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I'm in year 11 right now and I just had a brief discussion with my maths teacher about function notation in trigonometry.

For a test, I wrote this, 

sin(50)^2

I assumed that would be interpreted as sin(50)*sin(50)

But I was told the correct notation for this is is

sin^2 (50)

or optionally

(sin (50))^2

I'm curious if that is the 'proper' mathematical convention, or just how things are taught in high schools?

I'm Australian, in case there are some regional differences.

Thanks for the help!

Tomas
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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 02 '11 at 00:45

1 Answers1

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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 $(f(\log x))^2$.

Qiaochu Yuan
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