Question : If $\theta$ is an acute angle and $\sin \theta=\frac{43}{47}$, what is the value of $\cot \theta$?
Option 1: $\frac{6 \sqrt{10}}{47}$
Option 2: $\frac{43}{6 \sqrt{10}}$
Option 3: $\frac{47}{6 \sqrt{10}}$
Option 4: $\frac{6 \sqrt{10}}{43}$
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Correct Answer: $\frac{6 \sqrt{10}}{43}$
Solution :
$\sin \theta = \frac{43}{47}$
$\therefore \cos \theta = \sqrt{1-\sin^2 \theta} = \sqrt{1-(\frac{43}{47})^2}$ = $\sqrt\frac{360}{47^2}$ = $\frac{6\sqrt{10}}{47}$
$\cot \theta = \frac{\cos\theta}{\sin \theta} = \frac{\frac{6\sqrt{10}}{47}}{\frac{43}{47}} = \frac{6\sqrt{10}}{43}$
Hence, the correct answer is $\frac{6\sqrt{10}}{43}$.
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