sin left( frac{1}{2} cos^{-1} frac{4}{5} righ | vedclass

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Question

(A) Let $\cos^{-1} \frac{4}{5} = x$,which implies $\cos x = \frac{4}{5}$.We need to find $\sin \left( \frac{x}{2} \right)$.Using the trigonometric ide

Vedclass · Accepted Answer

$\frac{1}{\sqrt{10}}$

Vedclass · Answer

(A) Let $\cos^{-1} \frac{4}{5} = x$,which implies $\cos x = \frac{4}{5}$.We need to find $\sin \left( \frac{x}{2} \right)$.Using the trigonometric identity $\cos x = 1 - 2 \sin^2 \left( \frac{x}{2} \right)$:$\frac{4}{5} = 1 - 2 \sin^2 \left( \frac{x}{2} \right)$$2 \sin^2 \left( \frac{x}{2} \right) = 1 - \frac{4}{5} = \frac{1}{5}$$\sin^2 \left( \frac{x}{2} \right) = \frac{1}{10}$Since $\cos^{-1} \frac{4}{5}$ is in the first quadrant,$\frac{x}{2}$ is also in the first quadrant,so $\sin \left( \frac{x}{2} \right)$ must be positive.Therefore,$\sin \left( \frac{x}{2} \right) = \sqrt{\frac{1}{10}} = \frac{1}{\sqrt{10}}$.

$\sin \left( \frac{1}{2} \cos^{-1} \frac{4}{5} \right) = $

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