The value of sin left[2 cos ^{-1} frac{sqrt{5 | vedclass

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(C) Let $	heta = \cos ^{-1} \frac{\sqrt{5}}{3}$,then $\cos 	heta = \frac{\sqrt{5}}{3}$.We need to find $\sin(2	heta)$.Using the identity $\sin(2	h

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$\frac{4 \sqrt{5}}{9}$

Vedclass · Answer

(C) Let $	heta = \cos ^{-1} \frac{\sqrt{5}}{3}$,then $\cos 	heta = \frac{\sqrt{5}}{3}$.We need to find $\sin(2	heta)$.Using the identity $\sin(2	heta) = 2 \sin 	heta \cos 	heta$.Since $\cos 	heta = \frac{\sqrt{5}}{3}$,we have $\sin 	heta = \sqrt{1 - \cos^2 	heta} = \sqrt{1 - \left(\frac{\sqrt{5}}{3}ight)^2} = \sqrt{1 - \frac{5}{9}} = \sqrt{\frac{4}{9}} = \frac{2}{3}$.Therefore,$\sin(2	heta) = 2 	imes \left(\frac{2}{3}ight) 	imes \left(\frac{\sqrt{5}}{3}ight) = \frac{4 \sqrt{5}}{9}$.

The value of $\sin \left[2 \cos ^{-1} \frac{\sqrt{5}}{3}\right]$ is

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