If sin ^{-1}left(frac{3}{5}right)+cos ^{-1}le | vedclass

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Question

(A) Let $\sin ^{-1}\left(\frac{3}{5}\right) = x$ and $\cos ^{-1}\left(\frac{12}{13}\right) = y$. Then $\sin x = \frac{3}{5} \implies \cos x = \sqrt{1

Vedclass · Accepted Answer

$\frac{56}{65}$

Vedclass · Answer

(A) Let $\sin ^{-1}\left(\frac{3}{5}ight) = x$ and $\cos ^{-1}\left(\frac{12}{13}ight) = y$. Then $\sin x = \frac{3}{5} \implies \cos x = \sqrt{1 - (\frac{3}{5})^2} = \frac{4}{5}$. And $\cos y = \frac{12}{13} \implies \sin y = \sqrt{1 - (\frac{12}{13})^2} = \frac{5}{13}$. We are given $x + y = \sin ^{-1} \alpha$,which means $\sin(x + y) = \alpha$. Using the identity $\sin(x + y) = \sin x \cos y + \cos x \sin y$: $\alpha = (\frac{3}{5}) 	imes (\frac{12}{13}) + (\frac{4}{5}) 	imes (\frac{5}{13})$ $\alpha = \frac{36}{65} + \frac{20}{65} = \frac{56}{65}$.

If $\sin ^{-1}\left(\frac{3}{5}\right)+\cos ^{-1}\left(\frac{12}{13}\right)=\sin ^{-1} \alpha$,then $\alpha=$

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