Evaluate [ ^ -1 ( -3 2 )+ 6 ] - Vidyadip

Question

Evaluate $\cos\Big[\cos^{-1}\Big(\frac{-\sqrt{3}}{2}\Big)+\frac{\pi}{6}\Big]$

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Answer

We have, $\cos\Big[\cos^{-1}\Big(\frac{-\sqrt{3}}{2}\Big)+\frac{\pi}{6}\Big]$
$=\cos\Big[\cos^{-1}\Big(\cos\frac{5\pi}{6}\Big)+\frac{\pi}{6}\Big]$
$\Big[\because\cos\frac{5\pi}{6}=\frac{-\sqrt{3}}{2}\Big]$
$=\cos\Big(\frac{5\pi}{6}+\frac{\pi}{6}\Big)$
$\{\because\ \cos^{-1}\cos\text{x}=\text{x};\ \text{x}\in[0,\pi]\}$
$=\cos\Big(\frac{6\pi}{6}\Big)$
$=\cos(\pi)=-1$

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