Question : If $\theta$ is an acute angle and $\sin \theta=\frac{13}{19}$, what is the value of $\cos \theta?$
Option 1: $\frac{6}{19}$
Option 2: $\frac{10 \sqrt{2}}{19}$
Option 3: $\frac{14}{19}$
Option 4: $\frac{8 \sqrt{3}}{19}$
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Correct Answer: $\frac{8 \sqrt{3}}{19}$
Solution :
Given: $\sin \theta=\frac{13}{19}$
We know, $\cos\theta=\sqrt{1-\sin^2\theta}$
$⇒\cos\theta=\sqrt{1-(\frac{13}{19})^2}$
$⇒\cos\theta=\sqrt{1-\frac{169}{361}}$
$⇒\cos\theta=\sqrt{\frac{192}{361}}$
$\therefore \cos\theta=\frac{8\sqrt3}{19}$
Hence, the correct answer is $\frac{8\sqrt3}{19}$.
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