Question : If $0^{\circ} < \theta < 90^{\circ}$ and $2 \sin^{2}\theta +3\cos\theta =3$, then the value of $\theta$ is:
Option 1: 30°
Option 2: 60°
Option 3: 45°
Option 4: 75°
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Correct Answer: 60°
Solution :
We know that $\sin^2 \theta + \cos ^2 \theta = 1$
Using the value of $\sin^2 \theta$ in the equation, we get,
⇒ $2(1- \cos ^2 \theta)+3\cos\theta=3$
⇒ $2-2\cos ^2 \theta+3\cos\theta=3$
⇒ $2\cos ^2 \theta-3\cos\theta+1=0$
⇒ $2\cos ^2 \theta-2\cos\theta-\cos\theta+1=0$
⇒ $(2\cos\theta-1)(\cos\theta-1)=0$
⇒ $\cos\theta=\frac{1}{2}$ or $\cos\theta=1$
Since $0^{\circ} < \theta < 90^{\circ}$,
So, $\cos\theta=\frac{1}{2}=\cos60^\circ$
$\therefore \theta = 60^{\circ}$
Hence, the correct answer is 60°.
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