the x-components of the vectors represented by the six sides taken in order. Use the fact that the resultant of these six vectors is zero, to prove that cos0 + cosπ/3 + cos2π/3 + cos3π/3 + cos4π/3 + cos5π/3 = 0. Use the known cosine values to verify the result.
