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Question
sin \[\left\{ 2 \cos^{- 1} \left( \frac{- 3}{5} \right) \right\}\] is equal to
Options
`6/25`
`24/25`
`4/5`
`-24/25`
Solution
(d) `-24/25`
Let \[\cos^{- 1} \left( - \frac{3}{5} \right) = x, 0 \leq x \leq \pi\]
Then,
`cosx=-3/5`
\[\therefore \sin{x} = \sqrt{1 - \cos^2 x} = \sqrt{1 - \left( - \frac{3}{5} \right)^2} = \sqrt{\frac{16}{25}} = \frac{4}{5}\]
Now,
\[\sin\left\{ 2 \cos^{- 1} \left( - \frac{3}{5} \right) \right\} = \sin\left( 2x \right)\]
\[ = 2\sin{x} \cos{x}\]
\[ = 2 \times \frac{4}{5} \times \frac{- 3}{5}\]
\[ = - \frac{24}{25}\]
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