Question

The value(s) of $\theta$, which satisfy $3-2\cos \theta -4\sin \theta -\cos 2\theta +\sin 2\theta =0$ is/are

• A. $\theta =2n\pi ;n\in I$
• B. $2n\pi +\frac{\pi }{2};n\in I$
• C. $2n\pi -\frac{\pi }{2};n\in I$
• D. $n\pi ;n\in I$

Answer 1

Answer: (A), (B)

$2n\pi +\frac{\pi }{2};n\in I$
$\theta =2n\pi ;n\in I$

$3-2\cos \theta -4\sin \theta -\cos 2\theta +\sin 2\theta =0$
$\sin 2\theta -2\cos \theta +3-4\sin \theta -(1-2{\sin }^2\theta )=0$
$2\cos \theta (\sin \theta -1)+2{\sin }^2\theta -4\sin \theta +2=0$
$2\cos \theta (\sin \theta -1)+2(\sin \theta -1{)}^2=0$
Or
$2(\sin \theta -1)(\cos \theta +\sin \theta -1)=0$
$\sin \theta =1$
Or
$\theta =2n\pi +\frac{\pi }{2}$.
And
$\cos \theta +\sin \theta -1=0$
Or
$\sqrt{2}\left[\sin \right(\theta +\frac{\pi }{4}\left)\right]=1$
Or
$\sin (\theta +\frac{\pi }{4})=\frac{1}{\sqrt{2}}$
Or
$\theta +\frac{\pi }{4}=2n\pi +\frac{\pi }{4}$
Or $\theta =2n\pi$.