Question

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

• A. $(2n\pi -\frac{3\pi }{4},2n\pi +\frac{\pi }{4}),nϵZ$
• B. $(2n\pi -\frac{\pi }{4},2n\pi +\frac{\pi }{6}),nϵZ$
• C. $(2n\pi -\frac{\pi }{3},2n\pi +\frac{\pi }{3}),nϵZ$
• D. $(2n\pi -\frac{\pi }{4},2n\pi +\frac{3\pi }{4}),nϵZ$

Answer 1

Answer: (B), (C)

The correct options are
B $(2n\pi -\frac{\pi }{4},2n\pi +\frac{\pi }{6}),nϵZ$
C $(2n\pi -\frac{\pi }{3},2n\pi +\frac{\pi }{3}),nϵZ$

$\cos 3\theta +\sin 3\theta +(2\sin 2\theta -3)(\sin \theta -\cos \theta )>0$ .....using the formula of $\cos 3\theta ,\sin 3\theta \sin 2\theta$
$4{\cos }^3\theta -3\cos \theta +3\sin \theta -4{\sin }^3\theta +4{\sin }^2\theta \cos \theta -3\sin \theta -4\sin \theta {\cos }^2\theta +3\cos \theta$

$\Rightarrow 4{\cos }^3\theta -4{\sin }^3\theta +2\cos \theta -2{\cos }^3\theta -2\sin \theta +2{\sin }^3\theta >0$

$\Rightarrow {\cos }^3\theta -{\sin }^3\theta +\cos \theta -\sin \theta >0\Rightarrow \left(\cos \theta -\sin \theta \right)(1+\sin \theta \cos \theta )+\cos \theta -\sin \theta >0$ ....[Using formula $(a^3-b^3)=(a-b)(a^2+ab+b^2)$ and ${\sin }^2\theta +{\cos }^2\theta =1$]

$\Rightarrow \left(\cos \theta -\sin \theta \right)(2+\sin \theta \cos \theta )>0$
$\Rightarrow \theta \in (2n\pi -\frac{\pi }{4},2n\pi +\frac{\pi }{6}),(2n\pi -\frac{\pi }{3},2n\pi +\frac{\pi }{3})$