Question

In $\triangle A B C, \tan A: \tan B: \tan C=3: 4: 5$, then the value of $\sin A \sin B \sin C$ is

• A. $\frac{2 \sqrt{5}}{7}$
• B. $\frac{\sqrt{5}}{7}$
• C. $\frac{3 \sqrt{5}}{7}$
• D. none of these

Answer 1

Answer: (A)

We have $\frac{\tan A}{3}=\frac{\tan B}{4}=\frac{\tan C}{5}=\lambda$
$\Rightarrow \tan A=3 \lambda, \tan B=4 \lambda, \tan C=5 \lambda$
Now, we have
$\tan A+\tan B+\tan C=\tan A \tan B \tan C$
$\Rightarrow 3 \lambda+4 \lambda+5 \lambda=60 \lambda^{3}$
$\Rightarrow 12 \lambda=60 \lambda^{3}$
$\Rightarrow \lambda=\frac{1}{\sqrt{5}}$ (As $\lambda$ cannot be negative)
$\Rightarrow \tan A=\frac{3}{\sqrt{5}}, \tan B=\frac{4}{\sqrt{5}}, \tan C=\sqrt{5}$
$\Rightarrow \sin A=\frac{3}{\sqrt{14}}, \sin B=\frac{4}{\sqrt{21}}, \sin C=\frac{\sqrt{5}}{\sqrt{6}}$
$\Rightarrow \sin A \sin B \sin C=\frac{3 \times 4 \times \sqrt{5}}{\sqrt{14} \times \sqrt{21} \times \sqrt{6}}$
$=\frac{2 \sqrt{5}}{7}$