Question

without using a calculator, evaluate, if possible, the following expression. $cos^{-1}left(cosleft(-\frac{4pi}{3}
ight)
ight)$ select the correct choice below and, if necessary, fill in the answer box to complete your choice. oa. $cos^{-1}left(cosleft(-\frac{4pi}{3}
ight)
ight)=square$ (type your answer in radians. use angle measures greater than or equal to 0 and less than or equal to $pi$. type an exact answer, using $pi$ as needed. use integers or fractions for any numbers in the expression.) ob. the expression is undefined

Explanation:

Step1: Recall cosine - inverse cosine property

The property of $y = \cos^{-1}(\cos(x))$ is that $y=x$ when $0\leq x\leq\pi$. First, find an equivalent angle of $-\frac{4\pi}{3}$ in the range $[0, 2\pi]$.
$-\frac{4\pi}{3}+2\pi=\frac{2\pi}{3}$, and $\frac{2\pi}{3}\in[0,\pi]$.

Step2: Evaluate the expression

Since $\cos^{-1}(\cos(x)) = x$ for $x\in[0,\pi]$, then $\cos^{-1}(\cos(-\frac{4\pi}{3}))=\frac{2\pi}{3}$.

Answer:

A. $\frac{2\pi}{3}$