I am teaching a Trigonometry class, and every year we get to this point my students start asking a lot of questions. And for good reason.
Here is my issue: We teach that $\arccos\frac{1}{2}=\frac{\pi}{3}$ and this is the only answer because our range is restricted to $0\le{y}\le{\pi}$ We've been over that cosine and sine are not one to one functions which is why we restrict the domain to find the inverses.
However, when we start solving trigonometric equations and solve problems such as:
Find the solutions of $2\cos x=1$ in the interval [$\pi,2\pi]$
We solve by dividing by 2, and we have $\cos x=\frac{1}{2}$. Now, the book states that we take $\arccos(\frac{1}{2})$ and we find that equals $\frac{5\pi}{3}$. The problem I run into is that students object that $\arccos(\frac{1}{2})=\frac{5\pi}{3}$ because it does not fit within the range of $0\le{y}\le{\pi}$. Is it incorrect notation to use arccos in this situation as the book does? Because I agree with students that this does not fall within the range of arccos. It confuses students every year.
Instead, is it more proper to leave the equation $\cos x=\frac{1}{2}$ and find the value of $x$ without showing the arccos step?