How do you find the exact value of $sin ({cos}^{- 1} (\frac{\sqrt{5}}{5}))$ $sin(cos^-1(sqrt5/5))$ ?

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How do you find the exact value of $sin ({cos}^{- 1} (\frac{\sqrt{5}}{5}))$ $sin(cos^-1(sqrt5/5))$ ?

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Answer 1 · 2018-02-23T05:43:24

$sin ({cos}^{- 1} (\frac{\sqrt{5}}{5})) = \frac{2 \sqrt{5}}{5}$ $sin(cos^-1(sqrt(5)/5))=(2sqrt(5))/5$

Explanation:

Let ${cos}^{- 1} (\frac{\sqrt{5}}{5}) = A$ $cos^-1(sqrt(5)/5)=A$ then $cos A = \frac{\sqrt{5}}{5}$ $cosA=sqrt(5)/5$

and $sin A = \sqrt{1 - {cos}^{2} A} = \sqrt{1 - {(\frac{\sqrt{5}}{5})}^{2}} = \frac{2 \sqrt{5}}{5}$ $sinA=sqrt(1-cos^2A)=sqrt(1-(sqrt(5)/5)^2)=(2sqrt(5))/5$
$\to A = {sin}^{- 1} (\frac{2 \sqrt{5}}{5})$ $rarrA=sin^-1((2sqrt(5))/5)$

Now, $sin ({cos}^{- 1} (\frac{\sqrt{5}}{5})) = sin ({sin}^{- 1} (\frac{2 \sqrt{5}}{5})) = \frac{2 \sqrt{5}}{5}$ $sin(cos^-1(sqrt(5)/5))=sin(sin^-1((2sqrt(5))/5))=(2sqrt(5))/5$

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How do you find the exact value of $sin ({cos}^{- 1} (\frac{\sqrt{5}}{5}))$ $sin(cos^-1(sqrt5/5))$ ?

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