# How do you evaluate cos^-1(cos(-1/2)) without a calculator?

${\cos}^{-} 1 \left(\cos \left(- \frac{1}{2}\right)\right) = {\cos}^{-} 1 \left(\cos \left(\frac{1}{2}\right)\right) = \frac{1}{2}$
${\cos}^{-} 1 \left(\cos \left(- \frac{1}{2}\right)\right)$
Since the argument $- \frac{1}{2}$ is within the domain $- 1 \le x \le 1$ we can apply the property f^-1(f(x)=x and the property of even function $\cos \left(- x\right) = \cos x$. Hence,
${\cos}^{-} 1 \left(\cos \left(- \frac{1}{2}\right)\right) = {\cos}^{-} 1 \left(\cos \left(\frac{1}{2}\right)\right) = \frac{1}{2}$