Solve : `2 cos^(2)theta + sin theta le 2`, where `pi/2 le theta le (3pi)/2.`

Options

A. `{theta:pi//2le thetale 5 pi //6}`
B. `{theta:pilethetale 3pi//2}`
C. `{theta:pi//2le theta le 5pi/6} cup {theta: pi le theta 3pi //2}`
D. none of these

Explanation

we have ,
`2cos^(2)theta + sin thetale2`
`implies 2(1-sin^(2) theta ) + sin theta le 2`
`implies 2 sin ^(2) theta - sin theta ge 0`
`implies sin theta ( 2 sin theta -1)ge0`
`implies sin theta ge 0 and 2 sin theta -1ge 0`or ,
`implies sin theta le0 and 2 sin theta -1 le 0`
Case I when sin `theta gt 0 and 2 sin theta -1 gt0` ltbr in this case , we have
` sin theta ge 0 and 2 sin thetatheta-1 ge0`
`implies sin theta ge 0 and sin theta ge (1)/(2) `
`implies (pi)/(6) le theta le (5pi)/(6)`
`therefore A cap B ={ theta : pi //2 le theta le 5pi //6}[:' B= pi//2 theta le 3pi //2}]`
Case II when `sin theta le 0 and theta -1le0`
in this case , we have
`sin theta le 0 and 2 sin theta -1 le0`
` implies sin theta le 0 and sin le (1)/(2)`
`implies sin theta le 0`
`implies pi le theta le 2 pi`
`therefore Acap B=[ theta : pi //2le theta le 5 pi//6] cup [ theta : pi le 3p//2]`