meaning of powers on trig functions

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I always forget this, when a trig function has an exponent does that mean multiply itself or apply itself to the result recursivly?

e.g. does $\sin(x)^2=\sin(x)\sin(x)$ or $=\sin(\sin(x))$? What about $\sin^2x$?

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4 Answers

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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.)

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As you can see, the notation isn't same for all of those bellow: $$ \sin(x)^2 $$ $$ \sin^2(x) $$ $$ \sin(\sin(x)) $$ $$ \sin(x)\sin(x) $$

Now, we should just find out, which one is corresponding with another. And the right answer is: $$ \sin^2(x) = (\sin(x))^2 = \sin(x)\sin(x) $$ $$ \sin(\sin(x)) \text{ is forever alone and never simplified} $$

And this is, how you can interpret $\sin(x)^2$: $$ \sin(x)^2 = \sin((x)^2) = \sin(x^2) $$

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It 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.

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• 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 $\sin(x^2)$).

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

  When I wish to express $\left[\sin(x)\right]^2,$ I prefer to write “$\sin^2(x)$”, knowing that this is unlikely to be interpreted as $\sin\left[\sin(x)\right]$ and is at least less ambiguous than “$\sin^{-1}(x)$”.

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