What is Cos^(-1) [cos(-5pi/3)]?

Sep 21, 2015

${\cos}^{-} 1 \left[\cos \left(\frac{- 5 \pi}{3}\right)\right] = \frac{\pi}{3}$

Explanation:

First get the value of $\cos \left(\frac{- 5 \pi}{3}\right)$

For this we need to find the acute angle associated with $\frac{- 5 \pi}{3}$

Since $\cos x$ is of period $2 \pi$,
the angle $\frac{- 5 \pi}{3}$ is equivalent to $\left(2 \pi + \frac{- 5 \pi}{3}\right) = \textcolor{red}{\frac{\pi}{3}}$

Hence, ${\cos}^{-} 1 \left[\cos \left(\frac{- 5 \pi}{3}\right)\right]$ is the same as $\textcolor{g r e e n}{{\cos}^{-} 1 \left[\cos \left(\frac{\pi}{3}\right)\right]}$

$\cos \left(\frac{\pi}{3}\right)$ is $\frac{1}{2}$

So, ${\cos}^{-} 1 \left[\cos \left(\frac{\pi}{3}\right)\right] = {\cos}^{-} 1 \left(\frac{1}{2}\right)$

The function ${\cos}^{-} 1 \left(a\right)$ is just asking us to give the angle whose cosine is $a$

Similarly, cos^-1(1/2)=color(blue)(pi/3

In other words, ${\cos}^{-} 1 \left(\frac{1}{2}\right)$ verbally means: "the (acute) angle whose cosine is $\frac{1}{2}$