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

Answer

2

Explanation

`(1)/ ( 4 cos^(2) theta + 1 + (3)/(2) sin 2 theta) = (1)/(2[1+ cos 2 theta] + 1+ (3)/(2) sin 2theta) `
` " " = (1)/( 2cos 2theta + (3)/(2) sin 2 theta + 3)`
Now,
`" " - sqrt(2^(2) + ((3)/(2))^(2)) le 2 cos 2theta + (3)/(2) sin 2 theta le sqrt (2^(2) + ((3)/(2))^(2))`
or ` - (5)/(2) le 2 cos 2 theta + (3)/(2) sin 2 theta le (5)/(2)`
`rArr (1)/(2) le 2 cos 2 theta + (3)/(2) sin 2 theta + 3 le (11)/(2)`
`rArr (2)/(11) le (1)/( 2 cos 2 theta + (3)/(2) sin 2 theta + 3) le 2`
Hence, the maximum value is 2.