Evaluate the following: ^ -1 (4 3 ) - Vidyadip

Question

Evaluate the following:
$\cos^{-1}\Big\{\cos\Big(\frac{4\pi}{3}\Big)\Big\}$

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Answer

We know that,
$\cos^{-1}\big(\cos\theta\big)=\begin{cases}-\theta,&\text{if }\theta\in[-\pi,0]\\\theta,&\text{if }\theta\in[0,\pi]\\2\pi-\theta,&\text{if }\theta\in[\pi,2\pi]\\-2\pi+\theta,&\text{if }\theta\in[2\pi,3\pi]\end{cases}$
We have
$\cos^{-1}\Big\{\cos\Big(\frac{4\pi}{3}\Big)\Big\}=\cos^{-1}\Big\{\cos\Big(2\pi-\frac{2\pi}{3}\Big)\Big\}$
$\cos^{-1}\Big\{\cos\Big(\frac{2\pi}{3}\Big)\Big\}$
$=\frac{2\pi}{4}$

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