In a $Δ A B C, i f tan A : tan B : tan C = 3 : 4 : 5,$ Then the value of $sin A . sin B . sin C,$ is equal to

\frac{2}{\sqrt{5}}

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\frac{2 \sqrt{5}}{7}

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\frac{2 \sqrt{5}}{9}

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\frac{2}{3 \sqrt{5}}

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Solution

The correct option is A $\frac{2 \sqrt{5}}{7}$
Given

In $Δ A B C$

$tan A : tan B : tan C = 3 : 4 : 5$

$⟹ tan A = 3 k, tan B = 4 k, tan C = 5 k$

We know that

In $Δ A B C$

$\sum tan A = Π tan A$

$⟹ 3 k + 4 k + 5 k = (3 k) (4 k) (5 k)$

$⟹ 12 k = 60 k^{3}$

$⟹ 5 k^{2} = 1$

$⟹ k = \frac{1}{\sqrt{5}}$

$⟹ tan A = \frac{3}{\sqrt{5}}, tan B = \frac{4}{\sqrt{5}}, tan C = \frac{5}{\sqrt{5}}$

$⟹ sin A = \frac{3}{\sqrt{5 + 9}}, sin B = \frac{4}{\sqrt{5 + 16}}, sin C = \frac{5}{\sqrt{5 + 25}}$

$⟹ sin A = \frac{3}{\sqrt{14}}, sin B = \frac{4}{\sqrt{21}}, sin C = \frac{5}{\sqrt{30}}$

Consider

$sin A sin B sin C$

$⟹ \frac{(3) (4) (5)}{\sqrt{14} \sqrt{21} \sqrt{30}}$

$⟹ \frac{60}{\sqrt{8820}}$

$⟹ \frac{10}{7 \sqrt{5}}$

$⟹ \frac{2 \sqrt{5}}{7}$

Suggest Corrections

Similar questions

\frac{2^{\frac{1}{2}} \times 3^{\frac{1}{3}} \times 4^{\frac{1}{4}}}{10^{- \frac{1}{5}} \times 5^{\frac{3}{5}}} \div \frac{4^{- \frac{2}{3}} \times 5^{- \frac{7}{5}}}{4^{- \frac{3}{5}} \times 6^{- \frac{1}{3}}}

5 + 3 + 2 = 151022

9 + 2 + 4 = 183652

8 + 6 + 3 = 482466

5 + 4 + 5 = 202541

7 + 2 + 5 = ? ? ? ? ?

x = \frac{5}{(2!) 3} + \frac{5 \cdot 7}{(3!) 3^{2}} + \frac{5 \cdot 7 \cdot 9}{(4!) 3^{3}} + . . . . .

x^{2} + 4 x

Fill in the blanks:

$(i) - 4 \times \frac{7}{9} = \frac{7}{9} \times$ ___

$(i i) \frac{5}{11} \times \frac{- 3}{8} = \frac{- 3}{8} \times$ ___

$(i i i) \frac{1}{2} \times (\frac{3}{4} + \frac{- 5}{12}) = \frac{1}{2} \times$ ____ $+ (\times) \frac{- 5}{12}$

$(i v) \frac{- 4}{5} \times \frac{5}{7} \times$ ___ $= (\frac{- 4}{5} \times \frac{5}{7}) \times \frac{- 8}{9}$

\frac{2}{3} \times \frac{4}{5} = \frac{4}{5} \times \frac{2}{3}

(\frac{- 3}{4}) \times 1 = \frac{- 3}{4} = 1 \times (\frac{- 3}{4})

(\frac{- 17}{28}) \times (\frac{- 28}{17}) = 1

\frac{1}{5} \times (\frac{7}{8} \times \frac{4}{3}) = (\frac{1}{5} \times \frac{7}{8}) \times \frac{4}{3}

\frac{2}{7} \times (\frac{9}{10} + \frac{2}{5}) = \frac{2}{7} \times \frac{9}{10} + \frac{2}{7} \times \frac{2}{5}

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