In , the sides , and are the roots of the equation Then find the value of Express your answer in the form of where and are coprime positive integers. Enter the value of .
This problem is part of the set Trigonometry .
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The given equation factorizes as follows:
( x − 4 ) ( x − 6 ) ( x − 2 ) = 0 ⟹ x = 4 , 6 , 2
Take the given cyclic sum as S . Now, using the Law of Cosines, we have,
S = c y c l i c ∑ a cos A = c y c l i c ∑ 2 a b c b 2 + c 2 − a 2
It can clearly be seen that the denominator of the general term of the cyclic sum is symmetric since 2 a b c = 2 b c a = 2 c a b . The sum can hence be expanded out as:
S = 2 a b c ( b 2 + c 2 − a 2 ) + ( c 2 + a 2 − b 2 ) + ( a 2 + b 2 − c 2 ) = 2 a b c a 2 + b 2 + c 2
The final result obtained is a symmetric expression in a , b , c and so, the answer remains the same for all the three cases of sides that can be given by { a , b , c } = { 4 , 5 , 2 } . Considering any one of the three cases (like a = 5 , b = 4 , c = 2 ), we have,
S = 8 0 1 6 + 2 5 + 4 = 1 6 9
A diagram with the case taken: