 Arccosine

For real $-1\le x\le 1$, the principal arccosine is the unique angle $\theta \in [0,\pi ]$ satisfying $\cos \theta =x$. Restricting to this interval makes the cosine one-to-one. Differentiating the identity on $-1<x<1$ gives $d(\arccos x)/dx=-1/\sqrt{1-x^2}$. Other solutions of $\cos \theta =x$ are $\theta =\pm \arccos x+2\pi n$, with integer $n$; an initial condition can select one of these branches.