Question : If $\cos^2 \theta=\frac{3}{4}$, where $\theta$ is an acute angle, then the value of $\sin \left(\theta+30^{\circ}\right)$ is:
Option 1: $1$
Option 2: $\frac{1}{\sqrt{2}}$
Option 3: $\frac{1}{2}$
Option 4: $\frac{\sqrt{3}}{2}$
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Correct Answer: $\frac{\sqrt{3}}{2}$
Solution :
Given:
$\cos^2 \theta=\frac{3}{4}$
⇒ $\cos \theta=\pm\frac{\sqrt3}{2}$
$\theta$ is an acute angle i.e. it will be less than 90°.
$\therefore \cos \theta=-\frac{\sqrt3}{2}$ will be rejected.
⇒ $\cos \theta=\frac{\sqrt3}{2}$
⇒ $\theta = 30°$
Now we have to find $\sin \left(\theta+30^{\circ}\right)$
$ = \sin(30° + 30°) = \sin 60° = \frac{\sqrt{3}}{2}$
Hence, the correct answer is $\frac{\sqrt{3}}{2}$.
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