$\arcsin$ written as $\sin^{-1}(x)$

Solution 1:

The notation for trigonometric functions is "traditional", which is to say that it is not the way we would invent notation today.

• $\sin^{-1}(x)$ means the inverse sine, as you mentioned, rather than a reciprocal. So $\sin^{-1}(x)$ is not an abbreviation for $(\sin(x))^{-1}$. Instead it's notation for $(\sin^{-1})(x)$, in the same way that $f^{-1}(x)$ means the inverse function of $f$, applied to $x$.

• But $\sin^2(x)$ means $(\sin(x))^2$, rather than $\sin(\sin(x))$. In other contexts, like dynamical systems, if I have a function $f$, the notation $f^2$ means $f \circ f$. This is compatible with the $f^{-1}$ notation, if we take juxtaposition of functions to mean composition: $f^{-1}f^{3}$ will be $f^{2}$ as desired.

So the traditional notation for sine is actually a mixture of two different systems: $-1$ denotes an inverse, not a power, while positive integer exponents denote powers, not iterated compositions.

This is simply a fact of life, like an irregular conjugation of a verb. As with other languages, the things that we use most often are the ones that are likely to remain irregular. That doesn't mean that they are incorrect, however, as long as other speakers of the language know what they mean.

Moreover, if you wanted to reform the system, there would be an equally strong argument for changing $\sin^2$ to mean $\sin \circ \sin$. This is already slowly happening with $\log$; I think that the usage of $\log^2(x)$ to mean $(\log(x))^2$ is slowly decreasing, because people tend to confuse it with $\log(\log(x))$. That confusion is less likely with $\sin$ because $\sin(\sin(x))$ arises so rarely in practice, unlike $\log(\log(x))$.