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Trigonometrical Equations
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$2 \sin ^2 \theta=\cos 2 \theta$ અને $2 \cos ^2 \theta=3 \sin \theta$ નું સમાધાન કરતી $\theta \in[0,2 \pi]$ ની તમામ કિંમતોનો સરવાળો _______ છે.
A$\frac{\pi}{2}$
B$4 \pi$
C$\frac{5 \pi}{6}$
D$\pi$
(JEE MAIN-2025)
Solution
$2 \sin ^2 \theta=\cos 2 \theta$
$2 \sin ^2 \theta=1-2 \sin ^2 \theta$
$4 \sin ^2 \theta=1$
$\sin ^2 \theta=\frac{1}{4}$
$\sin \theta= \pm \frac{1}{2}$
$2 \cos ^2 \theta=3 \sin \theta$
$2-2 \sin 2 \theta+3 \sin \theta-2=0$
$(2 \sin \theta-1)(2 \sin \theta-2)=0$
$\sin \theta=\frac{1}{2}$
so common equation which satisfy both equations is $\sin \theta=\frac{1}{2}$
$\theta=\frac{\pi}{6}, \frac{5 \pi}{6} \quad(\theta \in[0,2 \pi])$
$\operatorname{Sum}=\pi$
$2 \sin ^2 \theta=1-2 \sin ^2 \theta$
$4 \sin ^2 \theta=1$
$\sin ^2 \theta=\frac{1}{4}$
$\sin \theta= \pm \frac{1}{2}$
$2 \cos ^2 \theta=3 \sin \theta$
$2-2 \sin 2 \theta+3 \sin \theta-2=0$
$(2 \sin \theta-1)(2 \sin \theta-2)=0$
$\sin \theta=\frac{1}{2}$
so common equation which satisfy both equations is $\sin \theta=\frac{1}{2}$
$\theta=\frac{\pi}{6}, \frac{5 \pi}{6} \quad(\theta \in[0,2 \pi])$
$\operatorname{Sum}=\pi$
Standard 11
Mathematics
