Question : If $\sin \theta = \frac{3}{11}$, then what is the value of $\cot \theta$?
Option 1: $\frac{3 \sqrt{7}}{11}$
Option 2: $\frac{4 \sqrt{7}}{11}$
Option 3: $\frac{4 \sqrt{7}}{3}$
Option 4: $\frac{3 \sqrt{7}}{28}$
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Correct Answer: $\frac{4 \sqrt{7}}{3}$
Solution :
Given, $\sin \theta=\frac{3}{11}$
We know, $\sin\theta = \frac{\text{Perpendicular}}{\text{Hypotenuse}}$
Using Pythagoras theorem, we get,
$\small (\text{Hypotenuse})^2 = (\text{Base})^2 + (\text{Perpendicular})^2$
Let base = $b$
⇒ $11^2 = b^2 + 3^2$
⇒ $b^2 = 121 - 9$
⇒ $b = \sqrt{112} = 4\sqrt7$
⇒ $\cot\theta = \frac{\text{base}}{\text{perpendicular}} = \frac{4\sqrt7}{3}$
Hence, the correct answer is $\frac{4\sqrt7}{3}$.
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