Proposition: Inverse Sine of a Real Variable

The sine of a real variable $\mathbb R\to\mathbb R,~x\to \sin(x)$, is invertible on all real intervals $x\in[-\pi/2+k\pi,\pi/2+k\pi],$ where $\pi$ denotes the $\pi$ constant, and $k\in\mathbb Z$ denotes an integer. Its inverse function $\arcsin$, called the inverse sine, is continuous, strictly monotonically increasing, and defined by \[\arcsin:[-1,1]\to\mathbb R.\]

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

Proofs: 1

  1. Proposition: Derivative of the Inverse Sine
  2. Proposition: Integral of Inverse Sine

Propositions: 1 2

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