Let A 1 A 2 A 3 A 4 A 5 A 6 A 1 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.

Question

Philoid · Accepted Answer

According to the polygon law of vector addition, the resultant of these six vectors is zero.
Here, a = b = c = d = e = f (magnitudes), as it is a regular hexagon. A regular polygon has all sides equal to each other.

[As the resultant is zero, the x component of resultant R_x is zero]

Similarly, it can be proven that

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.

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.