trig
posted by usha .
if alpha and beta are 2 different values of θ lying between 0 and 2π which satisfy the equation 6cosθ+8 sinθ=9 find the value of sin alpha + beta

0.6 cosè + 0.8 sinè = 0.9
Let sinx = 0.6; then
cosx = 0.8 and x = 36.87 degrees
sinx cosè + cosx sinè = 0.9
sin (x + è) = 0.9
x + è = 64.16 degrees or 115.84 degrees
alpha = è1 = 27.29 degrees
beta = è2 = 78.97
alpha + beta = 106.26 degrees
sin (alpha + beta) = 0.9600 
drwls,
that is amazingly clever. 
thank you very much. can u solve the same by quadratic equation method

yes, it can be done that way, but I'd rather leave that to you.
Write it as
6cosθ= 9  8 sinθ
Square both sides and the left side can be written 36(1  sin^2 θ)
Then you have a quadratic in sin theta to solve. Have fun.