Question

In $△ABC,\tan A:\tan B:\tan C=3:4:5$, 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)
$\Rightarrow \tan A=\frac{3}{\sqrt{5}},\tan B=\frac{4}{\sqrt{5}},\tan C=\sqrt{5}$
$\Rightarrow \sin A=\frac{3}{\sqrt{14}},\sin B=\frac{4}{\sqrt{21}},\sin C=\frac{\sqrt{5}}{\sqrt{6}}$
$\Rightarrow \sin A\sin B\sin C=\frac{3\times 4\times \sqrt{5}}{\sqrt{14}\times \sqrt{21}\times \sqrt{6}}$
$=\frac{2\sqrt{5}}{7}$