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Evaluate the following : ( { 1 }{ 2 } { }^{ -1 }( { 4 }{ 5

Question

Evaluate the following :
$sin (\frac{1}{2} cos^{- 1} (\frac{4}{5}))$

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Evaluate the following :

sin (\frac{1}{2} cos^{- 1} (\frac{4}{5}))

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Text solutionVerified

Given:

sin (\frac{1}{2} cos^{- 1} \frac{4}{5})

Let

x = cos^{- 1} \frac{4}{5}

\Rightarrow cos x = \frac{4}{5}

\Rightarrow sin x = 1 - cos^{2} x = 1 - \frac{16}{25} = \frac{25 - 16}{25} = \frac{9}{25} = \frac{3}{5}

\Rightarrow x = cos^{- 1} \frac{4}{5} = sin^{- 1} \frac{3}{5}

Now,

sin (\frac{1}{2} cos^{- 1} \frac{4}{5})

= sin (\frac{1}{2} x)

= sin (\frac{x}{2})

We Know that

cos x = 1 - 2 sin^{2} \frac{x}{2}

\Rightarrow \frac{4}{5} = 1 - 2 sin^{2} \frac{x}{2}

\Rightarrow 2 sin^{2} \frac{x}{2} = 1 - \frac{4}{5} = \frac{5 - 4}{5} = \frac{1}{5}

\Rightarrow sin^{2} \frac{x}{2} = \frac{1}{10}

\Rightarrow sin \frac{x}{2} = \frac{1}{10}

∴ sin (\frac{1}{2} cos^{- 1} \frac{4}{5})

= sin \frac{x}{2} = \frac{1}{10}

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Question Text	Evaluate the following : $sin (\frac{1}{2} cos^{- 1} (\frac{4}{5}))$
Answer Type	Text solution:1
Upvotes	150