How do you find the exact value of cos^-1(cos((-pi)/3))?

Apr 15, 2016

${\cos}^{-} 1 \left(\cos \left(- \frac{\pi}{3}\right)\right) = {\cos}^{-} 1 \left(\cos \left(\frac{\pi}{3}\right)\right) = \frac{\pi}{3}$

Explanation:

Even Function
If f(-x) = f(x), then f is called an even function

Since cosine is an even function f(-x)=f(x) hence cos(-pi/3)=cos(pi/3)
Then
${\cos}^{-} 1 \left(\cos \left(- \frac{\pi}{3}\right)\right) = {\cos}^{-} 1 \left(\cos \left(\frac{\pi}{3}\right)\right)$

Now we use the property
${f}^{-} 1 f \left(x\right) = x$, for all x in the appropriate domain

therefore,

${\cos}^{-} 1 \left(\cos \left(- \frac{\pi}{3}\right)\right) = {\cos}^{-} 1 \left(\cos \left(\frac{\pi}{3}\right)\right) = \frac{\pi}{3}$