Sometimes notation is confusing. One way this can happen is when the same notation is used for multiple different concepts. Usually, if the same notation is used for different things, they occur in different contexts which allows them to be distinguished. In at least one case, the same notation is used for different things in the same context. I am speaking of exponents on trigonometric functions.

Since sinx is frequently squared, it is inconvenient to write parentheses every time like (sinx)², however just omitting parentheses like sinx² is ambiguous between (sinx)² and sin(x²). As a result, we have a convention of placing the 2 between the sin and the x: sin²x.

Unrelatedly, we have a convention of writing the inverse of a function f(x) as f^-1(x). We can think of this in a couple ways. First, an exponent of -1 normally means reciprocal, which we can think of as meaning inverse since a reciprocal is a multiplicative inverse. On the other hand, if we think of the number -1 as the number of times we’re applying the function, then it means “undoing” the function.

Here’s the problem: sine has an inverse function called arcsine, and by the convention above we write it as sin^-1x. Thus we have the following inconsistent notation:

(sinx)² = sin²x

(sinx)^-1 ≠ sin^-1x

Now, this may not be the most confusing notation in existence. However, I think it’s the worst. The reason is that we have a ready alternative to sin^-1 in arcsin. It is never necessary to use the notation that looks like an exponent, and yet that is still generally the most mainstream way to write arcsine. In my opinion, either sin^-1x should refer to cosecant or sin²x should mean sin(sin(x)).