Question

cos-1(-1/2)

Answer 1

$cos^{-1}$ is a function such that $cos^{-1}(t)$ is an angle x from $[0;2\pi)$ such that $cos(x)=t$.

We know that $cos(\pi/3)=1/2$. Also from trigonometry $cos(\pi-y)=-cos(y)$ for every real $y$.

Then

$cos(\pi/3)=1/2=-cos(\pi-\pi/3)=-cos(2\pi/3)$

$cos(2\pi/3)=-1/2$

Now we get for angle $2\pi/3$ equality $cos(2\pi/3)=-1/2$. Then by definition $cos^{-1}(-1/2)=2\pi/3$