In ` A B C ,a , ca n dA` are given and `b_1,b_2` are two values of the third side `b` such that `b_2=2b_1dot` Then prove that `sinA=sqrt((9a^2-c^2)/(8c^2))`

Explanation

We have `cos A = (b^(2) + c^(2) -a^(2))/(2bc)`
or `b^(2) - 2 bc cos A + (c^(2) -a^(2)) = 0`
It is given that `b_(1) and b_(2)` are the roots of this equation. Therefore,
`b_(1) + b_(2) = 2c cos A and b_(1) b_(2) = c^(2) - a^(2)`
`rArr 3b_(1) = 2c cos A, 2b_(1)^(2) = c^(2) = a^(2) " " ( :' b_(2) = 2b_(1) " given")`
or `2((2c)/(3) cos A)^(2) = c^(2) -a^(2)`
or `8c^(2) (1 - sin^(2) A) = 9c^(2) -9a^(2)`
or `sinA = sqrt((9a^(2) -c^(2))/(8c^(2)))`