Let's say I have
$$a > b$$
Now we know that the inequality sign will switch if:
We take the reciprocal of both sides
Multiply $-1$ to both sides.
But, are there more? What if we do more "complicated" things to it?
Will it change if I apply inverse trig on it? For instance is it still true then if
$$\arccos(a) > \arccos(b)$$
Note that $a,b \in\mathbb R$.
So what I really want to ask is, are there any other operations that would make you change the inequality signs?