Question

The most general value of $\theta$ satisfying $3-2\cos \theta -4\sin \theta -\cos 2\theta +\sin 2\theta =0$:

• A. $2n\pi$
• B. $2n\pi +\frac{\pi }{2}$
• C. $4n\pi$
• D. $2n\pi +\frac{\pi }{4}$

Answer 1

Answer: (A), (B)

The correct options are
A $2n\pi$
B $2n\pi +\frac{\pi }{2}$

$3-2\cos \theta -4\sin \theta -\cos 2\theta +\sin 2\theta =0$
$\Rightarrow 3-2\cos \theta -4\sin \theta -1+2{\sin }^2\theta +2\sin \theta \cos \theta =0$
$\Rightarrow 2{\sin }^2\theta -2\cos \theta -4\sin \theta +2\sin \theta \cos \theta +2=0$
$\Rightarrow ({\sin }^2\theta -2\sin \theta +1)+\cos \theta (\sin \theta -1)=0$
$\Rightarrow (\sin \theta -1)[\sin \theta -1+\cos \theta ]=0$
either $\sin \theta =1$
$\Rightarrow \theta =2n\pi +\pi /2$ where $nϵI$
or, $\sin \theta +\cos \theta =1$
$\cos (\theta -\pi /4)=\cos \left(\pi /4\right)\Rightarrow \theta -\pi /4=2n\pi \pm \pi /4$
$\Rightarrow \theta =2n\pi ,2n\pi +\pi /2$ where $nϵI$
Hence $\theta =2n\pi ,2n\pi +\pi /2$