The expression `cos 3 theta + sin 3 theta + (2 sin 2 theta-3) (sin theta- cos theta)` is positive for all `theta` in

Options

A. `(2 n pi-(3pi)/4, 2npi+pi/4), n in Z`
B. `(2npi-pi/4, 2npi +pi/6), n in Z`
C. `(2npi-pi/3, 2npi +pi/3), n in Z`
D. `(2npi-pi/4, 2npi+(3pi)/4), n in Z`

Answer

A, B

Explanation

`cos 3 theta+sin 3 theta+(2 sin 2 theta-3) (sin theta - cos theta) gt 0`
or `4(cos theta - sin theta) (cos^(2) theta+ sin theta cos theta+ sin^(2) theta-sin theta cos theta) gt 0`
or `-4sqrt(2) sin (theta - pi/4) gt 0`
or `sin (theta - pi/4) lt 0`
`rArr (2n-1) pi lt theta - pi/4 lt 2npi, n in I`
`rArr 2n pi - (3pi)/4 lt theta lt 2n pi + pi/4, n in I`