Write the principal value of ^ -1 ( ^ -1 1 2 ) - Vidyadip

Question

Write the principal value of $\sin^{-1}\Big\{\cos\Big(\sin^{-1}\frac{1}{2}\Big)\Big\}$

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Answer

$\sin^{-1}\Big\{\cos\Big(\sin^{-1}\frac{1}{2}\Big)\Big\}$
$=\sin^{-1}\Big\{\cos\Big[\sin^{-1}\Big(\sin\frac{\pi}{3}\Big)\Big]\Big\}$
$=\sin^{-1}\Big[\cos\Big(\frac{\pi}{3}\Big)\Big]$
$=\sin^{-1}\Big[\frac{1}{2}\Big]$
$=\sin^{-1}\Big[\sin\Big(\frac{\pi}{3}\Big)\Big]$
$=\frac{\pi}{3}$

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