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

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

Trigonometric Functions

Solution:

$3-2 \cos \theta-4 \sin \theta-\cos 2 \theta+\sin 2 \theta=0$
$\sin 2 \theta-2 \cos \theta+3-4 \sin \theta-\left(1-2 \sin ^2 \theta\right)-0$
$2 \cos \theta(\sin \theta-1)+2 \sin ^2 \theta-4 \sin \theta+2=0$
$\cos \theta(\sin \theta-1)+(\sin \theta-1)^2=0$
$(\sin \theta-1)(\cos \theta+\sin \theta-1)=0$
$\sin \theta=1$ or $\cos \left(\theta-\frac{\pi}{4}\right)=\frac{1}{\sqrt{2}}$
$\theta=2 n \pi+\frac{\pi}{2}$ or $\theta-\frac{\pi}{4}=2 m \pi+\frac{\pi}{4}, 2 m \pi-\frac{\pi}{4}$
$\theta=2 m \pi+\frac{\pi}{2}, 2 m \pi$