Find the value of $sin ({cos}^{- 1} (\frac{x}{2}))$ $sin(cos^(-1)(x/2))$ ?

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Answer 1 · 2017-02-27T09:08:57

$sin ({cos}^{- 1} (\frac{x}{2})) = \frac{\sqrt{4 - x^{2}}}{2}$ $sin(cos^(-1)(x/2))=sqrt(4-x^2)/2$

Let $cos A = \frac{x}{2}$ $cosA=x/2$ , then $A = {cos}^{- 1} (\frac{x}{2})$ $A=cos^(-1)(x/2)$

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

= $\sqrt{\frac{4 - x^{2}}{4}} = \frac{\sqrt{4 - x^{2}}}{2}$ $sqrt((4-x^2)/4)=sqrt(4-x^2)/2$

Hence $sin ({cos}^{- 1} (\frac{x}{2})) = \frac{\sqrt{4 - x^{2}}}{2}$ $sin(cos^(-1)(x/2))=sqrt(4-x^2)/2$

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