# Let A_1 A_2 A_3 A_4 A_5 A_6 A_1 be a regular hexagon.

Question:

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 $\cos 0+\cos \pi / 3+\cos 2 \pi / 3+\cos 4 \pi / 3+\cos 5 \pi / 3=0$

Use the known cosine values to verify the result.

Solution:

From polygon law of vector addition, the resultant of the six vectors can be affirmed to be zero. Here their magnitudes are the same.

That is, $\mathrm{A}=\mathrm{B}=\mathrm{C}=\mathrm{D}=\mathrm{E}=\mathrm{F}$.

$R x=A \cos \rightarrow+A \cos \Rightarrow / 3+A \cos 2 \Rightarrow / 3+A \cos 3 \Rightarrow / 3+A \cos$

$4 \Rightarrow / 4+A \cos 5 \Rightarrow / 5=0$ [As resultant is zero, $x$ component of

resultant is also 0]

Now taking $\mathrm{A}$ common and putting $\mathrm{Rx}=0$,

$\cos \Rightarrow+\cos \Rightarrow / 3+\cos 2 \Rightarrow / 3+\cos 3 \Rightarrow / 3+\cos 4 \Rightarrow / 3+\cos 5 \Rightarrow / 3=0$