How do you find the exact value of arccos(cos(pi/3))?

Jul 28, 2015

$0 \le \frac{\pi}{3} \le \pi$

So $\frac{\pi}{3}$ is in the range of $\arccos$ and $\arccos \left(\cos \left(\frac{\pi}{3}\right)\right) = \frac{\pi}{3}$

Explanation:

If we restrict the domain of $\cos$ to $\left[0 , \pi\right]$ then it is a one-one function onto its range $\left[- 1 , 1\right]$ with inverse $\arccos$.

So $\arccos$ is defined to have range $\left[0 , \pi\right]$,

If $\theta \in \left[0 , \pi\right]$ then $\arccos \left(\cos \left(\theta\right)\right) = \theta$

If $\theta \in \left[- \pi , 0\right]$ then $\arccos \left(\cos \left(\theta\right)\right) = - \theta$

If $\theta = \varphi + 2 n \pi$ for some $n \in \mathbb{Z}$ then $\arccos \left(\cos \left(\theta\right)\right) = \arccos \left(\cos \left(\varphi\right)\right)$