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Hey there world, I'm in year 11 right now and I just had a brief discussion with my maths teacher about function notation, specifically, how to write the result of a function squared, as apposed to the argument to a function squared.

Say we were using trigonomics functions

For a test, I wrote this,


Assuming, as the programmer I am that it would be taken to mean sin(50)*sin(50)

I was subsequently told that the correct notation is

sin^2 (50)

or optionally

(sin (50))^2

I'm going to get the marks back anyway as the calculator used the notation I thought correct, but I'm curious, is that actually 'proper' mathematical convention?

It seems a bit weird to bother using open close brackets to denote the argument to a function if your going to ignore them anyway. I suppose I should mention I'm Australian, just in case its some kind of English v American system.

Thanks for the help!

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I don't remember where I read it, but it seems Gauss strongly rejected the $\sin^2(x)$ notation, because it should mean $\sin(\sin(x))$ and that only, so you're not in bad company if you don't like it. -- By the way, tell your teacher to type "sin(1/2)^2" and "sin^2(1/2)" in google and check what comes up... – Myself Mar 2 '11 at 0:45
up vote 10 down vote accepted

This is a weird notational bug specific to trigonometric functions; chalk it up to historical inertia. We write $\sin^2 x$ for $(\sin x)^2$ but for a generic function $f$, more often than not $f^2(x)$ means $f(f(x))$ and does not mean $(f(x))^2$ (or $f(x^2)$). On the other hand, $\sin^{-1} x$ means $\arcsin x$ rather than $\csc x$...

It is preferable to include extra parentheses when in doubt. Generally I would interpret $f(x)^2$ as $(f(x))^2$ but it is less clear whether $f(\log x)^2$ means $f((\log x)^2)$ or $(f(\log x))^2$.

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