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

Options

A. 2
B. 3
C. 4
D. 6

Answer

A

Explanation

`sin^(2)theta+3sintheta costheta+5cos^(2)theta`
`=1/2{1-cos2theta+3sin2theta+5(1+cos2theta}`
`=3+2cos2theta+3/2sin2theta`
Clearly,
`-sqrt(4+(9)/(4))le2cos2theta+3/2sin2thetalesqrt(4+(9)/(4))`
`implies-5/2le2cos2theta+3/2sin2thetale5/2`
`implies2/11le(1)/(sin^(2)theta+3sinthetacostheta+5cos^(2)theta)le2`
Hence, the maximum value of the given expression is 2.