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If cos 3theta+sin3theta+(2sin2theta-3)(s...

If `cos 3theta+sin3theta+(2sin2theta-3)(sin theta-cos theta)gt0`, then `theta` lies in

A

`(2n pi-(3pi)/(4),2n pi+(pi)/(4)),n in Z`

B

`(2n pi-(pi)/(2),2n pi+(pi)/(6)), n in Z`

C

`(2n pi-(pi)/(3),2n pi+(pi)/(3)), n in Z`

D

`(2n pi-(pi)/(4),2n pi+(3pi)/(4)), n in Z`

Text Solution

Verified by Experts

The correct Answer is:
A, B
`cos 3theta + sin 3theta +(2sin 2theta-3)(sin theta - cos theta) gt 0`
`rArr 4(cos theta - sin theta) (cos^(2)theta + sin theta + sin theta cos theta + sin^(2) theta -sin theta cos theta) gt 0`
`rArr sin(theta-(pi)/(4))lt 0`
`rArr sin(theta - (pi)/(4))` is negative when
`rArr (2n-1)pi lt theta - pi//4 lt 2n pi, n in Z`
`rArr 2n pi-(3pi)/(4)lt theta lt 2n pi + pi//4, n in Z`
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