Solve the following

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, $A=B=C=D=E=F$.

$\mathrm{Rx}=\mathrm{A} \cos \rightarrow+\mathrm{A} \cos \Rightarrow / 3+\mathrm{A} \cos 2 \Rightarrow / 3+\mathrm{A} \cos 3 \Rightarrow / 3+\mathrm{A} \cos$

$R x=A \cos \rightarrow+A \cos \Rightarrow / 3+A \cos 2 \Rightarrow / 3+A \cos 3 \Rightarrow / 3+A \cos$ $4 \Rightarrow / 4+A \cos 5 \Rightarrow / 5=0$ [As resultant is zero, $x$ component of resultant is also 0]

Now taking $A$ common and putting $R x=0$,

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

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