Question : If $\tan \mathrm{A}=\frac{3}{4}$, then find the value of the following expression $\frac{6 \sin A}{1-\sin A}$.
Option 1: 18
Option 2: 9
Option 3: 24
Option 4: 12
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Correct Answer: 9
Solution : Given: $\tan \mathrm{A}=\frac{3}{4}$ We know that $\tan \theta=\frac{\text{Perpendicular}}{\text{Base}}=\frac{3}{4}$ Let perpendicular = $3k$ and base = $4k$ [where $k$ is a non zero constant] So, Hypotenuse $=\sqrt{(3k)^2+(4k)^2}=\sqrt{25k^2}=5k$ ⇒ $\sin A =\frac{\text{Perpendicular}}{\text{Hypotenuse}}=\frac{3k}{5k}=\frac{3}{5}$ Now, $\frac{6 \sin A}{1-\sin A}$ $= \frac{6 ×\frac{3}{5}}{1-\frac{3}{5}}$ $= \frac{\frac{18}{5}}{\frac{2}{5}}$ $=9$ Hence, the correct answer is 9.
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