Question

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

• A. $θ=2nÏ€;n∈I$
• B. $2nÏ€+\frac{Ï€}{2};n∈I$
• C. $2nπ−\frac{Ï€}{2};n∈I$
• D. $nÏ€;n∈I$

Answer 1

Answer: (A), (B)

$3−2\cos θ−4\sin θ−\cos 2θ+\sin 2θ=0$
$\sin 2θ−2\cos θ+3−4\sin θ−\left(1−2{\sin }^2θ\right)−0$
$2\cos θ(\sin θ−1)+2{\sin }^2θ−4\sin θ+2=0$
$\cos θ(\sin θ−1)+(\sin θ−1{)}^2=0$
$(\sin θ−1)(\cos θ+\sin θ−1)=0$
$\sin θ=1$ or $\cos \left(θ−\frac{π}{4}\right)=\frac{1}{\sqrt{2}}$
$θ=2nπ+\frac{π}{2}$ or $θ−\frac{π}{4}=2mπ+\frac{π}{4},2mπ−\frac{π}{4}$
$θ=2mπ+\frac{π}{2},2mπ$