What are the meanings of $\operatorname{trig}(x)^n$ and $\operatorname{trig}^n(x)$?

Question

When a trigonometric function has an exponent does that mean multiply itself or apply itself to the result recursively?

For example, does $\sin(x)^2$ denote $\sin(x)\sin(x)$ or does it denote $\sin(\sin(x))$? What about $\sin^2x$?

Answer 1

$\sin(x)^2$ means it multiplies itself, although I always thought that was weird since $(\sin(x))^5$ is already easy to write, although writing $\sin(\sin(\sin(\sin(\sin(x)))))$ is a lot harder. I remember it because I think it is weird.

Answer 2

The notation is a mess, and we’re stuck with it for purely historical reasons. As everybody has noted, $\sin^2x$ means $(\sin(x))^2$. But nobody pointed out that $\sin^{-1}x$ does not mean the reciprocal of the sine function, but rather its inverse with respect to composition. That is, for the right range of inputs, $\sin\bigl(\sin^{-1}(x)\bigr)=x$ and $\sin^{-1}\bigl(\sin(x)\bigr)=x$.

(In my own work, I have to refer to the $n$-fold composition of $f$ with itself, and (less often) the $n$-th power of $f$. I’ve chosen to write $f^{\circ n}$ for the multiple composition, and $f^n$ for the product of $f$ with itself $n$ times, but this is nonstandard. I still don’t know, when people in analytic number theory write $\log^2(x)$, which they mean.)

Answer 3

1. The currently accepted answer claims that $$ \color{blue}{\sin(x)^2} := \sin(x^2), \tag{1}$$ but in fact, possibly the most common interpretation is that $$\color{blue}{\sin(x)^2} := \left[\sin(x)\right]^2 \tag{2}$$ (the standard argument being that it is silly to move the exponent out of the parenthesis if one actually means $\sin(x^2)$).

2. However, I avoid writing $\text“\color{blue}{\sin(x)^2}\text”$ altogether, as I instinctively parse it as $(1)$ rather than $(2).$

   For $\left[\sin(x)\right]^2,$ I prefer to write $\text“\color{violet}{\sin^2(x)}\text”,$ knowing that this is the most conventional choice, is unlikely to be interpreted as the rarely occurring function composition $\sin\left(\sin(x)\right),$ and is at least less potentially controversial than $\text“\sin^{-1}(x)\text”,$ which can be clearly and compactly rewritten as $\arcsin(x)$ or $\operatorname{cosec}(x),$ depending on the intended meaning (most likely the former).

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Addendum to restore the OP's Accepted (green-ticked) Answer by PObdr (referenced above; to be clear: I disagree with its tone of definitiveness), which, together with the comments under it, have been deleted by the community:

$$ \color{blue}{\sin(x)^2} = \sin((x)^2) = \sin(x^2)\\ \color{violet}{\sin^2(x)} = (\sin(x))^2 = \sin(x)\sin(x)\\ \sin(\sin(x)) \text{ is forever alone and never simplified} $$

• Xander Henderson: $\quad$ This is simply not the way that most of the mathematical community interprets $\color{blue}{\sin(x)^2}$—indeed, I only see this interpretation when grading student work, and I mark it down. This answer is simply incorrect.

• me: $\quad$ @XanderHenderson Unless your required interpretation has been stressed in class, it's unfair to mark students down for being unaware that a significant subset of the mathematical community opts to understand $\text“\color{blue}{\sin(x)^2}\text”$ as the square of a function output.

  After all, reading $\text“\color{blue}{\sin(x)^2}\text”$ as $\sin(x^2)$ is neither nonsensical (reading it as $\sin\cdot\sin\cdot xx$) nor outre (reading it as $\sin(\sin(x))$ ) nor more unnatural than reading $\text“t(p)^2\text”$ as $t(p^2)$ instead of $(tp)^2.$

  Since $\text“\color{violet}{\sin^2(x)}\text”$ is by far the most prevalent expression for $[\sin(x)]^2,$ and since reading $\text“\color{blue}{\sin(x)^2}\text”$ as $[\sin(x)]^2$ is not clearly more logically/mathematically valid than as $\sin(x^2)$ (for example, no precedence convention says whether function application or exponentiation binds stronger), $\text“\boldsymbol{\color{blue}{\sin(x)^2}}\text”$ is actually more ambiguous than $\text“\color{violet}{\sin^2(x)}\text”.$ Certainly, there is no firm convention for interpreting $\text“\color{blue}{\sin(x)^2}\text”;$ for example, see the discussions here and here and here.