Inverse Trigonometric Functions

A function has an inverse only when it is one-to-one (injective) and onto (surjective) on the domain in use — an inverse simply cannot be defined for a function that is not one-to-one, because then a single output would trace back to more than one input. The book's own illustration is $y=x^2$: over all of $\mathbb R$ it fails to be one-to-one (both $x=2$ and $x=-2$ give $y=4$), but the moment the domain is restricted to $x\ge0$ (or, separately, to $x\le0$), the function becomes one-to-one and onto, and its inverse $f^{-1}(x)=\sqrt x,\ x\ge0$ exists.

Every one of the six trigonometric functions is periodic, so none is one-to-one over its full natural domain — infinitely many angles share the same sine, the same cosine, and so on. To speak of "the" inverse sine, inverse cosine, and so on, each function's domain must first be restricted to an interval on which it is one-to-one and whose image is the whole of the original range. Because of the periodicity this restriction can be chosen in more than one way, but the conventional choice is always made so that it (i) contains $0$, (ii) contains some positive angles, and (iii) has image equal to the entire range of the unrestricted function.

Building $\sin^{-1}$ as the model case. Take $f(x)=\sin x$ restricted to $x\in\left[-\tfrac\pi2,\tfrac\pi2\right]$. On this interval sine is one-to-one and onto $[-1,1]$, so its inverse exists. By the meaning of an inverse function, $f^{-1}(y)=x \iff f(x)=y$, and this inverse is written $\sin^{-1}x$. So

In words: $\sin^{-1}t$ is the angle lying in $\left[-\tfrac\pi2,\tfrac\pi2\right]$ whose sine equals $t$. Each of the other five inverse functions is built exactly the same way, on its own conventional restricted domain.

The six restricted-domain / range pairs.

Principal value. For $y=\sin x$, infinitely many angles satisfy $\sin x=t$ for a given $t\in[-1,1]$; exactly one of them lies in $\left[-\tfrac\pi2,\tfrac\pi2\right]$, and that one is the principal angle, written $\sin^{-1}t$. In general, the principal value of an inverse trigonometric function is, among all angles satisfying the equation, the one that is numerically the least (smallest in magnitude) — positive or negative. When two candidate values are numerically equal but of opposite sign, the convention is to take the positive one as the principal value.

Three points worth keeping straight:

• $\sin^{-1}x$ does not mean $\dfrac1{\sin x}$ — the $-1$ labels an inverse function, it is not an exponent.
• $\sin^{-1}x$ is also written $\arcsin x$ (the notation is credited to Sir John F. W. Herschel, 1813); similarly $\arccos x,\arctan x,\ldots$
• The graph of $f^{-1}$ is the reflection of the graph of $f$ in the line $y=x$: if $(a,b)$ lies on the graph of $f$, then $(b,a)$ lies on the graph of $f^{-1}$.

Principal-value sign table. Because the interval each inverse function's values land in is centred at or adjacent to $0$, the sign of the argument $x$ decides which half of that interval the principal value falls in:

Worked illustration (Example 3.72 — find the principal value). …