Proposition: Inverse Cosine of a Real Variable

The cosine of a real variable $\mathbb R\to\mathbb R,~x\to \cos(x)$, is invertible on all real intervals $x\in[k,(k+1)\pi],$ where $\pi$ denotes the $\pi$ constant, and $k\in\mathbb Z$ denotes an integer. Its inverse function $\arccos$, called the inverse cosine, is continuous, strictly monotonically decreasing, and defined by \[\arccos:[-1,1]\to\mathbb R.\]

Since the above proposition holds for all $k\in\mathbb Z$, the special case $k=0$ is called the principal branch of $\arccos$. The inverse cosine has a graph shown in the following figure:

Proofs: 1

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  1. Forster Otto: "Analysis 1, Differential- und Integralrechnung einer Veränderlichen", Vieweg Studium, 1983