Question

the maximum value of expression 1/(sin^2 theta + 3sincostheta + 5 cos^2 theta )is ?

[AIEEE 2010]

Answer 1

Let f(Ф) = 1/ [ sin^2 Ф + 3 sinФcosФ + 5 cos^2 Ф]

and, g(Ф) =  sin^2 Ф + 3 sinФcosФ + 5 cos^2 Ф

                = (1 - cos2Ф) / 2 + 5 (1+cos2Ф)/2 + 3/2 sin2Ф

               = 3 + 2 cos2Ф + 3/2sin2Ф

Now, Minimum value of g(Ф) = 3 - √(4+9/4) = 1/2

Then, maximum value of f(Ф) = 1/1/2 = 2