The Inverse Sine Function and its Properties

Sine fails to be one-to-one over $\mathbb{R}$: every horizontal line $y=b$, $-1\le b\le1$, crosses the sine curve infinitely many times, so sine does not pass the horizontal line test.

Restricting the domain. If sine is restricted to $\left[-\tfrac{\pi}2,\tfrac{\pi}2\right]$, it becomes both one-to-one AND onto $[-1,1]$ — a genuine bijection.

Definition 4.3. For $-1\le x\le1$, $\sin^{-1}x$ is defined as the UNIQUE number $y\in\left[-\tfrac{\pi}2,\tfrac{\pi}2\right]$ such that $\sin y=x$. In symbols, $\sin^{-1}:[-1,1]\to\left[-\tfrac{\pi}2,\tfrac{\pi}2\right]$ is defined by $\sin^{-1}(x)=y\iff\sin y=x$ and $y\in\left[-\tfrac{\pi}2,\tfrac{\pi}2\right]$.

Notes.

1. Sine is one-to-one on $\left[-\tfrac{\pi}2,\tfrac{\pi}2\right]$ but on NO larger interval containing the origin.
2. Cosine is non-negative on $\left[-\tfrac{\pi}2,\tfrac{\pi}2\right]$ — the range of $\sin^{-1}x$ — a fact that matters later for trigonometric substitutions in integral calculus.
3. $\sin:\left[-\tfrac{\pi}2,\tfrac{\pi}2\right]\to[-1,1]$ and $\sin^{-1}:[-1,1]\to\left[-\tfrac{\pi}2,\tfrac{\pi}2\right]$.
4. Sine could equally well be restricted to any ONE of the intervals $\ldots,\left[-\tfrac{5\pi}2,-\tfrac{3\pi}2\right],\left[-\tfrac{3\pi}2,-\tfrac{\pi}2\right],\left[-\tfrac{\pi}2,\tfrac{\pi}2\right],\left[\tfrac{\pi}2,\tfrac{3\pi}2\right],\left[\tfrac{3\pi}2,\tfrac{5\pi}2\right],\ldots$ and remain one-to-one with range $[-1,1]$ on each — but $\left[-\tfrac{\pi}2,\tfrac{\pi}2\right]$ is the one CHOSEN by convention.

(vi) $\left[-\tfrac{\pi}2,\tfrac{\pi}2\right]$ is called the principal domain of sine, and the values of $y=\sin^{-1}x$ are called the principal values of $\sin^{-1}x$.

From the definition, four immediate consequences:

(i) $y=\sin^{-1}x\iff x=\sin y$, for $-1\le x\le1$ and $-\tfrac{\pi}2\le y\le\tfrac{\pi}2$.

(ii) $\sin(\sin^{-1}x)=x$ if $x\le1$ (i.e. $x\in[-1,1]$), and is meaningless if $x>1$.

(iii) $\sin^{-1}(\sin x)=x$ if $-\tfrac{\pi}2\le x\le\tfrac{\pi}2$. NOTE: $\sin^{-1}(\sin2\pi)=0\ne2\pi$ — a direct illustration that this identity needs $x$ inside the principal range.

(iv) $\sin^{-1}(\sin x)=\pi-x$ if $\tfrac{\pi}2\le x\le\tfrac{3\pi}2$. Note that then $-\tfrac{\pi}2\le\pi-x\le\tfrac{\pi}2$, as required.

(v) $y=\sin^{-1}x$ is an ODD function.