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 representeed by the six sides taken in order. Use the fact that the resultant of these six vectors is zero, to prove that `cos0+cospi/3+cos(2pi)/3+cos(4pi)/3+cos(5pi)/3=0` Use the known cosine values of verify the result.
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Explanation

According to polygon law of vector additin, the resultant of these six vectors is zero. Here `a=b=c=d=e=f`( magnitude)
So`R_x= cos 0+A cos pi/3=A cos (2pi)/3+ Acos (3pi)/3+A cos (4pi)/3+A cos pi/3 =0` As resultant is zero
x component of reesultant `R_x=0], rarr cos 0+cos pi/3+cos (2pi)/3+cos(3pi)/3+cos (4pi)/3+ cos (5pi)/3=0`
[Note: Similarly it can be proved that `sin 0+ sin pi/3+sin (2pi)/3+sin(3pi)/3+sin(4pi)/3
+sin (5pi)/3=0`