Question
Evaluate the following:
$\cos^{-1}\Big\{\cos\Big(-\frac{\pi}{4}\Big)\Big\}$
$\cos^{-1}\Big\{\cos\Big(-\frac{\pi}{4}\Big)\Big\}$
✓
Answer
We know that,
$\cos^{-1}\big(\cos\theta\big)=\begin{cases}-\theta,&\text{if }\theta\in[-\pi,0]\\\theta,&\text{if }\theta\in[0,\pi]\\2\pi-\theta,&\text{if }\theta\in[\pi,2\pi]\\-2\pi+\theta,&\text{if }\theta\in[2\pi,3\pi]\end{cases}$
$\therefore\ \cos^{-1}\Big\{\cos\Big(-\frac{\pi}{4}\Big)\Big\}$
$=-\Big(-\frac{\pi}{4}\Big)$ $\Big\{\because-\frac{\pi}{4}\in[-\pi,0]\Big\}$
$=\frac{\pi}{4}$
Hence,
$\therefore\ \cos^{-1}\Big\{\cos\Big(-\frac{\pi}{4}\Big)\Big\}=\frac{\pi}{4}$
$\cos^{-1}\big(\cos\theta\big)=\begin{cases}-\theta,&\text{if }\theta\in[-\pi,0]\\\theta,&\text{if }\theta\in[0,\pi]\\2\pi-\theta,&\text{if }\theta\in[\pi,2\pi]\\-2\pi+\theta,&\text{if }\theta\in[2\pi,3\pi]\end{cases}$
$\therefore\ \cos^{-1}\Big\{\cos\Big(-\frac{\pi}{4}\Big)\Big\}$
$=-\Big(-\frac{\pi}{4}\Big)$ $\Big\{\because-\frac{\pi}{4}\in[-\pi,0]\Big\}$
$=\frac{\pi}{4}$
Hence,
$\therefore\ \cos^{-1}\Big\{\cos\Big(-\frac{\pi}{4}\Big)\Big\}=\frac{\pi}{4}$
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