# What is the value of cos^-1(cos(-pi/3))?

Jul 23, 2016

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

#### Explanation:

We have, $\cos \theta = \cos \left(- \theta\right) , \forall \theta \in \mathbb{R} \ldots \ldots \ldots \ldots . \left(1\right)$.

Next let us recall the following defn. of the ${\cos}^{-} 1$ function :-

${\cos}^{-} 1 x = \theta , | x | \le 1 \iff \cos \theta = x , \theta \in \left[0 , \pi\right] \ldots \ldots \ldots \ldots \ldots \left(2\right)$

Now, using $\left(2\right)$ in $\leftarrow$ direction, keeping in mind that $\cos \theta \le 1$ ,

we have, ${\cos}^{-} 1 \left(\cos \theta\right) = \theta , \mathmr{if} , \theta \in \left[0 , \pi\right] \ldots \ldots \ldots \ldots \ldots \ldots . \left(2 '\right)$

Now, cos^-1(cos(-pi/3))=cos^-1(cos(pi/3))................[by (1)],

&, here, since $\frac{\pi}{3} \in \left[0 , \pi\right]$, we get, by $\left(2 '\right)$,

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