Find the value: ^ -1 ( 13 6 ) - Vidyadip

Question

Find the value:
$\cos^{-1}\bigg(\cos\frac{13\pi}{6}\bigg)$

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Answer

We know that $\cos^{-1}(\cos x)=x$ if $x\in[0,\pi]$, which is the principal value branch of $\cos^{-1}x.$
Here, $\frac{13\pi}{6}\notin[0,\pi].$
Now, $\cos^{-1}\bigg(\cos\frac{13\pi}{6}\bigg)$ can be written as:
$\cos^{-1}\left(\cos\frac{13\pi}{6}\right)=\cos^{-1}\left[\cos\left(2\pi+\frac{\pi}{6}\right)\right]$
$=\cos^{-1}\left[\cos\left(\frac{\pi}{6}\right)\right],\text{where}\frac{\pi}{6}\in\left[0,\pi\right]$
$\therefore\cos^{-1}\bigg(\cos\frac{13\pi}{6}\bigg)=\cos^{-1}\bigg[\cos\bigg(\frac{\pi}{6}\bigg)\bigg]=\frac{\pi}{6}$

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