Question : If $8 \cot A = 7$, then find $\sin A$.
Option 1: $\frac{7}{15}$
Option 2: $\frac{8}{\sqrt{113}}$
Option 3: $\frac{7}{8}$
Option 4: $\frac{8}{7}$
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Correct Answer: $\frac{8}{\sqrt{113}}$
Solution :
Given: $8 \cot A = 7$
⇒ $\cot A = \frac{7}{8}$
We know that $\operatorname{cosec^2 A}-\cot^2A=1$
⇒ $\operatorname{cosec^2 A}- (\frac{7}{8})^2=1$
⇒ $\operatorname{cosec^2 A}=1+\frac{49}{64}$
⇒ $\operatorname{cosec A}=\sqrt{\frac{113}{64}}=\frac{\sqrt{113}}{8}$
⇒ $\sin A=\frac{8}{\sqrt{113}}$
Hence, the correct answer is $\frac{8}{\sqrt{113}}$.
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